University of TabrizComputational Methods for Differential Equations2345-398213320250701Analysis of a chaotic and a non-chaotic 3D dynamical system: the Quasi-Geostrophic omega equation and the Lorenz-96 model7217411824010.22034/cmde.2024.61295.2634ENNikolaosGkrekasDepartment of Mathematics, University of Thessaly, Lamia, 35100 Fthiotis, Greece.Journal Article20240420This paper delves into analyzing two 3D dynamical systems of ordinary differential equations (ODEs), namely the Quasi-Geostrophic Omega Equation and the Lorenz-96 Model. The primary objective of this paper is to analyze the chaotic and non-chaotic behavior exhibited by the QG Omega Equation and the Lorenz-96 Model in three dimensions. Through numerical simulations and analytical techniques, the author aims to characterize the existence and properties of attractors within these systems and explore their implications for atmospheric dynamics. Furthermore, we investigate how changes in initial conditions and system parameters influence the behavior of the dynamical systems. Employing a combination of numerical simulations and analytical methods, including stability analysis and Lyapunov functions,the author uncovers patterns and correlations that shed light on the mechanisms driving atmospheric phenomena. This analysis contributes to the understanding of atmospheric dynamics and has implications for weather forecasting and climate modeling, offering insights into the predictability and stability of atmospheric systems. Finally, the author presents the phase portrait of the chaotic system and visualizations of the attractors of both systems.https://cmde.tabrizu.ac.ir/article_18240_9acb8882afc8a8fd27539fd10029b03d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Jacobi collocation method for numerical solution of nonlinear weakly singular Volterra integro-differential equations: fractional and stochastic cases7427571861710.22034/cmde.2024.63023.2800ENQassim HadiHaddamDepartment of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.EsmaeilNajafiDepartment of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.SaeedSohrabiDepartment of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.Journal Article20240819This paper deals with the numerical solution of a class of nonlinear multi-term weakly singular fractional Volterra integro- differential equations by the Jacobi collocation method based on Jacobi orthogonal polynomials. Since the solution of the proposed equation is not smooth enough at the origin, the idea of a smoothing transformation is used to increase the smoothness of the solution. We represent an operator-based discussion of the smoothing transformation and Gauss Jacobi quadrature for Riemann-Liouville integral operators and weakly singular integral operators using their similar constructions and extend it to the error analysis of the proposed method and obtain an error bound for the discrete collocation solution. In addition, we propose an improved stochastic method, based on the efficient sum-of-exponentials (SOE) approximation, to address the low computational efficiency of the proposed method. To test the efficiency and accuracy, various numerical examples are solved by the proposed method and the obtained error results are in accordance with the convergence analysis of the method. Finally, we present an example regarding the stochastic Volterra integro-differential equations with one singular kernel function. https://cmde.tabrizu.ac.ir/article_18617_6fcb9a9ccb279a57ea715ae9da0c9233.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701A higher-order kernel approach for linear fourth-order boundary value problems7587671826110.22034/cmde.2024.61569.2669ENW. J.XingDepartment of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, PR China.FazhanGengDepartment of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, PR China.Journal Article20240508This paper aims at finding high-order convergent numerical approach to solve fourth-order linear boundary value problems (BVPs). By employing the good property of reproducing kernel functions (RKFs), a new collocation technique is proposed. The present approach can give highly accurate numerical solutions to fourth-order BVPs. Some numerical experiments are performed and compared with other approaches to indicate the validity of the proposed technique.https://cmde.tabrizu.ac.ir/article_18261_8623aaa8509568fbe4425a47ce14d460.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Analysis of quarantine and liberate effects on viral infection using SEIR and Caputo $ \alpha $-fractional-order model7687821854310.22034/cmde.2024.60950.2607ENAkbarDehghan NezhadSchool of Mathematics and Computer Science, Iran University of Science and Technology,
Narmak, Tehran, 1684613114, Iran.ArezooMoslemi GhadikolaeiSchool of Mathematics and Computer Science, Iran University of Science and Technology,
Narmak, Tehran, 1684613114, Iran.Journal Article20240319Of the various control measures available, lockdown is widely considered to be the most reliable method for containing the spread of Coronavirus. This study presents two mathematical models utilizing $\alpha$-fractional derivatives to investigate the significance of lockdown in reducing the spread of the virus. In this article, the entire population is divided into four groups:<br />\begin{enumerate}<br /> \item The first group comprises the susceptible population who are not under lockdown.<br /> \item The second group consists of susceptible individuals who are under lockdown.<br /> \item The third group comprises infected individuals who are not under lockdown.<br /> \item The fourth group consists of infective individuals who are under lockdown.<br /> \end{enumerate}<br />One of the aforementioned methods examines the dynamics of COVID-19 by generalizing the SEIR model using $\alpha$-fractional derivatives. The second model comprises five nonlinear differential equations of $\boldsymbol{\alpha}$-fractional order. In both methods, $ \boldsymbol{\alpha} = (\alpha_1,\cdots,\alpha_n) $, where $ 0 < \alpha_i \leq 1 $ for all $ 1 \leq i \leq n$. In other words, if $ \mathbb{T} = (0,1]$, then $ \boldsymbol{\alpha} \in \mathbb{T}^n$.https://cmde.tabrizu.ac.ir/article_18543_639cb52becca720a437d105e83e14758.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701A study on the fractional Ebola virus model by the semi-analytic and numerical approach7838011827010.22034/cmde.2024.61485.2653ENSachinKelagere NarayanaDepartment of Mathematics, Bangalore University, Bengaluru-560056, India.Suguntha DeviKannadasanDepartment of Mathematics, Bangalore University, Bengaluru-560056, India.KumbinarasaiahSrinivasaDepartment of Mathematics, Bangalore University, Bengaluru-560056, India.Journal Article20240502 In this study, an Ebola virus model involving fractional derivatives in the Caputo sense is considered and studied through three different techniques called the homotopy analysis method (HAM), the Haar wavelet method (HWM), and the Range-Kutta method (RKM). The HAM is a semi-analytical approach proposed for solving fractional-order nonlinear systems of ordinary differential equations (ODEs), the Haar wavelet technique (HWT) is a numerical approach for both fractional and integer order, and the RKM is a numerical method used to solve the system of ODEs. We have drawn a semi-analytical solution in terms of a series of polynomials and numerical solutions for the model. First, we solved the model through the HAM by choosing the preferred control parameter. Secondly, the HWT is considered; through this technique, the operational matrix of integration is used to convert the given fractional differential equations (FDEs) into a set of algebraic equation systems, and then the RKM is applied. The model is studied through all three methods, and the solutions are juxtaposed with ND Solver solutions. The nature of the model is analyzed with different parameters, and the calculations are performed using Scilab and Mathematica software. The obtained results are expressed in graphs and tables. Convergence analysis has been discussed in terms of theorems.https://cmde.tabrizu.ac.ir/article_18270_a308dbf3c53fd300190f6290e8098d20.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Enhancing the Legendre-Gauss-Radau pseudospectral method with sigmoid-based control parameterization for solving bang-bang optimal control problems8028141974810.22034/cmde.2025.65162.2982ENFereshtehSamadiDepartment of Mathematics, Payame Noor University (PNU), Tehran, P.O. Box 19395-3697, Iran.Journal Article20241225In bang-bang optimal control problems, the control function is inherently piecewise constant. This feature creates substantial difficulties for the standard Legendre-Gauss-Radau pseudospectral method, which relies on polynomial approximation for the control function. This study introduces a simplified approach that seamlessly integrates sigmoid-based control parameterization with the traditional Legendre-Gauss-Radau pseudospectral method. This integration enables precise approximation of discontinuous control profiles while maintaining the polynomial approximation for state variables. The proposed method significantly minimizes the number of decision variables in the optimization problem while precisely determining both the number and locations of switching points. This leads to notable enhancements in computational efficiency and solution accuracy. Numerical experiments conducted on two benchmark problems, a bridge crane system and a robotic arm control problem, demonstrate the exceptional precision and efficiency of the proposed method. Despite its simplicity, the method delivers results that are on par with those produced by more advanced and intricate techniques. https://cmde.tabrizu.ac.ir/article_19748_8f6ee3df5ad898641ae3d729469ff723.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Numerical solving of multi-term time fractional diffusion-wave equations using shifted Gegenbauer spectral collocation method8158271862110.22034/cmde.2024.61509.2660ENMahboubehMolavi-ArabshahiSchool of Mathematics and Computer Science, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.JalilRashidiniaSchool of Mathematics and Computer Science, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.ShivaTanoomandSchool of Mathematics and Computer Science, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.Journal Article20240504In this paper, we present a numerical method to approximate the solution of the multi-term time fractional diffusion-wave equation (M-TFDWE). The proposed method represents the solution as a sum of shifted Gegenbauer polynomials (SGPs) with unknown coefficients. By using the operational matrix of fractional integration and integer derivatives based on SGPs, the M-TFDWE is converted into a system of algebraic equations. The convergence analysis of this numerical method is also discussed. Finally, we provide two examples to illustrate the accuracy of the proposed method. https://cmde.tabrizu.ac.ir/article_18621_2bbded2e7b787742607637ea0f5f3e3a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Flow of gas-condensate system in a time-dependent deformable reservoir with rock creep effect in the well Bottomhole zone8288381874310.22034/cmde.2024.62354.2745ENFikret A.AlievInstitute of Applied Mathematics, Baku State University, Baku, Azerbaijan.MahammadJamalbayovSOCAR Oil Gas Scientific Research Project Institute, Baku, Azerbaijan.NargizAlizadehMilitary Institute named after Heydar Aliyev, Baku, Azerbaijan.NazileHajiyevaInstitute of Applied Mathematics, Baku State University, Baku, Azerbaijan.DavoodAhmadianFaculty of Mathematics, Statistics and Computer Science, University of Tabriz, Tabriz, Iran.Journal Article20240705 The problem of ow of a gas-condensate system to a draining from a time-dependent deformable formation is considered in the case where the laws of compressibility of reservoir rocks in the bottomhole zone and in the part of the reservoir remote from the well differ. Using the idea of a binary representation of a gas-condensate system, a semi-analytical solution to the considered problem is obtained, an algorithm is proposed for calculating the main indicators of depletion of a gas-condensate reservoir, for the case when near the well (inner zone) the formation undergoes creep, and in the distant part of the reservoir (outer zone) elastic deformation occurs. Based on this algorithm, a computer simulator of the considered process created. The results of a study of the influence of the noted factor on the main indicators of the depletion process of gas-condensate deposits represented by time-dependent (relaxing) reservoirs showed that taking into account the rheological characteristics of the reservoir in the well bottom-hole zone significantly refines the forecasting of the main development indicators. It has been established that when the creep effect of formation rocks in the near-wellbore zone is taken into account, the maximum difference in the current values of formation pressures reaches up to 12.53%. It corresponds to a gas recovery factor value of 0.55.https://cmde.tabrizu.ac.ir/article_18743_bd49c015a85204d7fbbb03db016cfb5d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Some existence and nonexistence results for a class of Kirchhoff-double phase systems in bounded domains8398491825610.22034/cmde.2024.61823.2689ENNguyen ThanhChungFaculty of Mathematics and Informatics, University of Danang, University of Science and Education, 459 Ton Duc Thang Street, Danang, Vietnam.ZohrehNaghizadehFaculty of Sciences, Department of Mathematics, University of Science and Technology of Mazandaran,
P.O. Box 48518-78195, Behshahr, Iran.Journal Article20240526In this paper, the existence and nonexistence of multiple solutions for a class of Kirchhoff-double phase systems depending on one parameter in bounded domains are considered. Our main tools are essentially based on variational techniques. To our best knowledge, there seem to be few results on Kirchhoff-double phase type systems in the existing literature.https://cmde.tabrizu.ac.ir/article_18256_cc3d441aa6b3df124d2d7c8297429273.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations8508691862810.22034/cmde.2024.62926.2793ENHaniyeDehestaniDepartment of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.YadollahOrdokhaniDepartment of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.Journal Article20240812In this work, the multi-term variable-order fractional multi-dimensional differential equations are studied based on Gegenbauer wavelet functions. The main aim of this paper is to develop the spectral method with the help of modified operational matrices, which are directly effective in the numerical process. Therefore, we discuss the novel method of obtaining the modified operational matrices (MOMs) of integration and variable-order (VO) fractional derivative. Then, the overall algorithm for solving multi-term VO-fractional differential equations and partial differential equations is introduced. We also discuss error analysis in detail. At last, we implement the numerical scheme in several examples that involve the damped mechanical oscillator equation, the VO-fractional mobile-immobile advection-dispersion equation, and the VO-fractional nonlinear Galilei invariant advection-diffusion equation. Also, to confirm the theoretical results and demonstrate the accuracy and efficiency of the method, we compare our numerical results with analytical solutions and other existing methods.https://cmde.tabrizu.ac.ir/article_18628_1258462bc0af0af24501816950783a6d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Mathematical modeling of COVID-19 with a constant spatial diffusion term in Ghana8708841832010.22034/cmde.2024.59036.2504ENVivianOsei-BuabengDepartment of Mathematics, College of Science, Kwame Nkrumah University of Science and Technology, Ghana.Albert AttakoraFrimpongKwame Nkrumah University of Science and Technology Senior High School, Ghana.BenedictBarnesDepartment of Mathematics, College of Science, Kwame Nkrumah University of Science and Technology, Ghana.Journal Article20231030The purpose of this study is to develop a mathematical model that incorporates a diffusion term in one dimension in the dynamics of coronavirus disease-19 (COVID-19) in Ghana. A reaction-diffusion model is derived by applying the law of conservation of matter and Fick's law, which are fundamental theorems in fluid dynamics. Since COVID-19 is declared to be a pandemic, most African countries are affected by the negative impacts of the disease. However, controlling the spread becomes a challenge for many developing countries like Ghana. A lot of studies about the dynamics of the infection do not consider the fact that since the disease is pandemic, its model should be spatially dependent, therefore failing to incorporate the diffusion aspect. In this study, the local and global stability analyses are carried out to determine the qualitative solutions to the SEIQRF model. Significant findings are made from these analyses as well as the numerical simulations and results. The basic reproduction number ($R_o$) calculated at the disease-free fixed point is obtained to be $R_o\approx2.5$, implying that, an infectious individual is likely to transmit the coronavirus to about three susceptible persons. A Lyapunov functional constructed at the endemic fixed point also explains that the system is globally asymptotically stable, meaning that COVID-19 will be under control in Ghana for a long period of time. https://cmde.tabrizu.ac.ir/article_18320_d7b12c3c9ac30ca041dae63797bfe24b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701On the wavelet Galerkin method for solving the fractional Fredholm integro-differential equations8859031860910.22034/cmde.2024.62193.2725ENShararehRanjbariDepartment of Mathematics, Ta.C., Islamic Azad University, Tabriz, Iran.MahdiBaghmishehDepartment of Mathematics, Ta.C., Islamic Azad University, Tabriz, Iran.MohammadJahangiri RadDepartment of Mathematics, Ta.C., Islamic Azad University, Tabriz, Iran.BehzadNemati SarayDepartment of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran.Journal Article20240620An effective scheme is presented to estimate the numerical solution of fractional integro-differential equations (FIDEs). In the present method, to obtain the solution of the FIDEs, they must first be reduced to the corresponding Volterra Fredholm integral equations (VFIEs) with a weakly singular kernel. Then, by applying the matrix that represents the fractional integral (FI) based on biorthogonal Hermite cubic spline scaling bases (BHCSSb), and using the wavelet Galerkin method, the reduced problem can be solved. The combination of singularity and the challenge related to nonlinearity poses a formidable obstacle in solving the desired equations, but our method overcomes them well. An investigation of the method's convergence is provided, and it verifies that the convergence rate is $O(2^{-J})$ where $J\in \mathbb{N}_0$ is the refinement level. The verification of convergence has also been demonstrated through the presentation of several numerical examples. Compared to other methods, the results obtained show better accuracy.https://cmde.tabrizu.ac.ir/article_18609_8ce686c49d69bcb7719d69decf72b974.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Existence and uniqueness theorems for fractional differential equations with proportional delay9049181816710.22034/cmde.2024.57553.2415ENPrajaktaRajmaneDepartment of Mathematics, Shivaji University, Kolhapur - 416004, India.JayvantPatadeDepartment of Mathematics, Jaysingpur College, Jaysingpur (Affiliated to Shivaji University, Kolhapur) - 416101, India.Machchhindra T.GophaneDepartment of Mathematics, Shivaji University, Kolhapur - 416004, India.Journal Article20230722In this paper, we apply the successive approximation method (SAM) to solve nonlinear differential equations (DEs) with proportional delay. Utilizing SAM, we establish results on existence and uniqueness. Differential equations (DEs) with proportional delay represent a particular case of time-dependent delay differential equations (DDEs). We demonstrate that the equilibrium solution of time-dependent DDEs is asymptotically stable over finite time intervals. We obtained a series solution for the pantograph and Ambartsumian equations and proved their convergence. Furthermore, we prove that the zero solution of the pantograph and Ambartsumian equations is asymptotically stable. The outcomes of integer order obtained for DEs with proportional delay and time-dependent DDEs have been extended to the initial value problem (IVP) for fractional DDEs and a system of fractional DDEs involving the Caputo fractional derivative. Finally, we illustrate SAMs efficacy using particular non-linear DEs with proportional delay. The results obtained for non-linear DEs with proportional delay by SAM are compared with exact solutions and other iterative methods. It is noted that SAM is easier to use than other techniques, and the solutions obtained using SAM are consistent with the exact solution.https://cmde.tabrizu.ac.ir/article_18167_638d2216c509bed7e91415db249689de.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701A finite difference approach to solve the nonlinear model of electro-osmotic flow in nano-channels9199261854210.22034/cmde.2024.62349.2742ENNasserAghazadehDepartment of Mathematics, Izmir Institute of Technology, Izmir, Türkiye. \\
Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan.0000-0003-2705-8942KianooshRabbaniDepartment of Aerospace Science and Technology, Politecnico di Milano, Milan, Italy.Seyyed HemmatollahTaheri OtaghsaraDepartment of Physics, Sar. C., Islamic Azad University, Sari, Iran.MohsenRabbaniDepartment of Applied Mathematics, Sar. C., Islamic Azad University, Sari, Iran.Journal Article20240704This article considers a system of coupled equations constructed by the nonlinear model of electro-osmotic flow through a one-dimensional nano-channel. Functions that belong to this system include distributions of mole fraction of cation and anion, electrical potential, and velocity. We try to find an accurate closed-form solution. To this end, some mathematical approaches are concurrently used to convert the equations to a nonlinear differential equation in terms of the mole fraction of anion. The latter nonlinear differential equation is transformed into a nonlinear algebraic system by the finite difference method, and the system's solution is obtained using Newton's iterative algorithm. Furthermore, equations for the mole fraction of cation, electrical potential, and velocity in terms of the mole fraction of anion are obtained. We calculate errors by substituting the proposed solution into the equations to validate the results. Comparing the results with some other numerical research works demonstrates an acceptable accuracy.https://cmde.tabrizu.ac.ir/article_18542_d102e98baee279b4d571da04caed26b2.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701A nonlinear mathematical model of the delayed predator-prey system that incorporates intraspecific predator competition and fear effect in prey9279391826910.22034/cmde.2024.61840.2692ENKambalaVenkataiahDepartment of Mathematics, Anurag University, Venkatapur, Hyderabad-500088, Telangana, India.RameshKandalaDepartment of Mathematics, Anurag University, Venkatapur, Hyderabad-500088, Telangana, India.Journal Article20240528In ecological systems, predator-prey contact is seen as something that happens naturally. How does the density of prey populations affect predators? This is a naturally occurring issue in ecosystems. Even though it plays a little role in population dynamics, predators in most ecological models lower prey numbers by direct killing. Research on vertebrates has shown that predator aversion may impact prey population dynamics and reproductive rates. There has been new research on mathematical models of predator-prey systems that include a range of predator-functional responses that include the fear effect. Researchers in this research failed to account for the impact of fear on prey mortality rates. In light of the above, our study focuses on analyzing a predator-prey system that incorporates the cost of perceived fear into reproductive processes using a Holling type-IV functional response. The scheme also includes intraspecific competition within the predators and a gestation delay to make the interactions more realistic and natural. The increase of the predator population is constrained by a high predator-to-prey density ratio by this extra intraspecific competition term. These dynamic model's fundamental aspects such as non-negative, boundedness of solutions, and viability of equilibria are investigated, and adequate conditions are discovered. Both the local and global stability of the system are obtained with sufficient conditions on its functionals and parameters. This study makes a major impact in that it creates a novel technique to quantify some important, regulating system resilience parameters, it also studies the presence of Hopf bifurcation when the time lag parameters exceed the critical values by looking at the related characteristic equation. Furthermore, we addressed how time delay factors reaching thresholds cause the Hopf bifurcation. Numerous numerical examples are used to validate all of these theoretical inferences, and simulations are given to help visualize the examples. https://cmde.tabrizu.ac.ir/article_18269_23a41e2996820bbf818373add662239d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Solving the optimizing parameters problem for non-linear datasets using the high-order general least deviations method (GLDM) algorithm9409671829510.22034/cmde.2024.62441.2751ENMostafa OAbotalebSchool of Electronic Engineering and Computer Science, Department of System Programming, South Ural State University, 454080 Chelyabinsk, Russia.Journal Article20240710This study presents an innovative approach to determining the coefficients of a high-order quasilinear autoregressive model using the Generalized Least Deviations Method (GLDM). The model aims to capture the dynamics of observed state variables over time, employing a set of given functions to relate past observations to current values. The errors in the observations are considered unknown. The core innovation lies in addressing the Cauchy problem within the GLDM framework, which enhances the robustness and precision of parameter estimation for non-linear datasets. GLDM is achieved by incorporating a loss function based on the arctangent function, improving resilience against outliers and non-standard error distributions. Comprehensive computational experiments and statistical validation determine optimal model orders for various datasets, including small NDVI (Normalized Difference Vegetation Index) time series, extensive temperature time series, and large wind speed datasets. The second-order model is most effective for small NDVI datasets, while the fifth-order model excels for large temperature datasets. For wind speed data, despite its large size, the second-order GLDM model demonstrates superior performance due to its ability to balance model complexity with the need for capturing essential dynamics without overfitting. Furthermore, a comparative analysis of GLDM-based models with classical forecasting models demonstrates the superior adaptability and accuracy of GLDM models across different dataset characteristics. This highlights their robustness against outliers and data anomalies. The study underscores the versatility and efficacy of high-order GLDM models as powerful tools in predictive modeling, offering significant improvements over traditional methods.https://cmde.tabrizu.ac.ir/article_18295_895671e1b23c66cbef9dd4d5cb747ae5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701On the numerical solution of the Bagley-Torvik equation using the M\"{u}ntz-Legendre wavelet collocation method9689791942910.22034/cmde.2025.65631.3029ENAnber Abraheem ShlashMohammadDigital Marketing Department, Faculty of Administrative and Financial Sciences, Petra University, Jordan.Suleiman IbrahimMohammadElectronic Marketing and Social Media, Economic and Administrative Sciences Zarqa University, Jordan.AsokanVasudevanFaculty of Business and Communications, INTI International University, 71800 Negeri Sembilan, Malaysia.Muhammad TurkiAlshuridehDepartment of Marketing, School of Business, The University of Jordan, Amman, Jordan.DingNanFaculty of Liberal Arts, Shinawatra University, Thailand.Journal Article20250125 The main goals of this work are to solve the Bagley–Torvik (BT) equation using an effective scheme and to find its numerical solution. The scheme uses the collocation method based on the Müntz-Legendre (ML) wavelets. To apply the method, after approximating the unknown solution by mapping it to the wavelet space, we replace it in the desired equation and then obtain the residual using the operational matrices of the derivative and the Caputo<br />fractional derivative (CFD).<br />Applying the collocation method results in a linear algebraic system. To implement the collocation method, either Chebyshev or Legendre roots serve as collocation points, or uniformly spaced grids are used. The error analysis is investigated, and some numerical examples are presented to show the scheme’s accuracy and effectiveness. Thanks to the flexibility of ML wavelets and the method’s structure, we can sometimes obtain the exact solution.https://cmde.tabrizu.ac.ir/article_19429_675fb2a1f3f72afe6e5214b2c66f17c4.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Solving a system of fractional Volterra integro-differential equations using cubic Hermit spline functions9809941962010.22034/cmde.2025.62477.2756ENMehrdadLakestaniFaculty of Mathematics, Statistics and Computer Science, University of Tabriz, Tabriz, Iran.RoyaGhasemkhaniDepartment of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.TofighAllahviranlooResearch Center of Performance and Productivity Analysis, Istinye University, Istanbul, Türkiye.Journal Article20240712In this article, we solve systems of fractional Volterra integro-differential equations in the sense of the Caputo fractional derivative, using cubic Hermite spline functions. We first construct the operational matrix for the fractional derivative of the cubic Hermite spline functions. Then, using this matrix and key properties of these functions, we transform systems of fractional Volterra integro-differential equations into a system of algebraic equations, which can be solved numerically to obtain approximate solutions. Numerous examples show that the results obtained by this method align closely with the results presented by some previous works.https://cmde.tabrizu.ac.ir/article_19620_600dc39159170f283e4a8b812785e85f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Modelling the transmission of Mpox with a case study in Nigeria and the Democratic Republic of the Congo (DRC)99510111832210.22034/cmde.2024.62086.2711ENOlumuyiwa JamesPeterDepartment of Mathematical and Computer Sciences, University of Medical Sciences, Ondo City, Ondo State, Nigeria.OluwatosinBabasolaCenter for Ecology of Infectious Diseases, Department of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America.Mayowa M.OjoDepartment of Mathematical Sciences, University of South Africa, South Africa.AndrewOmameDepartment of Mathematics, Federal University of Technology Owerri, Nigeria.Journal Article20240613This paper focuses on the dynamics of Mpox, a viral disease, in Nigeria and the Democratic Republic of Congo (DRC) by employing mathematical modeling and parameter estimation techniques. Utilizing optimization methods, the model parameters were calibrated to match the observed Mpox cases and deaths. The basic reproduction number (R0) was calculated for each region, indicating the disease’s transmission potential, and a sensitivity analysis was conducted to identify key parameters influencing disease outcomes. Subsequently, numerical simulations were performed to assess the impact of intervention scenarios on Mpox cases and deaths. The primary goal is to create mathematical methods that can evaluate the risk of Mpox transmission and implement control measures in Nigeria and the DRC, potentially extending the findings to other countries. Results show that reducing parameters related to transmission and progression significantly decreases disease burden, highlighting the importance of preventive measures. These findings provide valuable insights for policymakers and public health officials in designing effective strategies to mitigate Mpox’s impact on human populations.https://cmde.tabrizu.ac.ir/article_18322_9282da8283a49ca4aa94c6191df664e5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Numerical solution of the two-dimensional nonlinear schrödinger equation using an alternating direction implicit method101210211836810.22034/cmde.2024.61040.2619ENEndalew GTsegaDepartment of Mathematics, College of Science, Bahir Dar University, Bahir Dar, Ethiopia.Journal Article20240330In this paper, an alternating direction implicit (ADI) finite difference scheme is proposed for solving the two-dimensional time-dependent nonlinear Schrödinger equation. In the proposed scheme, the nonlinear term is linearized by using the values of the wave function from the previous time level at each iteration step. The resulting block tridiagonal system of algebraic equations is solved using the Gauss-Seidel method in conjunction with sparse matrix computation. The stability of the scheme is analyzed using matrix analysis and is found to be conditionally stable. Numerical examples are presented to demonstrate the efficiency, stability, and accuracy of the proposed scheme. The numerical results show good agreement with exact solutions.https://cmde.tabrizu.ac.ir/article_18368_e8b8b9189ee17a86f8b6c4de9d2dc46e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701On the stability analysis and the solitonic wave structures for the Fordy-Gibbons-Jimbo-Miwa equation102210361855810.22034/cmde.2024.62009.2706ENFazalBadshahSchool of Electrical and Information Engineering, Hubei University of Automotive Technology, Shiyan 442002, People's Republic of China.Kalim U.TariqDepartment of Mathematics, Mirpur University of Science and Technology, Mirpur-10250 (AJK), Pakistan.HadiRezazadehFaculty of Engineering Modern Technologies, Amol University of Special Modern Technologies, Amol, Iran.MedhatIlyasDepartment of Mathematics, Mirpur University of Science and Technology, Mirpur-10250 (AJK), Pakistan.Mir SajjadHashemiDepartment of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran.Mohammad AliHosseinzadehFaculty of Engineering Modern Technologies, Amol University of Special Modern Technologies, Amol, Iran.Journal Article20240607In this article, the Fordy-Gibbons-Jimbo-Miwa equation is analyzed, a special form of the Kadomtsev-Petviashvili hierarchy equation, which is one of the most prominent nonlinear dynamical models with two spatial and a temporal coordinate that represents the evolution of long, nonlinear, small-amplitude waves with a gradual dependence on the transverse coordinate. The governing model is investigated analytically by employing the extended generalized Riccati equation mapping approach (GREM). Furthermore, the dynamics of several wave structures are visualized in 3D, 2D, and contour forms for a given set of parameters using Mathematica 13.0 to demonstrate their characteristics, which has been achieved by selecting appropriate values of the relevant parameters. These solutions exhibit the characteristics of v-shaped, singular, and multi-bell-shaped, singular periodic, and multi-periodic solitons. Additionally, it has been confirmed that the model under consideration is a stable nonlinear structure by validating the established results. A range of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics and other relevant domains and research areas can be solved using these approaches, as demonstrated by their simplicity, clarity, and effectiveness, as well as the computational complexities and results.https://cmde.tabrizu.ac.ir/article_18558_2ed45522b1214de24f9ad733246f8848.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Control of fractional-order chaotic systems under perturbations103710461958810.22034/cmde.2025.59973.2556ENSaeedMirzajaniDepartment of Basic Sciences, Technical and Vocational University (TVU), Tehran, Iran.Journal Article20240107In this paper, an appropriate fractional-integer integral sliding mode method for the control of fractional-order chaotic systems with perturbations such as uncertainties and external disturbances is addressed. When the upper bound of the perturbations is determined, a sliding mode controller is presented. Also, when the upper bound of the perturbations is unknown, an adaptive sliding mode control is designed. Analysis of the stability of the sliding mode surface is presented using the Lyapunov stability theory. Eventually, the results were carried out for the control of the complex fractional order chaotic T system.https://cmde.tabrizu.ac.ir/article_19588_a9e42a0b8673d008492674d5d07e0db6.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701Two-dimensional temporal fractional advection-diffusion problem resolved through the Sinc-Galerkin method104710581838310.22034/cmde.2024.60039.2560ENAliSafaeiDepartment of Mathematics, Faculty of Science, University of Maragheh, Box 55136-553, Maragheh, Iran.Amir HosseinSalehi ShayeganDepartment of Mathematics n, Faculty of Basic Science, Khatam-ol-Anbia (PBU) University, Tehran, Iran.MohammadShahriariDepartment of Mathematics, Faculty of Science, University of Maragheh, Box 55136-553, Maragheh, Iran.Journal Article20240112The Sinc-Galerkin method, even for issues spanning infinite and semi-infinite intervals, is known as exponentially fading mistakes and, in certain circumstances, as the optimum convergence rate. Additionally, this approach does not suffer from the normal instability issues that often arise in other methods. Therefore, a numerical technique based on the Sinc-Galerkin method is devised in this study to solve the two-dimensional time fractional advection diffusion problem. To be precise, the spatial and temporal discretizations of the Sinc-Galerkin and finite difference methods are coupled to provide the suggested approach. Additionally, the suggested method’s convergence is looked at. Two numerical examples are provided in depth in the conclusion to demonstrate the effectiveness and precision of the suggested approach.https://cmde.tabrizu.ac.ir/article_18383_6799e70d422f9a043c8d3d8c3417579c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701High-order numerical solution for a class of nonlinear Fredholm integro-differential equations105910731874110.22034/cmde.2024.58818.2489ENSadeghAmiriDepartment of Basic Sciences, Shahid Sattari Aeronautical University of Science and Technology P.O. Box: 13846-63113, Tehran, Iran.MohammadEshaghnezhadDepartment of Basic Sciences, Shahid Sattari Aeronautical University of Science and Technology P.O. Box: 13846-63113, Tehran, Iran.Journal Article20231013The main objective of this work is to present a high-order numerical method to solve a class of nonlinear Fredholm integro-differential equations. By multiplying appropriate efficient factors and constructing an appropriate approximate function, as well as employing a numerical integration method of order $\gamma$, the above-mentioned problem can be simplified to a nonlinear system of algebraic equations. Furthermore, we discuss the convergence analysis of the presented method in detail and demonstrate that it converges with an order $\mathcal{O}(h^{3.5})$ in the $L^2$-norm. Some test examples are provided to demonstrate that the claimed order of convergence is obtained.https://cmde.tabrizu.ac.ir/article_18741_f0653983dd9f3daaa10ae630a86b90e4.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213320250701On the efficiency of the algorithm for solving complex quadratic double–ratio minimax optimization problem107410841909310.22034/cmde.2025.64854.2951ENArezuZareDepartment of Computer Science, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.BahramSadeghi BighamDepartment of Computer Science, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.Journal Article20241128Quadratic fractional optimization problems frequently arise in wireless communications. This paper introduces an enhanced semidefinite optimization relaxation approach for tackling signal design challenges associated with quadratic double–ratio minimax optimization in complex space. It results in two algorithms that offer a global optimum solution for the problem.https://cmde.tabrizu.ac.ir/article_19093_ba46f9a075316b4bfe12cf8f8cc55e23.pdf