University of TabrizComputational Methods for Differential Equations2345-398214120261101A reproducing kernel method for solving nonlocal functional differential equations with delayed or advanced arguments1131939910.22034/cmde.2025.63743.2850ENHajarRasekhinezhadDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.SaeidAbbasbandyDepartment of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.TofighAllahviranlooFaculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey.EsmailBabolianDepartment of Computer Science, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.Journal Article20240928This paper discusses an effective approach for solving non-local functional differential equations with delayed or advanced arguments. The reproducing kernel method is utilized to avoid the need for an orthogonalization process. The main objective of this technique is to successfully apply this method to solve singular multi-point boundary value problems with non-local conditions, resulting in an accurate approximate solution and a valid error analysis. This method greatly improves the accuracy of the solutions obtained.https://cmde.tabrizu.ac.ir/article_19399_a9f44f4c64489380638066684c4820cc.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101An efficient numerical scheme for solving a competitive Lotka-Volterra system with two discrete delays14221873510.22034/cmde.2024.55194.2293ENElcinGokmenDepartment of Mathematics, Faculty of Science, Muğla Sıtkı Koçman University, Muğla, Turkey.Osman RaşitIşıkElementary Mathematics Education Program, Faculty of Education, Muğla Sıtkı Koçman University, Muğla, Turkey.Journal Article20230131In this study, the Euler series solution method is developed to solve the Lotka–Volterra predator-prey model with two discrete delays. The improved method depends on a matrix-collocation method and Euler polynomials. While creating the method, all terms in the system are converted into matrix forms. Hence, the fundamental matrix equation of the system is obtained. A nonlinear algebraic equation system is achieved by inserting the collocation points into the fundamental system. Then, the unknown coefficients that arise from the Euler series expansion are calculated by solving the final system. Two different error estimation procedures are used to estimate the error of the approximation; the first one is the residual correction procedure, and the second one is a technique similar to RK45. In numerical examples, the variations in the population of both species are presented by figures regarding time. Also, the method’s validity is checked by using residual error analysis.https://cmde.tabrizu.ac.ir/article_18735_064de79406dd6325580c1f4a6a471bfa.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Existence result for the fuzzy form of the equation governing the unsteady motion of solid particles in a fluid medium23351907810.22034/cmde.2024.63793.2862ENMasoumehZeinaliFaculty of Mathematics, Statistics and Computer sciences, University of Tabriz, Tabriz, Iran.GhiyamEslamiDepartment of Mechanical Engineering, Ahar branch, Islamic Azad University, Ahar, Iran.Journal Article20241005The unsteady drag force in the equation governing the dynamics of small solid particles in the fluid medium appears as an integral Volterra operator in the equation, which is known as the history force. The history force has a kernel whose exact and general form is not known to date. In this article, the very general form of this equation is considered so that both the kernel of the history force and the fields affecting the particle motion can have a general linear or non-linear form. In the present work, the fuzzy form of this equation is proposed as a new method for uncertainty analysis of the problem. Using the Shoulder’s fixed point theorem in the semi-linear Banach space, it is proved that the fuzzy form of this equation has a solution.https://cmde.tabrizu.ac.ir/article_19078_abc43ac36aef73c84511961125f5d5ba.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Quintic B-spline method for numerical solution of second-order singularly perturbed delay differential equations36491875910.22034/cmde.2024.61474.2650ENShilpaMalgeSymbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India.Ram KishunLodhiSymbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India.Journal Article20240501This article presents a quintic B-spline method to find an approximate solution of the second-order singularly perturbed differential equation in which the convection term occurs with a negative shift. The proposed method gives rise to a pentadiagonal linear system. Thomas' algorithm is employed to solve the obtained system of equations. The method’s convergence is examined through truncation error analysis, and the existence and uniqueness of the solution are also established. Maximum absolute error is tabulated for two numerical examples, proving the proposed method’s efficiency. Graphs are drawn to show the behavior of the solution. A comparative study shows that the obtained solution is better than the previous solutions in the literature. The method is found to be fourth-order convergent. The effect of the delay parameter on the boundary region is also discussed in the example.https://cmde.tabrizu.ac.ir/article_18759_fd898012f3fbdbae82ad7f3dd7667b31.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Investigation of convergence analysis of the stochastic Heston model with one singular point50591931510.22034/cmde.2025.64394.2921ENAliShandal HashimDepartment of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.EsmaeilNajafiDepartment of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.DavoodAhmadianFaculty of Mathematics, Statistics and Computer Sciences, University of Tabriz, Tabriz, Iran.OmidFarkhonde RouzFaculty of Mathematics, Statistics and Computer Sciences, University of Tabriz, Tabriz, Iran.Journal Article20241107The Heston model is a popular stochastic volatility model used in financial mathematics for option pricing. This paper focuses on the stochastic Heston model (SHM) with one singular point. In this way, we first consider the existence, uniqueness, and boundedness of the numerical solution under the global Lipschitz condition and the linear growth condition. In addition, the stochastic ${\theta}$-scheme is developed to solve the equation numerically, and we obtain a convergence rate with $\min \{2-2\alpha, 1-2\beta \}$ which depends on the kernel parameters. Moreover, Monte Carlo (M.C.) simulation is implemented for this kind of problem in the 95 percent confidence interval, which reveals that it verifies the stochastic ${\theta}$-scheme results. Finally, a numerical example is given to show the validity and effectiveness of the theoretical results.https://cmde.tabrizu.ac.ir/article_19315_c0daf075936b2545e4e9495c60e27db4.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101A mathematical analysis of a non-linear smoking model via fractional operators60801876210.22034/cmde.2024.62331.2738ENSavitaPanwarAmity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India.Rupakshi MishraPandeyAmity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India.Sunil DuttPurohitDepartment of HEAS (Mathematics), Rajasthan Technical University, Kota, India.Shyamsunder-Department of Mathematics, SRM University Delhi-NCR, Sonepat-131029, Haryana, India.Journal Article20240703Smoking is one of the most significant public health hazards that adversely affects all organs in the body and has a detrimental effect on general health. In this work, we investigate a mathematical smoking model by considering a singular and non-local Caputo operator as well as a non-singular modified Atangana-Baleanu Caputo derivative. We propose a Yang transform decomposition technique, which combines the Yang transform with the Adomian decomposition method, to obtain the analytical solution of the model. The existence of a unique solution to the model is established using the Lipschitz condition and fixed-point theory. The local asymptotic stability of the equilibrium point is also discussed. Furthermore, graphical analysis is carried out in order to demonstrate the impact of fractional order.https://cmde.tabrizu.ac.ir/article_18762_78c4a7fd529b6f3da0403bd7587d993f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Compact finite difference scheme for numerical solution of Caputo-Fabrizio fractional Riccati differential equations81941916210.22034/cmde.2025.60832.2602ENMansourehSattariDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, 98155-987, Iran.MaryamArab AmeriDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, 98155-987, Iran.Journal Article20240306A Riccati differential equation (RDE) is a nonlinear differential equation used in many fields, like Newtonian dynamics, quantum mechanics, stochastic processes, propagation, reactor engineering, and optimal control. In this work, we consider the fractional RDE (FRDE) with the Caputo-Fabrizio derivative and use the compact finite difference scheme to solve it numerically. To solve this equation, we initially approximate the first-order derivative appearing in the definition of the Caputo-Fabrizio derivative through the compact finite difference method. By substituting the obtained approximation formula into the original equation, we derive a system of algebraic equations containing unknown values of the solution of the Riccati equation corresponding to specific discrete points in the domain. Solving this system of non-linear equations yields the solution of the Riccati differential equation at the discrete points. We provide some examples to examine the efficiency and accuracy of the suggested method.https://cmde.tabrizu.ac.ir/article_19162_c4651cd9200862314a43c1ca841e4896.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Numerical treatment and optimal control of the hepatitis B virus spatio-temporal model951091863810.22034/cmde.2024.60011.2557ENKhalid K.Ali1. Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman.\\ 2. Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt.KamalRaslsnMathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt.Ahmed F.KouraBasic Science Department, Al-Safwa High Institute of Engineering, Egypt.Mohamed AbozeidShaalanHigher Technological Institute, Tenth of Ramadan City, Egypt.Journal Article20240109In this article, we propose an optimal control of the hepatitis B virus (HBV) infection model. We use four control functions in this model to show the effect of quarantine, vaccination, treatment, and rapid testing in minimizing the infection between individuals. We apply Pontryagin’s maximum principle to study these four controls. We solve the mathematical model without control and after adding control functions using by finite difference scheme. We show the results graphically. In addition, we study the HBV spatio-temporal model numerically and discuss the truncation error and the stability of its numerical scheme.https://cmde.tabrizu.ac.ir/article_18638_74c8ed348e575d8d14f8e655ace33af0.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Numerical solution of stochastic fractional integro-differential and Itô-Volterra integral equations via fractional Genocchi wavelets1101281919710.22034/cmde.2024.64161.2891ENParisaRahimkhaniFaculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran.0000-0002-1286-3087YadollahOrdokhaniDepartment of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.Pedro MLimaCentro de Matemática Computacional e Estocástica, Instituto Superior Técnico, Universidade de Lisboa, Portugal.Journal Article20241024In this research, a novel approach based on the fractional-order Genocchi wavelets (FGWs), inverse hyperbolic functions, and collocation technique is introduced for obtaining numerical solutions of stochastic fractional integro differential equations (SFIDEs) and Itô-Volterra integral equations (IVIEs). Initially, we utilize the Laplace trans form approach to approximate the Caputo fractional derivative. Then, the unknown solution is approximated via a combination of the FGWs and inverse hyperbolic functions. We replace this approximation and its derivatives in the resulting stochastic equation (SE). By the Gauss-Legendre quadrature rule (GLQR) and collocation method, we obtain a system of nonlinear algebraic equations. The derived algebraic system can be readily solved through application of Newton’s iterative scheme. Also, we show the convergence of the mentioned scheme. Ultimately, several test problems are examined to demonstrate the applicability and effectiveness of the suggested technique.https://cmde.tabrizu.ac.ir/article_19197_bdd62bb25b28176610e2fc58cc666ef6.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101On the resolvent operator of a singular impulsive Hahn–Dirac system1291441899110.22034/cmde.2024.59198.2515ENBilender PAllahverdiev1. Department of Mathematics, Khazar University, AZ1096 Baku, Azerbaijan.\\
2. Research Center of Econophysics, UNEC-Azerbaijan State University of Economics, Baku, Azerbaijan.HuseyinTuna1. Research Center of Econophysics, UNEC-Azerbaijan State University of Economics, Baku, Azerbaijan. \\
2. Department of Mathematics, Mehmet Akif Ersoy University, 15030 Burdur, Turkey.Hamlet AIsayevDepartment of Mathematics, Khazar University, AZ1096 Baku, Azerbaijan.Journal Article20231112 In this study, we consider impulsive singular Hahn–Dirac systems. Green’s function and a spectral function for these systems are constructed. Finally, an integral representation of the resolvent operator is obtained.https://cmde.tabrizu.ac.ir/article_18991_4c5520a6422e7744d2e1f464018cbe68.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101An effective Legendre wavelet technique for the time-fractional Fisher equation1451641922010.22034/cmde.2025.63725.2849ENFatihİdizInstitute of Applied Mathematics, Middle East Technical University, Ankara, Turkey.GamzeTangoluDepartment of Mathematics, Izmir Institute of Technology, Izmir, Turkey.NasserAghazadeh1. Department of Mathematics, Izmir Institute of Technology, Izmir, Turkey.\\
2. Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan.0000-0003-2705-8942AmirMohammadiDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.Journal Article20240927This study modifies the time-fractional Fisher equation by adding a source term and generalizing the non-linear power to an arbitrary order. A numerical technique is proposed for the modified time-fractional Fisher equation using Legendre wavelets and the quasilinearization technique. The non-linear term is iteratively linearized using the quasilinearization technique. The convergence analysis and error estimates of the proposed method are studied. Several test problems are solved using the proposed technique, and numerical outcomes are contrasted with those obtained using some other approaches existing in the literature.https://cmde.tabrizu.ac.ir/article_19220_6770194c1ce2d756328dbe80c7bb5c57.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101An accurate finite-difference scheme for the numerical solution of a fractional differential equation1651871861610.22034/cmde.2024.61919.2699ENAniruddhaSealDepartment of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India.SrinivasanNatesanDepartment of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India.0000-0001-7527-1989Journal Article20240602In this article, a steady-state fractional-order boundary-value problem is considered with a fractional convection term. The highest-order derivative term involves a mixed-fractional derivative, which appears as a combination of a first-order classical derivative and a Caputo fractional derivative. We propose an L1 scheme over a uniform mesh for the numerical solution of the fractional differential equation. With the help of a properly chosen barrier function, we discuss error analysis and prove that the proposed method converges with almost first-order. The proposed scheme is also applied to a semilinear fractional differential equation. Numerical experiments are presented to validate the proposed method.https://cmde.tabrizu.ac.ir/article_18616_dd20e460562f61b56f6736c7fb46dfec.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Numerical simulation of a nonlinear system of space-fractional Klein-Gordon-Zakharov equations using the Fourier spectral method1882001947010.22034/cmde.2025.65052.2970ENZohrehYaghooblooDepartment of Mathematics, Ahvaz branch, Islamic Azad University, Ahvaz, Iran.SedighehToubaeiDepartment of Mathematics, Ahvaz branch, Islamic Azad University, Ahvaz, Iran.MehdiJalalvandDepartment of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.JafarEsmailyDepartment of Mathematics, Ahvaz branch, Islamic Azad University, Ahvaz, Iran.SafiyehMohammadianDepartment of Mathematics, Osku Branch, Islamic Azad University, Osku, Iran.Journal Article20241216An accurate and efficient numerical approach for the nonlinear space-fractional Klein-Gordon-Zakharov (KGZ) system of equations incorporating the fractional Laplacian operator is proposed in this study. The method is designed to preserve both mass and energy, which is crucial for accurately solving such complex systems. The spatial discretization is carried out using the Fourier spectral method. In contrast, temporal discretization is achieved through the fourth-order exponential time-differencing Runge-Kutta (ETDRK4) technique, ensuring both efficiency and stability. We prove the convergence of the proposed method, establishing a theoretical foundation for its application. To assess the efficiency and versatility of the proposed method, we report on a series of numerical simulations. The outcomes of these simulations are displayed in tables and graphs, illustrating the performance of the method regarding the approximation error, convergence order, and execution time for various fractional values.https://cmde.tabrizu.ac.ir/article_19470_9511721492d2f910dfd3b3595941a44c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Exact and iterative solutions for DEs, including Fokker-Planck and Newell-Whitehead-Segel equations, using Shehu transform and HPM2012221863210.22034/cmde.2024.61746.2683ENZahidaPerveenDepartment of Mathematics, Lahore Garrison University, Lahore, Pakistan.Afraz HussainMajeedSchool of Energy and Power Engineering, Jiangsu University, Zhenjiang 212013, China.AhmedRefaie AliDepartment of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El Kom 32511, Menofia, Egypt.Journal Article20240521This article proposes an iterative method using the Shehu Transform (ST) and the He’s Homotopy Perturbation Method (HPM). Integrating HPM with ST, this study addresses linear and nonlinear instances of equations like Fokker-Planck and Newell-Whitehead-Segel. The method shows reliability and precision through comparisons between exact and approximate results. The Shehu Transform Homotopy Perturbation Method (STHPM) is applied to these equations for the first time, with numerical and graphical comparisons made to HPM and the Elzaki Projected Differential Transform Method (EPDTM). Results demonstrate quick and accurate convergence, offering a robust alternative to traditional numerical methods. Future research explores extending this method to complex systems and real-world applications.https://cmde.tabrizu.ac.ir/article_18632_cb719fb40169adf63191edca9b5a9acf.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Numerical solutions and error analysis of a system of two-dimensional Volterra integral equations via Block-Pulse functions2232341914210.22034/cmde.2024.60254.2570ENAkramKarimiDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.KhosrowMaleknejadDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.RezaEzzatiDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.Journal Article20240124This paper tries to provide an attractive framework based on Block-Pulse functions for the numerical solution of a system of two-dimensional Volterra integral equations of the second kind. These types of systems are created through the modeling of physics or engineering phenomena. By constructing operational matrices based on Block Pulse functions and the reduction of variables, a simpler algorithm is built. The block-pulse method is affordable because it converts algebraic systems to a matrix system and reduces the amount of computation. Some numerical examples and error analysis, which are in detail, support the method.https://cmde.tabrizu.ac.ir/article_19142_ffeab8dc22d9a6ed220e0364d6fbaa6e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Existence results for nonlinear elliptic $\overrightarrow{p}(\cdot)-$equations2352461901510.22034/cmde.2024.61426.2644ENMokhtarNaceriENS of Laghouat; Box 4033 Station post avenue of Martyrs, Laghouat, Algeria.Journal Article20240429This paper investigates the existence of distributional solutions for a class of nonlinear elliptic $\overrightarrow{p}(\cdot)-$equations, the analysis focuses on the right-hand side, which comprises of a datum $f\in L^{\overrightarrow{p'}(\cdot)}(\Omega)$ that is independent of $u$, and a compound nonlinear term involving a given function $g \in L^{\overrightarrow{p}(\cdot)}(\Omega)$, the solution $u$ and its partial derivatives $\partial_iu,\,i\in\{1,\ldots,N\}$, where $L^{\overrightarrow{p}(\cdot)}(\Omega)$ and $L^{\overrightarrow{p'}(\cdot)}(\Omega)$ represent the variable exponents anisotropic Lebesgue spaces.https://cmde.tabrizu.ac.ir/article_19015_7d3d3403bdd2618f7aae1cc00f74b98d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Efficient numerical solution of singularly perturbed two-point boundary value problems using double exponential non-classical sinc-collocation method2472601939310.22034/cmde.2025.64105.2885ENRozhanMahamad HajiDepartment of Mathematics, Faculty of Sciences, University of Kurdistan, Sanandaj, Iran.AmjadAliPanahDepartment of Mathematics, Faculty of Sciences, University of Kurdistan, Sanandaj, Iran.Journal Article20241021In this paper, we present the application of the double exponential non-classical sinc method for solving a specific class of singularly perturbed two-point boundary value problems. This method is particularly effective for problems with singularities, infinite domains, or boundary layers. We discuss the convergence and error estimation of our approach. Using three illustrative examples, we demonstrate the superior performance of our method compared to existing approaches.https://cmde.tabrizu.ac.ir/article_19393_b4251abf7c6c2e009fda1560c776ad40.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Dynamic insights into gaseous diffusion: analytical soliton and wave solutions via Chaffee-Infante equation in homogeneous media2612731880310.22034/cmde.2024.62543.2765ENSamahMabroukDepartment of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt.MahyMahdyDepartment of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt.RashaSalehDepartment of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt.AhmedRashed1. Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt.
2. Basic Science department, Faculty of Engineering, Delta University for Science and Technology, 11152, Gamasa, Egypt.Journal Article20240718Gaseous diffusion (GD) has been used in various fields, including electromagnetic wave fields, high-energy physics, fluid dynamics, coastal engineering, ion-acoustic waves in plasma physics, and optical fibers. GD involves random molecular movement from areas of high partial pressure to areas of low partial pressure. Researchers have developed models to describe this phenomenon, among these models is the (2 + 1)-dimensional Chaffee–Infante (CI)-equation. This research explores analytical soliton and wave solutions of Gaseous diffusion through a homogeneous medium, considering two analytical methods, the Riccati equation and F-expansion methods. Thirty-seven different solutions have been identified and some of these solutions have been illustrated graphically. The figures show a range of bright, dark, singular, singular-periodic, and kink-type soliton wave solutions.https://cmde.tabrizu.ac.ir/article_18803_2642f7a19deebfadf308d68806267022.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101A novel approach using a new generalization of Bernoulli wavelets for solving fractional integro-differential equations with singular kernel2742911909210.22034/cmde.2024.62229.2728ENSomayehNematiDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran.Journal Article20240623In recent years, numerous fractional-order basis functions have been developed and applied for solving different classes of fractional problems. In this work, a new generalization of fractional-order Bernoulli wavelets is introduced. These new basis functions are used to provide a numerical solution for Hammerstein-type fractional integro-differential equations with a weakly singular kernel. To achieve this, the Riemann-Liouville integral operator is applied to the basis functions, and the result is computed exactly using the analytic form of Bernoulli polynomials. Through this process, key properties of the Riemann-Liouville integral and Caputo derivative are utilized to define two remainders associated with the main problem. After that, using an appropriate set of collocation points, the problem is converted to a system of algebraic equations. Due to the efficiency and high accuracy of this new technique, we extend the method for solving fractional Fredholm-Volterra integro-differential equations. Then, an upper bound of the error is discussed for the approximation of a function based on the fractional-order Bernoulli wavelets. Finally, the method is utilized for solving some illustrative examples to check its performance.https://cmde.tabrizu.ac.ir/article_19092_abc71f45234823f64f1c0dd94b9a64fd.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101A comprehensive review on quantum computing and algorithms2923041891210.22034/cmde.2024.63085.2806ENAbdulrahmanSultanDepartment of System Programming, South Ural State University, Chelyabinsk 454080, Russia.IgorKlebanovDepartment of System Programming, South Ural State University, Chelyabinsk 454080, Russia.Journal Article20240822Quantum computers and simulations are creating new prospects by applying quantum physics concepts in innovative ways to generate and process information. It is expected that such computations will have a positive impact on several fields, from daily tasks to the discovery of new scientific findings. Quantum computing has become much more feasible in recent years, thanks to significant advances in both quantum software and hardware development. In fact, the confirmation of quantum supremacy marks a crucial turning point in the Noisy Intermediate-Scale Quantum (NISQ) era. To comprehend the current state of this developing field and identify unresolved issues that the quantum computing community has to address in the upcoming years, a thorough analysis of the current literature on quantum computing will be of immeasurable value. This article offers a thorough analysis of the literature on quantum computing, Qubits, quantum algorithms, and implementations.https://cmde.tabrizu.ac.ir/article_18912_0cfb9da3242231e46422f48e12b88b4d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101An efficient numerical solution of the optimal control mathematical models for pollutant spread through forest resources based on shifted Bernoulli polynomials3053151942710.22034/cmde.2025.63589.2838ENAsiyehEbrahimzadehDepartment of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran.ElhamKeshavarz HedayatiDepartment of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran.Journal Article20240918This study presents a computational method for solving a mathematical model of optimal control for pollutant spread through forest resources using shifted Bernoulli polynomials (SBPs). The model is formulated as an optimal control problem governed by a system of ordinary differential equations, which is then transformed into a nonlinear programming problem (NLP) using the collocation approach and operational matrix of derivatives based on SBPs. The NLP is employed to obtain numerical solutions, and the results demonstrate the proposed method’s acceptability for modeling pollutant spread through forest resources. The study advises three controls on both types of industries (wood-based and non-wood-based) and forest resources to reduce pollution.https://cmde.tabrizu.ac.ir/article_19427_a8f2574d3369c77d32344d3cf9a939d1.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Numerical solution of linear and nonlinear Schrödinger equations via the shifted Chebyshev collocation method3163361898310.22034/cmde.2024.57475.2405ENNidhiPrabhakarDepartment of Mathematics and Statistics, Gurukul Kangri (Deemed to be University), Haridwar, Uttarakhand.SeemaSharmaDepartment of Mathematics and Statistics, Gurukul Kangri (Deemed to be University), Haridwar, Uttarakhand.Journal Article20230710The present study focuses on numerical solutions of linear and nonlinear Schrödinger equations subject to initial and boundary conditions employing the shifted Chebyshev spectral collocation method (SCSCM). In the solution procedure, unknown function and its space derivatives have been approximated employing shifted Chebyshev polynomials and their derivatives, respectively, together with Chebyshev-Gauss-Lobatto points. The present collocation method transforms the Schrödinger equation into a system of ordinary differential equations (ODEs). Thereafter, the obtained system has been solved employing the fourth-order Runge-Kutta scheme. To demonstrate the accuracy and efficiency of the present method, a comparison of the present numerical solutions of different examples of the Schrödinger equation with exact and approximate solutions available in the literature has been discussed. The SCSCM can be implemented to solve second and higher-order linear and nonlinear partial differential equations (PDEs) arising in physics, mechanics, and biophysics.https://cmde.tabrizu.ac.ir/article_18983_e9059eb0fe9a8c45a0388c543f4ea386.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101A dynamic neuro-fuzzy approach for pattern classification3373471944610.22034/cmde.2025.65211.2984ENErfanVeisiContinuous Improvement Department, Mirab Valves Company, Tehran, Iran.BahramSadeghi BighamDepartment of Computer Science, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.MahdiVasighiDepartment of Computer Science and Information Technology
Institute for Advanced Studies in Basic Sciences, Zanjan, Iran.Journal Article20241229Nowadays, the application of neuro-fuzzy methods has been discovered more than ever for pattern recognition. These powerful tools are able to model the reality of data structure as it should be because, in the real world, datasets are defined in a fuzzy concept. In this research, we present a novel neuro-fuzzy method called Fuzzy Growing Map (FGM), combining the dynamic properties of the Growing Self-Organizing Map (GSOM) and fuzzy set theory. FGM is a dynamic neural fuzzy inference system based on if-then rules, which has the ability to generate fuzzy rules based on certain criteria during the learning phase. This approach can be used as a classifier and approximator. In addition, the trained FGM was used to visualize the fuzzy sets as a map, and the structure of the data can easily be revealed in the feature space. To investigate the effectiveness of FGM, several benchmark datasets were analyzed, and the experimental results for classification show improvements in terms of accuracy and topographic error compared to classification algorithms Fuzzy Self-Organizing Map (FSOM)and Counter Propagation Neural Networks (CPNN).https://cmde.tabrizu.ac.ir/article_19446_bfc81baa57d552b7161bed5a93289e5d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Exact solutions of the nonlinear heat conduction equation using an analytical approach3483601958010.22034/cmde.2025.64393.2920ENInaam RikanHassanUniversity of Information Technology and Communications, (UoITC), Baghdad, Iraq.Journal Article20241106This paper presents new solutions to the nonlinear heat equation using the Exp-function method. The method employs an exponential form to construct diverse solution models, including one-soliton, two-soliton, hyperbolic, and trigonometric soliton solutions. These solutions are crucial for modeling wave phenomena in studying the stress of water surfaces. By utilizing exponential structures, the complexity of the equation is reduced, and computational efficiency is enhanced. This approach offers a robust framework for solving higher-order nonlinear partial differential equations and explains the behavior of solitons in complex systems.https://cmde.tabrizu.ac.ir/article_19580_89a5d02c43fa41e9e4b7005b4b307760.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Numerical idea to solve three-dimensional nonlinear Volterra integral equations with 3D-Legendre polynomials3613731946810.22034/cmde.2025.65630.3028ENJalilManafianDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.PeymanBolgarDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.Journal Article20250125In this paper, a three-dimensional Legendre polynomial (3D-LPs) is used for solving the nonlinear three-dimensional Volterra integral equations (VIEs). Converting the main problem to a nonlinear algebraic system using 3D-LPs, which can be generalized to equations in higher dimensions, then the nonlinear system will be solved. Some results concerning the error analysis have been achieved. Several examples are included to demonstrate the validity and applicability of the method. Moreover, we prove a theorem and a corollary about a sufficient condition for the minimum of mean square error under the Legendre coefficients and the uniqueness of the solution of the nonlinear VIEs. In addition, illustrative examples are included to demonstrate the validity and applicability of the presented method.https://cmde.tabrizu.ac.ir/article_19468_db69e0943cea53efb0467e9848cafdaf.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101A U-Net framework using differential equations for enhanced computer vision in lung disease diagnosis3743911887510.22034/cmde.2024.64290.2905ENNaifAlmusallamDepartment of Management Information Systems (MIS), School of Business, King Faisal University (KFU), Al-Ahsa 31982, Saudi Arabia.VusalaMuradovaLankaran State University, Lankaran, Azerbaijan.Mostafa OAbotalebEngineering School of Digital Technologies, Yugra State University, Khanty Mansiysk, 628012, Russia.TatianaMakarovskikhDepartment of System Programming, South Ural State University, Chelyabinsk, 454080, Russia.HusseinAlkattanDepartment of System Programming, South Ural State University, Chelyabinsk, 454080, Russia.OmarGamal AhmedDepartment of Electric Drive, Mechatronics and Electromechanics, South Ural State University, Chelyabinsk, 454080, Russia.MaadMohsin MijwilCollege of Administration and Economics, Al-Iraqia University, Baghdad, Iraq.Journal Article20241031This study presents a U-Net-based approach for the classification of lung diseases using chest X-ray images. The model effectively leverages its encoder-decoder architecture and skip connections to capture both high-level semantic features and detailed spatial information, crucial for medical image analysis. The U-Net model was trained and tested on a dataset of 3,475 X-ray images, representing three classes: Normal, Lung Opacity, and Viral Pneumonia. The model achieved strong performance, with a weighted F1 score of 0.9770 and Cohen’s Kappa of 0.9653, demonstrating its high accuracy in classifying lung diseases. These results confirm the suitability of U-Net for medical imaging tasks, particularly in detecting subtle abnormalities in chest X-ray images. However, the study also identifies challenges, including class imbalance in medical datasets and the computational demands of training large models like the U-Net. Future improvements could focus on enhancing generalizability and reducing computational complexity through advanced data augmentation, domain adaptation, and architectural optimizations. Overall, this research highlights the potential of U-Net for developing reliable and efficient automated diagnostic tools in healthcare.https://cmde.tabrizu.ac.ir/article_18875_f414608f6df4f824869f03fe59b96a8f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Numerical analysis of the SEIR epidemic model with fractional order3924101907610.22034/cmde.2024.63428.2832ENZain Ul AbadinZafarDepartment of Mathematics, University of Central Punjab, Lahore.SadafIjazDepartment of Mathematics, University of Central Punjab, Lahore.CemilTuncDepartment of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Campus, Van, Turkey.0000-0003-2909-8753Journal Article20240910This study explores an SEIR epidemic model, aiming to achieve rapid stabilization of infectious disease dynamics. The model’s dynamic behavior is analyzed with an emphasis on both local and global stability of equilibria using a Lyapunov function. The existence and uniqueness of the model are confirmed. The theoretical findings are validated, and the controller’s effectiveness is illustrated through numerical simulations conducted in MATLAB/Simulink.https://cmde.tabrizu.ac.ir/article_19076_80c7f2e6aa321bb6b2bb33f6634f04e7.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101Iterative methods for large-scale problems4114202014310.22034/cmde.2025.67389.3208ENRafael H.GamidovBaku State University, Baku, Azerbaijan.Mutallim M.Mutallimov1. Institute of Applied Mathematics, BSU, Baku, Azerbaijan. \\
2. Institute of Information Technology, Ministry of Science and
Education of the Republic of Azerbaijan, Baku, Azerbaijan.\\
3. Azerbaijan Technical University, Baku, Azerbaijan.Journal Article20250521One linear bi-criterion mathematical program, which appears as a large-scale problem in practice, is considered. Problems related to the large size are usually solved with the help of methods based on the possibilities created by the zeros of the matrix of the problem. In this way, a large number of different separation schemes have been suggested in the scientific literature. However, the problems considered here have no such possibility due to their large size. In order to overcome the size problem during the solution of the problem, the possibility of reducing it to a smaller problem is investigated. The reduction is carried out without disturbing the original structure of the problem. The goal is to maintain the possibility of using the existing effective solution methods for the problems before the reduction, as well as for the problems received after the reduction. Suggested here method mainly uses sequential approximation schemes to fulfill.https://cmde.tabrizu.ac.ir/article_20143_6fd1de4359c5c3d34111df045c1316f3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101The Kolmogorov forward equations, information theory, and mathematical modeling of the Ising model4214691962710.22034/cmde.2025.63161.2814ENMehdiShamsDepartment of Mathematics, University of Kashan, Kashan, Iran.Mohammad AliMirzaieDepartment of Mathematics, University of Kashan, Kashan, Iran.Journal Article20240827In this paper, one of the applications of the Kolmogorov forward equations in solving the equations in the Ising model will be analyzed. Then, the limiting distribution and the stationary distribution of the corresponding birth and death process will be calculated. The Ising model is one of the famous physical models that is used in other sciences. In this paper, the three special cases of the Ising model, i.e., the square n × n grid, horizontal, and circular systems, will be analyzed and examined. We will show that, in general, Ising models based on the Boltzmann stationary distribution, the maximum likelihood estimator, and a method of moments strategy to estimate the reverse temperature characteristic of the heat baths are the same. The equation related to finding the maximum likelihood estimator and a method of moments strategy cannot be solved analytically, so these estimates will be calculated using the technique of fitting a linear regression model. By the way, it will be observed that the entropy values for the n×n grid system are smaller than the corresponding values in the horizontal and circular systems. Then, a general case for the convexity of the entropy of the Boltzmann probability function will be introduced. Also, considering an Ising model on two and three points, where the points take values independently and follow a Markov chain, the stationary distribution as well as the Boltzmann probability function, its induced probability function, and maximum likelihood estimator for the parameter will be calculated. Finally, we will review the Metropolis-Hastings algorithm and Gibbs sampling to simulate the one-dimensional and two-dimensional Ising model and also the Potts model in the R software. Finally, we will compare the n × n grid, horizontal, and circular systems. The induced probability vectors by system energies will be found for the three systems. The entropy of these three induced probability vectors and the entropy of Boltzmann probability and their two-by-two Kullback-Leibler divergences will be plotted and compared as a function of the parameter.https://cmde.tabrizu.ac.ir/article_19627_08587b86f7a81f77f8b43500736f438d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214120261101A new delayed SEIR-SEI model for dengue transmission control with sensitivity and competitive mathematical analysis4705012064510.22034/cmde.2025.67008.3176ENShahZebSchool of Distance Education, Universiti Sains Malaysia, USM, 11800, Penang, Malaysia.Siti AinorMohd Yatim1) School of Distance Education, Universiti Sains Malaysia, USM, 11800, Penang, Malaysia.
2) School of Mathematical Sciences, Universiti Sains Malaysia, USM, 11800, Penang, Malaysia.AyeshaKamranDepartment of Mathematics, University of Management and Technology, CII Johar Town, Lahore, 54770, Punjab, Pakistan.RizwanaKausarSchool of Biological Sciences, Universiti Sains Malaysia, USM, 11800, Penang, Malaysia.MuhammadRafiqDepartment of Mathematics, Namal
University 30km Talagang Road, Mianwali, 42250, Pakistan.Journal Article20250427Dengue fever is a viral illness affecting over 129 nations and more than 50% of the global population, causing approximately 400 million cases annually. This study explores the mathematical formulation and dynamics of dengue transmission using a structured SEIR-SEI (susceptible human, exposed human, infected human, recovered human, susceptible vector, exposed vector, and infected vector) model, focusing on immunological and delay-based control strategies. An existing nonlinear delayed SEIR-SEI epidemic model is extended to evaluate the effectiveness of awareness, mosquito deterrence, and therapeutic interventions. Rather than immediately resorting to pharmacological methods, the model emphasizes analyzing delay factors due to their significant role in disease control. Since reducing mosquito populations can harm ecological balance, this new approach applies delay-based strategies on human-related factors such as hospitalization, awareness, and travel restrictions to safeguard both public health and the environment. The findings show that the reproductive number alone is insufficient to predict outbreak persistence; recruitment patterns and mosquito biting rates play a more pivotal role. We analyze the model’s mathematical properties, including the reproduction number, equilibrium points, parameter sensitivity, and both local and global stability. Our results demonstrate that model-based strategies focusing on vector control and human behavior effectively reduce dengue transmission. Additionally, we show that the non-standard finite difference scheme outperforms traditional methods like the fourth-order Runge-Kutta in terms of accuracy, stability, and predictive capability. This study offers valuable insights for public health officials and policymakers in designing sustainable strategies to control endemic dengue transmission and prevent future outbreaks.https://cmde.tabrizu.ac.ir/article_20645_9b2927b10029f05147cab122e9a28ea3.pdf