University of TabrizComputational Methods for Differential Equations2345-398214320260701Numerical solution for an inverse source problem of a fractional order diffusion-wave equation99610091997910.22034/cmde.2025.61173.2630ENMaryamEbrahimiDepartment of Applied Mathematics, University of Mazandaran, Babolsar, Iran.MashallahMatinfarDepartment of Applied Mathematics, University of Mazandaran, Babolsar, Iran.AfshinBabaeiDepartment of Applied Mathematics, University of Mazandaran, Babolsar, Iran.Journal Article20240414In this paper, we consider an inverse source problem of a fractional order diffusion-wave equation (FDWE), in which the space-dependent source term is unknown. In order to obtain the numerical solution of the discussed problem and to find the unknown source function, a Chebyshev collocation method is proposed. Since this inverse problem is an ill-posed problem, a regularization scheme based on the mollification technique is used to find a stable problem. Subsequently, the stable problem is solved numerically by applying the collocation method. Furthermore, the convergence analysis is considered, and finally, the effectiveness of the studied algorithm is demonstrated by some test examples.https://cmde.tabrizu.ac.ir/article_19979_ed4e73bb419211ee824b4a8ce628f0db.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Fractional prey-predator model in biological Pest control101010241995310.22034/cmde.2025.64960.2960ENRam PratapChauhanDepartment of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Amaravati, Andhra Pradesh--522503, India.RavikantSinghDepartment of Mathematics, Amity School of Engineering and Technology, Amity University, Gwalior, Madhya Pradesh--474020, India.SrinivasaraoThotaDepartment of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Amaravati, Andhra Pradesh--522503, India.Journal Article20241210In developing countries, the agricultural industry is pivotal in economic development and sustaining rural livelihoods. However, one of the major challenges to achieving global food security is crop losses caused by pests. Pest control is a vital practice for safeguarding crops; it is often complicated by the need to balance pest reduction with the associated costs of operations, as well as the potential impacts on the environment and human health. This delicate balance is crucial for sustainable agriculture and long-term food security. In this paper, a mathematical model is developed to quantify the complex biological processes involved in the biological control of pests through prey-predator mechanisms. We provide trajectory analysis in the research discussion analysis.https://cmde.tabrizu.ac.ir/article_19953_71a90d4dc02ca138563646cc819ba94e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Introduction to quantum graphs and their stability analysis102510391990410.22034/cmde.2025.63178.2816ENSovanSamanta1. Department of Mathematics, Tamralipta Mahavidyalaya, Tamluk, W.B-721636, India.
2. Research Center of Performance and Productivity Analysis, Istinye University, Istanbul, Türkiye.TofighAllahviranloo1. Research Center of Performance and Productivity Analysis, Istinye University, Istanbul, Türkiye.
2. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.Journal Article20240828Quantum graphs have dynamic vertices and edges. The vertices and edges are assigned by functions. These types of representations of graphs can represent time-dependent network structures. The network characteristics can be visible at any point in time. The networks, such as brain networks or biological disease networks, can be represented by quantum graphs, and hence the characteristics and parameters of quantum graphs are an important research area. This study discusses the properties of quantum graphs. Additionally, real examples and the area of applications are provided.https://cmde.tabrizu.ac.ir/article_19904_c4caba62e2eeaf89d2b4eee8ef6786f7.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701A novel high-order approximation method for higher-dimensional time-fractional reaction-diffusion problems with weak initial singularity104010671992810.22034/cmde.2025.61994.2712ENRichaSinghDepartment of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh, India.AnshimaSinghDepartment of Computational and Data Sciences,
Indian Institute of Science, Bangalore, India.SunilKumarDepartment of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh,
India.JesusVigo-AguiarDepartment of Applied Mathematics, University of Salamanca, Salamanca, Spain.Journal Article20240613The objective of this manuscript is to construct and analyze a fully discrete method to approximate one- and two-dimensional time-fractional reaction-diffusion equations defined in the Caputo sense. The current approach combines Alikhanov’s L2-1θ formula on a non-uniform graded mesh to discretize the time-fractional Caputo derivative and the discretization of the space variables using a cubic spline difference scheme. The two-dimensional problem is then separated into two one-dimensional problems using the alternating direction implicit (ADI) approach. The theoretical analysis, which covers both stability and convergence, has been provided for one- and two-dimensional problems. Further, to illustrate the accuracy and efficiency of the proposed method, numerical results for two test examples have been presented.https://cmde.tabrizu.ac.ir/article_19928_83db35714bebfbda1a6c89f807c41e3e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Unsteady and velocity-slip effects on laminar boundary layer flow and forced convective heat transfer over a moving wedge106810792001110.22034/cmde.2025.64175.2893ENShrikantChavajRegional Research Centre, VTU, Belagavi-590018, India.Md. HanifPageDepartment of Mathematics, KLE Technological University, Hubballli-580 031, India.Krishna BChavaraddiDepartment of Mathematics, S. S. Government First Grade College and P. G. Studies Centre, Nargund-582 207, India.Priya MGouderDepartment of Mathematics, KLE Technological University, Dr. M. S. Sheshgiri Campus, Belagavi-590008, India.Journal Article20241025The focus of this study is to examine the effects of velocity-slip on the surface of the moving wedge on the laminar boundary layer flow of a viscous fluid, in addition to the heat transfer across the moving wedge. When fluid and solid interact, velocity-slip effects may have a major impact on most industrial applications. It is considered that the mainstream and wedge velocities and the shape of the velocity-slip depend on the distance along the boundary layer wall. These equations offer the essence of a set of ordinary differential equations for the momentum and thermal boundary layer systems. The numerical solutions reveal that when the velocity-slip and unstable parameters increase, the thermal and momentum boundary layers narrow. The momentum boundary layer domain also appears to be reduced due to pressure gradient effects. There is also little variation in the thermal boundary layers as the wall shear stress (skin friction) and temperature gradient curves grow flat with increasing velocity-slip parameter. The physical mechanisms underlying these remarkable results are further discussed.https://cmde.tabrizu.ac.ir/article_20011_595c23b332ee4c42c9c1a419504e46a6.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Semi-analytical study for Jeffery-Hamel flow using Shehu HPM and Elzaki HPM – A comparative study108011021958710.22034/cmde.2025.64721.2939ENMamtaKapoorMarwadi University Research Center, Department of Mathematics, Faculty of Engineering & Technology, Marwadi University, Rajkot, 360003, Gujarat, India.Journal Article20241126Via this study, a discussion about Jeffery-Hamel fluid flow is provided, including fluid flow in converging/diverging channels, influenced by Reynolds number and Hartmann number. Two innovative semi-analytical techniques are introduced, referred to as the Shehu-HPM and Elzaki-HPM methods, to analyze the solution profiles of a model governing Jeffrey-Hamel fluid flow. These methods are developed by combining the Shehu transform with the Homotopy Perturbation Method (HPM), referred to as Method I, and the Elzaki transform with HPM, referred to as Method II. The performance of these techniques is evaluated under varying parameters, including the Reynolds number and the Hartmann number. These approaches are straightforward to implement and avoid the errors typically associated with discretization or quasi-linearization. Given the increasing demand for reliable solutions to fluid mechanics models, the proposed methods offer a valuable and practical alternative for solving such problems. Their simplicity and accuracy make them particularly suitable for a wide range of applications in this field. The novelty of this work lies in the hybridization of Shehu and Elzaki transforms with HPM, addressing gaps in the latest literature by providing novel techniques to address complex nonlinear fluid flow problems with improved accuracy and efficiency.https://cmde.tabrizu.ac.ir/article_19587_5acea32987fe051e51ff2215113693b6.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701New numerical method via RBF approach to the price of fixed-rate mortgages110311111978110.22034/cmde.2025.65809.3045ENRaheleJaliliDepartment of Mathematics, Science and Research branch, Islamic Azad University, Tehran, Iran.AbdolsadehNeisyDepartment of Mathematics, Faculty of Mathematics Science and Computer, Allameh Tabataba’i University (ATU), Tehran, Iran.AlirezaVahidiDepartment of Mathematics, Shahr-e-rey branch, Islamic Azad University, Tehran, Iran.Journal Article20250205In this paper, we introduce a novel fixed-rate mortgage (FRM) pricing model that overcomes the limitations of existing approaches. Unlike traditional models, which rely on deterministic interest rate volatility and thus produce inaccurate valuations in volatile markets, our model incorporates stochastic volatility to more accurately reflect the dynamic nature of interest rate risk. This yields a pricing formula derived from a stochastic volatility framework, providing a strong theoretical basis for understanding volatility’s impact on FRM prices. We use the efficient and accurate Radial Basis Function (RBF) method to solve the resulting partial differential equation (PDE), effectively handling complex boundary conditions. Our numerical experiments demonstrate the model’s practical application and illustrate how FRM prices react to varying volatility across different market conditions. Our findings underscore the critical need for stochastic volatility in FRM valuation and offer valuable insights for improved hedging strategies, ultimately contributing to more realistic and accurate mortgage pricing and enhanced risk management for financial institutions and investors.https://cmde.tabrizu.ac.ir/article_19781_53125da05b839d3884e7abe02115cbbc.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701A new perspective for simulations of the equal-width wave equation111211291999010.22034/cmde.2025.62400.2747ENSelçukKutluayDepartment of Mathematics, Faculty of Arts and Sciences, Inonu University, Malatya, Turkey.Nuri MuratYağmurluDepartment of Mathematics, Faculty of Arts and Sciences, Inonu University, Malatya, Turkey.Ali SercanKarakaşDepartment of Mathematics, Faculty of Arts and Sciences, Inonu University, Malatya, Turkey.Journal Article20240708The fundamental aim of the present article is to numerically solve the non-linear Equal-Width Wave (EW) equation. For this purpose, the nonlinear term appearing in the equation is first linearized by a Rubin-Graves-type approach. After that, to reduce the equation into a solvable discretized linear algebraic equation system, which is the essential part of this study, the Crank-Nicolson-type approximation and cubic Hermite collocation method are respectively applied to obtain the integration in the temporal and spatial domain directions. To demonstrate how good the offered method is at generating approximate numerical results, six experimental problems exhibiting different wave profiles, known as the motion of a single, interacting two and three, the Maxwellian initial, undular bore, and colliding soliton waves given with different initial and boundary conditions of the EW equation, will be taken into consideration and solved. Since only the first model problem has an exact solution among these solitary waves, to measure error magnitudes, the widely used mean squared and maximum norms between analytical and approximate solutions are calculated and also compared with those from other existing works available in the literature. Furthermore, the three conservation constants known as mass, moment, and energy quantities are also computed and presented throughout the wave simulations with increasing time. In addition, a tabular comparison of the newly computed norms and conservation constants shows that the current scheme produces better and more compatible solutions than those of most of the previous works with the same parameters. Apart from that, the stability analysis for this present scheme has been illustrated using the von Neumann method.https://cmde.tabrizu.ac.ir/article_19990_a2a1bf0b9aa776298c1bed69e2842cec.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Solution of a non-homogeneous dynamic equation on a time scale113011461979610.22034/cmde.2025.65428.3009ENAhmedAzabaAl-Ryada University for Science and Technology, Sadat City, Menoufia, Egypt.Kamal R.RaslanMathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt.Khalid K.AliMathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman.Journal Article20250112We find the general solution to the non-homogeneous dynamic equation, which is a combination of discrete and continuous mathematics. The nature of this equation requires a careful approach, as it involves elements of both types of math, making it both versatile and challenging. To address this, we will derive the formula for the general solution of the non-homogeneous equation, incorporating the given initial conditions. During this process, we will define several critical points that may be either discrete, continuous, or a mix of both. By analyzing these points, we aim to capture the essence of the dynamic behavior of the system. Our approach involves finding an analytical solution to the equation and comparing it with a numerical approximation to evaluate their accuracy. We will graph both the analytical and numerical solutions to visualize their behavior and identify any discrepancies. Additionally, we will calculate the absolute error between the exact solution and the numerical solution to quantify the differences precisely. This comparison provides valuable insights into the accuracy and stability of numerical methods for solving such equations. Finally, we will demonstrate this approach by applying it to various examples, showcasing the methodology’s effectiveness in solving a range of non-homogeneous dynamic equations with different initial conditions and parameters.https://cmde.tabrizu.ac.ir/article_19796_70719c40349d480274bdf6c42865653c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Decoding computational complexity: a fractional-order clique-based approach for solving Hilfer fractal-fractional differential equations114711642061210.22034/cmde.2025.68006.3266ENParisaRahimkhaniFaculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran.0000-0002-1286-3087NasrinSamadyarDepartment of Basic Science, Kermanshah University of Technology, Kermanshah, Iran.MohsenRazzaghiDepartment of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi, USA.Journal Article20250707In the current investigation, we propose a novel and computationally efficient numerical framework for solving Hilfer fractal-fractional differential equations (HF-FDEs) by introducing a new class of basis functions termed fractional-order clique functions (FCFs). In contrast to conventional fractional models, which primarily rely on classical kernels and often fall short in encapsulating the intricate interplay between memory-dependent behavior and fractal geometries, the adopted Hilfer fractal-fractional derivative offers a unified formulation that inherently incorporates both non-locality and fractality-features essential for accurately modeling complex real-world processes. To the best of our knowledge, this work marks the first development and implementation of FCFs within a numerical solution framework. The distinctive analytical properties of FCFs facilitate precise, adaptable, and computationally stable representations of HF-FDE solutions. By integrating the FCFs-based approximation with a collocation technique and Newton’s iterative algorithm, the under-study problem is efficiently transformed into a system of nonlinear algebraic equations. A thorough convergence analysis is presented to ensure the theoretical soundness of the approach, and its practical performance is validated through five numerical examples. The results decisively demonstrate the enhanced accuracy and effectiveness of the proposed method in capturing the multifaceted behavior of fractional dynamic systems when compared to traditional approaches.https://cmde.tabrizu.ac.ir/article_20612_026e7dea43e7234966814dab9fe73542.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Relational graph convolutional networks for sentiment analysis116511791987010.22034/cmde.2025.65816.3048ENAsalKhosraviDepartment of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran.ZahedRahmatiDepartment of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran.AliVefghiDepartment of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran.Journal Article20250208With the growth of textual data across online platforms, sentiment analysis is essential for deriving insights from user-generated content. While traditional approaches and deep learning models have shown promise, they often cannot capture complex relationships between entities. In this paper, we propose using Relational Graph Convolutional Networks (RGCNs) for sentiment analysis, which provide better interpretability by modeling dependencies between data points represented as interconnected nodes in a graph structure. We demonstrate our method’s effectiveness through pretrained language models such as BERT and RoBERTa with RGCN architecture on product reviews from Amazon and Digikala datasets and analyze the resulting performance. Our experiments underscore the strength of RGCNs in capturing relational information for sentiment analysis tasks.https://cmde.tabrizu.ac.ir/article_19870_7fd3d9c86ef36506a2338bb12f8de215.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Exactness of the solution to the stochastic fractional impulsive differential equations118011921992510.22034/cmde.2025.61363.2639ENMLatha MaheswariDepartment of Mathematics, PSG College of Arts and Science, Coimbatore, 641 014, India.KarthikMuthusamyDepartment of Mathematics, PSG College of Arts and Science, Coimbatore, 641 014, India.Journal Article20240425This paper investigates the averaging principle for the solutions to stochastic fractional impulsive differential equations (SFIDEs) with nonlocal conditions. The main focus lies in deriving sufficient conditions for the convergence of the averaged SFIDEs. According to certain proposals, solutions to SFIDEs can be approximated by averaged stochastic systems using the mean square. Furthermore, two illustrative examples are provided to demonstrate the effectiveness of the proposed method in approximating the solutions to our model. The numerical simulations highlight the applicability and accuracy of the proposed approach in practical scenarios. This work contributes to the understanding and analysis of SFIDEs with complex conditions, paving the way for further research in the field of finance and industry.https://cmde.tabrizu.ac.ir/article_19925_f8d85c2a03f34b57bd2c4ec4fbebce38.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701A fractional stochastic differential modeling of cancer cells with an application to the immune response of tumor dynamics119312292008310.22034/cmde.2025.63609.2840ENFaezehTohidiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran.JavadDamirchiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran.MaryamRezaeiDepartment of Financial Mathematics, Faculty of Finance Sciences, Kharazmi University, Tehran, Iran.Journal Article20240920In this paper, we present a fractional stochastic model that examines the response of cancer cells to the immune system. The model combines the long-term memory dependence of fractional derivatives with the stochastic nature of cancer cell growth. The geometric Brownian motion is used to present the stochastic nature of this model. By applying the global derivative from different versions of Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, and converting them into the fractional integral version, we demonstrate the memory property of the model by maintaining the initial conditions. We also prove the stability of the model analytically in the two states of the ordinary differential equation and the fractional differential equation by obtaining the equilibrium points of the model in the disease-free state and the disease state. Additionally, we use the numerical method based on Lagrange polynomials and Newton’s polynomials to examine and compare the approximate solution of the model in two different states: the disease-free state and the disease state. Finally, using numerical simulation, we examine the stability of the model in the fractional-random state. We show that using Newton’s polynomial will preserve the stability condition better than Lagrange’s polynomial. Further, we demonstrate that the solutions of the stochastic fractional model are positive and bounded, and we also prove their uniqueness and existence.https://cmde.tabrizu.ac.ir/article_20083_b1f8de96f14a99d6d9f3084632379ed7.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Robust finite difference schemes for one-dimensional parabolic singularly perturbed problems with regular layers123012491977810.22034/cmde.2025.63398.2828ENKishun KumarSahDepartment of Mathematics, National Institute of Technology Patna, India.SubramaniamGowrisankarDepartment of Mathematics, National Institute of Technology Patna, India.Journal Article20240908In this article, we study the midpoint finite difference scheme for the singularly perturbed parabolic convection<br />diffusion problem on non-uniform meshes, which are determined by a generating function with the boundary layer<br />on the right side of the domain. The non-uniform meshes considered in this work are the classical Shishkin meshes,<br />Shishkin-Bakhvalov meshes, and Shishkin-Bakhvalov modified meshes. Uniform convergence is established with<br />respect to the perturbation parameter on the non-uniform meshes. The backward Euler scheme is applied in<br />the time direction and midpoint finite schemes in the space. Uniform convergence of up to second order in the<br />space and first order in time is obtained. Two numerical examples are considered to validate the numerical and<br />theoretical results obtained.https://cmde.tabrizu.ac.ir/article_19778_b03bd6a32b5b52476c1d1f5fb209847c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Unveiling accurate numerical solutions of time-dependent nonlinear models via a modified hyperbolic polynomial collocation approach125012662005610.22034/cmde.2025.66726.3147ENKolade M.Owolabi1. Department of Mathematical Sciences, Federal University of Technology Akure, PMB 704, Akure, Ondo State, Nigeria.
2. Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa 0208, South Africa.
3. Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences
University of the Free State, Bloemfontein 9300, South Africa.BeratKaraagacDepartment of Natural and Mathematical Sciences,
Faculty of Engineering, Tarsus University, Mersin, Turkey.Journal Article20250412This paper proposes a robust numerical strategy for solving the Zeldovich combustion model by employing a hybrid method that integrates hyperbolic polynomial B-spline collocation with finite difference techniques. The Zeldovich model, which arises in combustion theory, captures complex reactive dynamics such as flame propagation, thermal explosions, and detonation waves. In the proposed scheme, time discretization is performed using a finite difference method, while the spatial discretization is handled via a Crank–Nicolson scheme for improved stability and accuracy. The inherent nonlinear terms are linearized using the Rubin–Graves technique, leading to a tractable linear system at each time step. To approximate the spatial component, fourth-order hyperbolic polynomial B-spline basis functions are employed within a collocation framework rooted in finite element methodology. The method is applied to both one-dimensional and two-dimensional versions of the Zeldovich equation. To assess its performance, the proposed approach is compared with an existing fourth-order finite difference method. Numerical experiments show that the hybrid method yields superior accuracy, particularly in capturing sharp gradients and transient dynamics. Benchmark comparisons against exact solutions confirm the method’s improved precision, with detailed error analysis provided through both $L_2$ and $L_\infty$ norms.https://cmde.tabrizu.ac.ir/article_20056_806fe8fbe5266dd35c95a5da6a3a37ed.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Space-efficient algorithms for counting triangles in data streams using trained oracles126712791986710.22034/cmde.2025.65909.3060ENHosseinJowhariFaculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran.ArashRahmatiFaculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran.Journal Article20250216In this paper, we study data stream algorithms for approximating the number of triangles under the assumption that the algorithm has access to an oracle that answers certain queries about the input graph. Specifically, we present algorithms that process the input graph given as a sequence of edges (or vertices) and output an estimate of the number of triangles in the given graph. We consider algorithms that, while processing the input stream, have access to a degree oracle (given a vertex, the oracle provides the degree of the queried vertex) or an edge triangle oracle where the oracle answers whether an edge $(u,v)$ participates in a triangle or not. We implement two single-pass algorithms and the associated oracles in both the edge-arrival and the vertex-arrival models, and evaluate their performance on real-world datasets. Despite the inaccuracies of the oracles used in our experiments, our study shows that they can improve the performance of state-of-the-art triangle counting algorithms on some real-world graphs.https://cmde.tabrizu.ac.ir/article_19867_539869b0e069e72080b33034a2b05500.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Flexible fractional wavelet neural network for non-linear system identification128013001978010.22034/cmde.2025.64339.2915ENHasanDadashzadehDepartment of Mathematics, Payame Noor University, Tehran, Iran.GhasemAhmadiDepartment of Mathematics, Payame Noor University, Tehran, Iran.YousefEdrisi TabriziDepartment of Mathematics, Payame Noor University, Tehran, Iran.BehroozRezaeiFaculty of Electrical and Computer Engineering, Babol Noshirvani University of Technology, Babol, Iran.Journal Article20241104This paper presents a novel neural network architecture called the flexible fractional wavelet neural network (FFrWNN), which enhances traditional wavelet networks by introducing two additional fractional wavelet parameters. These fractional parameters, along with the translation and scale parameters, are dynamically adjusted during the learning process, offering greater flexibility and improved approximation power. The network is trained using a stochastic gradient descent algorithm, and iterative online training formulas are developed for optimizing both the wavelet parameters and network weights. The stability of the network is proven through the Lyapunov stability approach, ensuring reliable convergence. The proposed FFrWNN is evaluated in the context of both one-dimensional and multi-dimensional dynamic system identification. Results demonstrate that the fractional wavelet parameters significantly improve the network’s accuracy and efficiency. Compared to conventional neural networks, the FFrWNN shows superior performance in terms of precision and learning capability, making it a powerful tool for complex system modeling and signal processing applications.https://cmde.tabrizu.ac.ir/article_19780_4d1aa39e454e3ecec0791e9dd06d9827.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Analytical and numerical solutions of the convection-diffusion-reaction equations applying the differential transformation method and the Crank-Nicolson method, along with stability analysis and truncation error analysis130113191986610.22034/cmde.2025.64919.2958ENUmmuHabibahDepartment of Mathematics, Faculty of Science, Technology, and Mathematics, Brawijaya University, Indonesia.0000-0002-9460-6374Zalfa CamillaRohmanDepartment of Mathematics, Faculty of Science, Technology, and Mathematics, Brawijaya University, Indonesia.Yashinta NovenaDewantiDepartment of Mathematics, Faculty of Science, Technology, and Mathematics, Brawijaya University, Indonesia.Arfi NadhifaHanantiDepartment of Mathematics, Faculty of Science, Technology, and Mathematics, Brawijaya University, Indonesia.Journal Article20241208This study presents a unified approach for solving convection-diffusion-reaction equations by integrating the Differential Transformation Method (DTM) for analytical approximations with the Crank-Nicolson numerical scheme. The DTM is employed to derive an analytical solution, while the Crank-Nicolson method is used to compute the numerical solution. The results demonstrate that the analytical solution obtained via DTM is identical to the exact solution. Furthermore, the stability of the Crank-Nicolson numerical scheme is assessed using Von-Neumann stability analysis, confirming that the method is unconditionally stable. The local truncation error is determined via Taylor series expansion to establish its order of accuracy. This analysis reveals that the Crank-Nicolson scheme for the convection-diffusion-reaction equation exhibits a local truncation error of order $O(h^2+k^2)$, ensuring a second-order accurate scheme. Numerical simulations are conducted for various parameter values to examine their impact on the solution. The simulation results demonstrate the gradual transport of the substance from high to low concentration regions, observed through the diminishing displacement of material along the $x$-axis. Further numerical experiments investigate the effects of different values of $h$ and $k$. The results indicate a direct correlation between decreasing values of $h$ and $k$ and a reduction in the average error, underscoring the method’s accuracy and efficiency.https://cmde.tabrizu.ac.ir/article_19866_f64629cba0a6274396bbd8d9e01c8145.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701The double Ramadan group accelerated the Adomian decomposition method for solving nonlinear partial differential equations132013361993210.22034/cmde.2025.65368.3000ENMohamed A.RamadanMathematics and Computer Science Department, Faculty of Science, Menoufia University, Shibin El Kom, Menoufia, Egypt.Mariam M. A.MansourDepartment of basic science, Modern Academy of Computer Science and Management Technology in Maadi, Maadi, Cairo, Egypt.Naglaa A.El-ShazlyMathematics and Computer Science Department, Faculty of Science, Menoufia University, Shibin El Kom, Menoufia, Egypt.Heba A.OshebaMathematics and Computer Science Department, Faculty of Science, Menoufia University, Shibin El Kom, Menoufia, Egypt.Journal Article20250107This paper investigates an advanced method for solving partial differential equations (PDEs) by integrating the Double Ramadan Group Transform (DRGT) with a faster version of the Adomian Decomposition Method (ADM). Initially, the DRGT is applied to transform the PDEs, which simplifies the management of boundary conditions and linear elements. The resulting transformed PDEs are subsequently solved using the enhanced ADM, which is specially tailored to efficiently handle the nonlinear terms that typically make solutions more difficult. The acceleration of the ADM is achieved by utilizing improved decomposition techniques and optimized series expansion methods, leading to significant gains in both the speed of convergence and the accuracy in addressing nonlinearities. The effectiveness of this combined approach is illustrated through several examples involving complex PDEs with challenging nonlinear aspects. The findings demonstrate significant improvements in computational efficiency and solution accuracy, underscoring the potential of this method for solving a wide variety of PDE problems in scientific and engineering applications.https://cmde.tabrizu.ac.ir/article_19932_594bb01dac600864c5db0c23b4e8b3ac.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Application of tanh–coth method for combined and double combined sinh–cosh–Gordon equations arising from chemical reactions to water surface gravity waves133713522008410.22034/cmde.2025.67655.3234ENElvirAkhmetshin1. Department of Economics, Mamun University, Khiva, Uzbekistan. \\
2. Faculty of Economics, RUDN University, Moscow, Russia.IlyosAbdullayevDepartment of Business and Management, Urgench State University, Urgench, Uzbekistan.SamariddinMakhmudov1. Department of Finance and Tourism, Termez University of Economics and Service, Termez, Uzbekistan.\\
2. Department of Finance, Alfraganus University, Tashkent, Uzbekistan.KamilaDakhkilgovaDepartment of programming and infocommunication technologies, Kadyrov Chechen State University, Grozny, Russia.IrinaKorotaevaDepartment I-11 Foreign language for aerospace specialties, Moscow Aviation Institute, Moscow, Russia.GalinaYanovskayaDepartment I-11 Foreign language for aerospace specialties, Moscow Aviation Institute, Moscow, Russia.Journal Article20250602An application of the generalized tanh–coth method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for the combined and the double combined sinh-cosh-Gordon equations. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching for exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized $tanh-coth$ method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.https://cmde.tabrizu.ac.ir/article_20084_ba6d1ddce677d30cc8d107253d4e16b5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Modified Newton’s method for solving parametric ν-support vector regression with Universum data135313672008110.22034/cmde.2025.66530.3113ENFatemehBazikarDepartment of Computer Science, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.BahramSadeghi BighamDepartment of Computer Science, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.AtefehHemmatiDepartment of Computer Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran.Journal Article20250324Universum, representing a third category distinct from the two primary classes in classification tasks, facilitates the incorporation of prior knowledge into the learning process. Extensive studies have confirmed its effectiveness in improving both supervised and semi-supervised classification models. Recently, Universum data has been introduced into parametric $\nu$-support vector regression (UPar-$\nu$-SVR) to enhance generalization performance. In this paper, we present a Newton-based method for solving UPar-$\nu$-SVR, with the objective of further improving its efficiency and accuracy. Our approach reformulates the problem into an unconstrained convex optimization framework and employs a generalized Newton’s method for its solution. To assess the effectiveness of our proposed method, we conduct comprehensive experiments on multiple UCI benchmark data sets. The experimental results indicate that our algorithm outperforms existing techniques, providing superior generalization capabilities and computational efficiency.https://cmde.tabrizu.ac.ir/article_20081_401c554f12c70ec04032f2a93a728d77.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Ambarzumyan-type theorem with local derivative136813741989410.22034/cmde.2025.64979.2963ENBeyhanKemalogluFaculty of Science, Department of Mathematics
23119, Firat University, Elazig, Turkey.HasanBulutFaculty of Science, Department of Mathematics
23119, Firat University, Elazig, Turkey.Journal Article20241211We show the Ambarzumyan theorem in this paper by considering the Sturm-Liouville problem with separable boundary conditions and local derivatives. We prove that if the spectrum consists of the first eigenvalue, then the potential function can be found depending on the first eigenvalue. Also, we give some examples like periodic and anti-periodic boundary conditions. In the case of α = 1, the results of the classical case can be obtained. Although the concept of conformable fractional is debatable, we think the results will be useful for Sturm-Liouville theory.https://cmde.tabrizu.ac.ir/article_19894_b70d4172ef0c036909c93f94798df204.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Investigation of optical solitons in a weakly nonlocal Schrödinger equation with parabolic nonlinearity137513861926210.22034/cmde.2024.61598.2673ENMir SajjadHashemiDepartment of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran.MohammadMirzazadehDepartment of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar-Vajargah, Iran.HamoodUr RehmanDepartment of Mathematics, University Of Okara, Okara, Pakistan.Ahmed H.ArnousDepartment of Physics and Engineering Mathematics, Higher Institute of Engineering,El Shorouk Academy-11837, Cairo, Egypt.MustafaBayramComputer Engineering, Biruni University, Istanbul, Turkey.MostafaEslamiDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.Journal Article20240510The weakly nonlinear Schr¨odinger equation (NLSE) describes wave phenomena in media characterized by weakly nonlinear dispersion. As a versatile framework, it finds application across diverse fields such as plasma waves, water waves, fiber optics, and Bose-Einstein condensates, and this study focuses on investigating various solutions for the weakly nonlocal NLSE with parabolic law nonlinearity. By employing the Nucci reduction method (NRM), we extract exact solutions, including dark and bright solitons and other traveling wave solutions. This technique is particularly valuable for identifying nonlocal symmetries of differential equations, providing an efficient analytical tool for nonlinear problem-solving in engineering and related domains. Furthermore, we derive a first integral through the reduction method. These results are essential for understanding soliton wave propagation in weakly nonlocal media with parabolic law nonlinearity, providing insights into wave dynamics for the proposed model. Finally, two- and three-dimensional density plots are presented to illustrate the physical behavior of some obtained solutions within the governing model.https://cmde.tabrizu.ac.ir/article_19262_e9a93d8de59d58ab6e149851d34298c0.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Numerical investigation of smoking behavior dynamics using the spectral collocation method138714071995510.22034/cmde.2025.63796.2861ENKeerthanaGDepartment of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, India.SAGITHYATHIRUMALAIDepartment of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, India.Journal Article20241004Smoking poses a significant threat to global public health and remains one of the leading causes of health problems. To examine these smoking-related issues, this paper aims to study the modified smoking model, which represents a five-compartment system consisting of potential smokers, snuffing class, irregular smokers, regular smokers, and quitters. The model is evaluated computationally using the spectral collocation method. The core idea of the spectral collocation technique is to approximate the solution as a truncated series of basis functions using Chebyshev polynomials. By incorporating collocation points, the truncated series is transformed into an operational matrix form, which in turn converts the governing differential equations into a system of non-linear algebraic equations. Furthermore, the residual and absolute error for different collocation points are established. Additionally, the effects of various parameters such as transmission rate, recovery rate, quit rate, and death rate on the smoking model have been analyzed. All these computational investigations on the model are displayed in the form of figures. Finally, the effect of different combinations of parameters on the smoking dynamics and its impact is represented using contour plots.https://cmde.tabrizu.ac.ir/article_19955_2dace7aa64e9aa653a3637d7068125bd.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Novel $(\psi,\phi)$-fractional operators with exponential kernels: properties and applications to linear differential equations140814251983610.22034/cmde.2025.62416.2748ENLakhlifaSadek1. Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamil Nadu, India.
2. Department of Mathematics, Faculty of Sciences and Technology, BP 34. Ajdir 32003 Al-Hoceima, Abdelmalek Essaadi University, Tetouan, Morocco.Journal Article20240709This study introduces novel generalized fractional derivatives known as $(\psi,\phi)$-fractional derivatives of the Riemann-Liouville and Caputo types, each incorporating exponential function kernels. These new operators offer distinct advantages, including a semi-group property and a seamless extension of the Riemann-Liouville (RL-FD) and Caputo fractional derivatives (C-FD), as well as integrals (RL-FI). We explore the Laplace transform of these $(\psi,\phi)$-fractional derivatives and integrals, leveraging them to address linear $(\psi,\phi)$-fractional differential equations. Moreover, these fractional operators are general to classical fractional operators, cotangent fractional operators, and generalized proportional operators.https://cmde.tabrizu.ac.ir/article_19836_5eaf2551aea5b688b7764cadbcebe92b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701A multigrid solver for subdiffusion equations142614411978210.22034/cmde.2025.65520.3017ENRezaMokhtari1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran.
2. Department for Mathematics and Scientific Computing, University of Graz, Graz, Austria.MohadesehRamezaniDepartment of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran.GundolfHaase1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran.
2. Department for Mathematics and Scientific Computing, University of Graz, Graz, Austria.Journal Article20250118In this paper, we investigate the S3-FD method for solving time-fractional diffusion equations in both one-dimensional (1D) and two-dimensional (2D) spatial domains, achieving high-order temporal accuracy. We leverage the S3 formula, which has a temporal accuracy of \(4 - \alpha\), to approximate the Caputo fractional derivative of order \(\alpha \in (0,1)\), and we employ the finite difference approach for spatial discretization. We develop a fully discrete scheme for both uniform and non-uniform spatial meshes. Our analysis begins with the 1D subdiffusion problem, where we employ the cyclic reduction method alongside OpenMP-based parallel programming to reduce computational costs. Leveraging this groundwork, we extend our technique to the 2D subdiffusion problem using a multigrid method and domain decomposition strategy paired with MPI programming. This innovative method yields an impressive temporal convergence order of \(\mathcal{O}(\Delta t^{4-\alpha})\). The performance and efficiency of the proposed S3-FD algorithm are demonstrated through numerical experiments, highlighting its potential for large-scale fractional diffusion problems.https://cmde.tabrizu.ac.ir/article_19782_ec45b54311f1b4bfd67871f6ad8855d3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Deriving Novel wave solutions to the (2+1)-dimensional fractional Paraxial wave dynamical equation with Kerr law using the $({G}'/G^2) $-expansion function technique144214571997110.22034/cmde.2025.64296.2908ENFatmaE. Abd ElbaryFaculty of Engineering, MTI university, Cario, Egypt.Mourad S.SemaryBasic Sciences Department, Faculty of Engineering, Badr University in Cairo, Cairo 11829, Egypt.ِِAisha F.FareedDepartment of Electrical Engineering, College of Engineering, Prince Sattam bin Abdulaziz University, Al Kharj 16278, Saudi Arabia.Mohammed A.ElsesyDepartment of Basic Engineering Sciences, Faculty of Engineering Benha, Benha university, Egypt.Journal Article20241101The method is used to find traveling wave solutions of nonlinear evolution equations (NLEEs), not "certain types". Specify "to find exact traveling wave solutions to certain types of nonlinear partial differential equations (NLPDEs)". In this case, the Paraxial Wave Dynamical Equation with Kerr law (PWDE) in the sense of the truncated $ \mathsf{M} $-fractional derivative. This equation is important in the study of wave propagation and optical phenomena. By employing this method, the researchers were able to obtain new, previously unknown exact solutions to this equation. These solutions represent different types of wave solutions, each with their own unique characteristics and properties. The significance of these novel wave solutions lies in their potential applications in physics and engineering. Wave phenomena play a crucial role in various fields, such as optics, photonics, and electromagnetics. The researchers indicate that these specific wave solutions have important practical applications in these domains. Moreover, we provide visual representations of the obtained solutions in the form of 3D, contour, and 2D plots. These graphical illustrations serve to demonstrate the feasibility and reliability of our proposed technique, showcasing its ability to capture the essential characteristics and behaviors of the solutions.https://cmde.tabrizu.ac.ir/article_19971_baca8560e13babd7c91c52f5ca227312.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701$\varphi$-Caputo fractional integro-differential equations with nonlocal conditions via noncompactness measure145814701970510.22034/cmde.2025.62509.2761ENNajatChefnajApplied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University, FST, Beni Mellal, Morocco.AbdellahTaqbibtApplied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University, FST, Beni Mellal, Morocco.KhalidHilalApplied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University, FST, Beni Mellal, Morocco.M'hamedELomariApplied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University, FST, Beni Mellal, Morocco.Journal Article20240715This article aims to prove the existence and uniqueness of solutions to fractional integro-differential equations involving the $\varphi$-Caputo fractional derivative with nonlocal conditions. The results are obtained using the measure of noncompactness, probability density functions, M\"{o}nch's fixed point theorem, and semigroup theory. To demonstrate the applicability and effectiveness of the results, an illustrative example is presented.https://cmde.tabrizu.ac.ir/article_19705_f7747d98cbc4e1be90296b18468d2cc1.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701On the Green's function for the Poisson and the Helmholtz equations on the $n$-dimensional unit sphere147114771992610.22034/cmde.2025.61973.2701ENIlonaIglewska-NowakDepartment of Mathematics, West Pomeranian University of Technology in Szczecin, al. Piastow 17, PL-70-310 Szczecin, PolandJournal Article20240606In the paper, a new method is presented to obtain a closed form of the generalized Green function to the Poisson and the Helmholtz equations on the $n$-dimensional unit sphere.https://cmde.tabrizu.ac.ir/article_19926_aca9c227b3f753d27c20ad2763839205.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214320260701Mechanics of nanofluidic flow-induced nonlinear vibrations of single and multi-walled branched nanotubes in a thermal-magnetic environment147815251823910.22034/cmde.2024.56288.2353ENAhmed AmooYinusa1. Department of Mechanical Engineering, University of Lagos, Nigeria.
2. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, USA.Gbeminiyi MusibauSobamowoDepartment of Mechanical Engineering, University of Lagos, Nigeria.Adekunle OmoladeAdelajaDepartment of Mechanical Engineering, University of Lagos, Nigeria.Sunday JoshuaOjoloDepartment of Mechanical Engineering, University of Lagos, Nigeria.Mufutau AdekojoWaheedDepartment of Mechanical Engineering, Federal University of Agriculture, Abeokuta, Nigeria.RidwanOla-GbadamosiDepartment of Mechanical Engineering, Lagos State University, Nigeria.RidwanAdesesan AmokunDepartment of Mechanical Engineering, University of Lagos, Nigeria.Journal Article20230422The nonlinear vibration analysis of embedded multi-walled branching nanotubes with integrated nanofluids that are resting on a Winkler-Pasternak foundation in a thermal-magnetic environment is the main emphasis of this work. The coupled equations of motion controlling the transverse and longitudinal vibrations of the nanotube are derived using the Euler-Bernoulli theory, Hamilton’s principle, and nonlocal elasticity theory. Additionally, the pressure variation in the tubes and the equation for the deformation of the nanotubes are derived. Furthermore, the vibration models are coupled with the Navier-Stokes equation and the energy equation for the fluid and nanotube. Since the dynamics of multi-walled carbon nanotubes differ from the typical assumption of plug flow, careful investigation is needed when combining them with Navier-Stokes and energy equations. Thus, the generated coupled systems of nonlinear partial differential equations in this work are solved using the multi-dimensional differential transformation method. With the aid of the analytical solution, parametric studies are performed. The findings show that the system's stability reduces as the downstream angle increases. Furthermore, the system's dynamic behavior yielded results that show the magnetic effect has a 20% attenuating or damping effect. Additionally, there is a more than 11% discrepancy between the plug flow assumption and real functioning procedures. Existing analytical, numerical, and experimental results were used to verify and validate the analytical method. It is hoped that this study will provide further understanding of the design of nanotubes and act as a reference for further research in the field.https://cmde.tabrizu.ac.ir/article_18239_50210372b14459a9fe0747409c8745d1.pdf