Article
Mathematics, Applied
Chen Zhu, Bingyin Zhang, Hongfei Fu, Jun Liu
Summary: This paper considers a three-dimensional time-dependent Riesz space-fractional diffusion equation and proposes an ADI difference scheme, which is proven to be unconditionally stable and with second-order accuracy. Numerical experiments demonstrate the effectiveness and efficiency of the method for large-scale modeling and simulations. Additionally, a linearized ADI scheme for the nonlinear Riesz space-fractional diffusion equation is developed and tested.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Shujuan Lu, Tao Xu, Zhaosheng Feng
Summary: In this study, a second-order finite difference scheme is proposed for analyzing a class of space-time variable-order fractional diffusion equation. The scheme is demonstrated to be unconditionally stable and convergent with a convergence order of O(tau(2) + h(2)) under certain conditions, as validated by numerical examples.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Anatoly A. Alikhanov, Chengming Huang
Summary: This paper focuses on constructing L2 type difference analog of the Caputo fractional derivative, studying its fundamental features, and using it to generate difference schemes with different orders in space and time for time fractional diffusion equations. The stability and convergence of the schemes are proven, and numerical computations support the obtained results.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Muhammad Yousuf, Khaled M. Furati, Abdul Q. M. Khaliq
Summary: This article presents a time-stepping method for solving nonhomogeneous parabolic problems with Riesz-space-fractional, distributed-order derivatives. The method approximates the matrix exponential functions and uses Gaussian quadrature approach. Experimental results demonstrate the computational efficiency of the method.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Jianfeng Liu, Tingchun Wang, Teng Zhang
Summary: This paper presents a linearized second-order finite difference scheme for solving the nonlinear time-fractional Schrodinger equation. The optimal error estimate of the numerical solution is established under a weak assumption on the nonlinearity, without any restriction on the grid ratio. The analysis employs mathematical induction method, several inverse Sobolev inequalities, and a discrete fractional Gronwall-type inequality. The proposed scheme exhibits a convergence rate of O(tau(2) + h(2)) with time step tau and mesh size h.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Rahul Kumar Maurya, Vineet Kumar Singh
Summary: In this work, a novel numerical approximation method is designed for dealing with the Caputo fractional derivative in time. The proposed algorithm utilizes non-uniform and uniform grid points to improve the numerical accuracy and convergence speed. Rigorous analysis and numerical tests are conducted to verify the effectiveness of the method for different alpha values.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics
Jing Li, Yingying Yang, Yingjun Jiang, Libo Feng, Boling Guo
Summary: This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation, which transforms the space distributed-order term into multi-term fractional derivatives using the mid-point quadrature rule, discretizes the multi-term fractional equation by the finite volume method based on piecewise-quadratic polynomials, and uses the finite difference method for the time-fractional derivative. The iterative scheme is proven to be unconditionally stable and convergent with a certain accuracy level, as demonstrated by a numerical example.
ACTA MATHEMATICA SCIENTIA
(2021)
Article
Computer Science, Interdisciplinary Applications
Zi-Hang She, Li -Min Qiu, Wei Qu
Summary: In this paper, we employ the RSCSCS iteration method to solve Toeplitz-like linear systems arising from time-dependent Riesz space fractional diffusion equations. The method is shown to be convergent unconditionally and its convergence factor and iteration parameter are analyzed. Additionally, a fast induced RSCSCS preconditioner is designed to accelerate the convergence rate.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Mehdi Dehghan
Summary: The paper introduces a new high-order finite difference scheme with low computational complexity to solve the space-time fractional tempered diffusion equation. The stability analysis and convergence order proof demonstrate the effectiveness and feasibility of the technique.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Shuiping Yang, Fawang Liu, Libo Feng, Ian Turner
Summary: Distributed-order diffusion equations are more useful than general fractional diffusion equations when dealing with anomalous diffusion characterized by multiple scaling exponents. A novel finite volume method was developed in this paper to solve a nonlinear distributed-order diffusion equation with variable coefficients, demonstrating stability and convergence with second order accuracy in both space and time. Numerical examples were provided to show the efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Mohammad Hossein Derakhshan, Hamid Rezaei, Hamid Reza Marasi
Summary: This article proposes a numerical method for finding the numerical solutions to time-fractional diffusion equations involving the Caputo-type fractional distributed order operator. The method uses finite difference approach and applies semi-discrete method for the time variable and fully-discrete method for the spatial variable. It also utilizes Gauss-Legendre quadrature formula for the distributed integral part and the L2 -1 approach to estimate the multi-term time-fractional operator, including the Caputo fractional derivative. The article presents error analysis and stability of the proposed numerical method and provides numerical examples comparing it with previous methods.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Interdisciplinary Applications
Mohammadhossein Derakhshan, Ahmed S. Hendy, Antonio M. Lopes, Alexandra Galhano, Mahmoud A. Zaky
Summary: In this paper, we propose a novel numerical scheme to solve the time-fractional advection-dispersion equation with distributed-order Riesz-space fractional derivatives. This method reformulates the distributed-order Riesz-space fractional derivatives by means of a second-order linear combination of Riesz-space fractional derivatives. The resulting equation is transformed into a time-fractional ordinary differential equation using the matrix transform technique, and a discretization method is used to approximate the weakly singular kernel. The stability, convergence, and error analysis are presented, and simulations are performed to verify the theoretical findings.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Nikhil Srivastava, Aman Singh, Yashveer Kumar, Vineet Kumar Singh
Summary: This paper proposes two efficient numerical schemes for solving Riesz-space fractional diffusion equation and Riesz-space fractional advection-dispersion equation. These schemes transform the equations into systems of linear algebraic equations and achieve second-order accuracy in space. Numerical results show that the schemes are simple, easy to implement, yield high accuracy, and require less CPU time when based on SLP basis.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Pratibha Verma, Manoj Kumar
Summary: In this article, the two-step Adomian decomposition method (TSADM) is introduced for solving the multi-dimensional Riesz space distributed-order advection-diffusion (RSDOAD) equation. The TSADM is successfully applied to obtain the analytical solution without approximation or discretization of the Riesz fractional operator. The existence and uniqueness of the solution are investigated using fixed point theorems and the Banach contraction principle. A generalized example is included to demonstrate the effectiveness and applicability of the proposed method.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Mohammad Hossein Derakhshan
Summary: In this manuscript, a numerical method based on the difference-Legendre spectral method is proposed for obtaining the numerical solutions of the two-dimensional space-time distributed order fractional diffusion-wave equations with the Riesz space fractional derivative. The proposed method discretizes the distribution-order integral part using the difference method and estimates the distribution-order integral part using the Gauss quadrature formula. The stability and convergence analyses are also performed to analyze the numerical estimation.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Ali Ebrahimijahan, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: This article considers a meshfree method for the numerical solution of conversation law equations. By using the integrated radial basis function (IRBF) method and finite difference approximation, the governing models are discretized and converted into a system of nonlinear ordinary differential equations (ODEs). The obtained ODEs are then solved using the Runge-Kutta technique. Numerical examples demonstrate the feasibility and accuracy of the presented method.
ENGINEERING WITH COMPUTERS
(2022)
Article
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, Mehdi Dehghan
Summary: This contribution presents a new high-order numerical algorithm for solving cubic-quintic complex Ginzburg-Landau equations, which is based on problem decomposition and the application of different numerical techniques to obtain numerical approximations.
ENGINEERING WITH COMPUTERS
(2022)
Article
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, Mehdi Dehghan
Summary: The main purpose of this investigation is to develop an interpolating meshless numerical procedure for solving the stochastic parabolic interface problems. The PDE is discretized using the ISMLS approximation and reduced to a system of nonlinear ODEs. A fourth-order time discrete scheme known as ETDRK4 is used to achieve high-order numerical accuracy. Several examples with adequate complexity are examined to validate the new numerical procedure.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Mehdi Dehghan, Amirreza Khodadadian, Thomas Wick
Summary: In this work, we developed a Legendre spectral element method (LSEM) for solving stochastic nonlinear advection-reaction-diffusion models. The basis functions used in this method are based on a class of Legendre functions, with tridiagonal mass and diagonal diffusion matrices. We discretized the temporal variable using a Crank-Nicolson finite-difference formulation, and introduced a random variable W based on the Q-Wiener process for the stochastic direction. We validated the convergence rate and unconditional stability of the semi-discrete formulation, and then extended it to a full-discrete scheme using the Legendre spectral element technique. The error estimation of the numerical scheme was substantiated based on the energy method, and the numerical results confirmed the theoretical analysis.
APPLICABLE ANALYSIS
(2022)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Mehdi Dehghan, Amirreza Khodadadian, Clemens Heitzinger
Summary: A truly meshless numerical procedure has been developed for simulating stochastic elliptic interface problems, based on the generalized moving least squares approximation. This method is straightforward to implement and has high accuracy, with examples demonstrating its efficiency. Compared to other meshless methods, it requires less CPU time.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Mehdi Dehghan, Ali Ebrahimijahan, Mostafa Abbaszadeh
Summary: This research presents a new meshless approach based on integrated radial basis functions (IRBFs) to study the fractional modified anomalous sub-diffusion equation. The temporal direction is discretized using the finite difference method with second-order accuracy, and the spatial direction is approximated using the IRBFs methodology. The numerical results demonstrate the efficiency of this new method in solving time fractional PDEs on complex computational domains.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Mostafa Bayat, Mehdi Dehghan
Summary: This paper investigates the numerical solution of the magnetohydrodynamics equation. The time derivative is approximated using the Crank-Nicholson scheme, and the convergence order and unconditional stability are analyzed using the energy method. The spatial derivative is solved using the direct meshless local Petrov-Galerkin method to obtain the full-discrete scheme. Numerical results demonstrate the efficiency of this method.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Engineering, Multidisciplinary
Mostafa Abbaszadeh, Ali Ebrahimijahan, Mehdi Dehghan
Summary: This article presents a numerical technique based on the compact local integrated radial basis function (CLI-RBF) method for solving ill-posed inverse heat problems (IHP) with continuous/discontinuous heat source. The space derivative is discretized using the CLIRBF procedure, resulting in a system of ODEs related to the time variable. The final system of ODEs is solved using an adaptive fourth-order Runge-Kutta algorithm. The new numerical method is verified through challenging examples and found to be accurate for solving IHP with continuous/discontinuous heat source in one-and two-dimensional cases.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Engineering, Multidisciplinary
Mostafa Abbaszadeh, Yasmin Kalhor, Mehdi Dehghan, Marco Donatelli
Summary: The purpose of this research is to develop a numerical method for option pricing in jump-diffusion models. The proposed model consists of a backward partial integro-differential equation with diffusion and advection factors. Pseudo-spectral technique and cubic B-spline functions are used to solve the equation, and a second-order Strong Stability Preserved Runge-Kutta procedure is adopted. The efficiency and accuracy of the proposed method are demonstrated through various test cases.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Engineering, Multidisciplinary
Mostafa Abbaszadeh, AliReza Bagheri Salec, Alaa Salim Jebur
Summary: This paper investigates a time fractional distributed-order diffusion equation and analyzes its stability, convergence, and numerical accuracy.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Computer Science, Artificial Intelligence
Fatemeh Gholami, Zahed Rahmati, Alireza Mofidi, Mostafa Abbaszadeh
Summary: This research investigates and elaborates on graph machine learning methods applied to non-English datasets for text classification tasks. By utilizing different graph neural network architectures and ensemble learning methods, along with language-specific pre-trained models, the study shows improved accuracy in capturing the topological information between textual data, leading to better text classification performance.
Article
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, Mehdi Dehghan, Dunhui Xiao
Summary: This paper presents a new numerical formulation for simulating tumor growth. The proposed method utilizes the meshless Galerkin technique and a two-grid algorithm to improve accuracy and efficiency in obtaining simulation results.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics, Applied
Reza MohammadiArani, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: Lattice Boltzmann method is a powerful solver for fluid flow, but it is challenging to use it to solve other partial differential equations. This paper challenges the LBM to solve the two-dimensional DKS equation by finding a suitable local equilibrium distribution function and proposes a modification for implementing boundary conditions in complex geometries.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Alireza Bagheri Salec, Taghreed Abdul-Kareem Hatim Aal-Ezirej
Summary: In this paper, an improved Boussinesq model is studied. The existence, uniqueness, stability and convergence of the solution are analyzed through discretization and finite difference methods. The proposed scheme is validated through examples in 1D and 2D cases.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, AliReza Bagheri Salec, Shurooq Kamel Abd Al-Khafaji
Summary: This paper proposes a numerical method using spectral collocation and POD approach to solve systems of space fractional PDEs. The method achieves high accuracy and computational efficiency.
ENGINEERING COMPUTATIONS
(2023)
Article
Mathematics, Applied
Manh Tuan Hoang, Matthias Ehrhardt
Summary: In this paper, a simple approach for solving stiff problems is proposed. Through nonlinear approximation and rigorous mathematical analysis, a class of explicit second-order one-step methods with L-stability and second-order convergence are constructed. The proposed methods generalize and improve existing nonstandard explicit integration schemes, and can be extended to higher-order explicit one-step methods.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Jian Liu, Zengqin Zhao
Summary: In this article, we investigate p(x)-biharmonic equations involving Leray-Lions type operators and Hardy potentials. Some new theorems regarding the existence of generalized solutions are reestablished for such equations when the Leray-Lions type operator and the nonlinearity satisfy suitable hypotheses in variable exponent Lebesgue spaces.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Chengcheng Cheng, Rong Yuan
Summary: This paper investigates the spreading dynamics of a nonlocal diffusion KPP model with free boundaries in time almost periodic media. By applying the novel positive time almost periodic function and satisfying the threshold condition for the kernel function, the unique asymptotic spreading speed of the free boundary problem is accurately expressed.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Xia Wang, Xin Meng, Libin Rong
Summary: In this study, a multiscale model incorporating the modes of infection and types of immune responses of HCV is developed. The basic and immune reproduction numbers are derived and five equilibria are identified. The global asymptotic stability of the equilibria is established using Lyapunov functions, highlighting the significant impact of the reproduction numbers on the overall stability of the model.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Junpu Li, Lan Zhang, Shouyu Cai, Na Li
Summary: This research proposes a regularized singular boundary method for quickly calculating the singularity of the special Green's function at origin. By utilizing the special Green's function and the origin intensity factor technique, an explicit intensity factor suitable for three-dimensional ocean dynamics is derived. The method does not involve singular integrals, resulting in improved computational efficiency and accuracy.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Ying Dong, Shuai Zhang, Yichen Zhang
Summary: This paper investigates a 2D chemotaxis-consumption system with rotation and no-flux-Dirichlet boundary conditions. It proves that under certain conditions on the rotation angle, the corresponding initial-boundary value problem has a classical solution that blows up at a finite time.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Shuhan Yao, Qi Hong, Yuezheng Gong
Summary: In this article, an extended quadratic auxiliary variable method is introduced for a droplet liquid film model. The method shows good numerical solvability and accuracy.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Tong Wang, Binxiang Dai
Summary: This paper investigates the spreading speed and traveling wave of an impulsive reaction-diffusion model with non-monotone birth function and age structure, which models the evolution of annually synchronized emergence of adult population with maturation. The result extends the work recently established in Bai, Lou, and Zhao (J. Nonlinear Sci. 2022). Numerical simulations are conducted to illustrate the findings.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Dinghao Zhu, Xiaodong Zhu
Summary: This paper constructs the soliton solutions of the KdV equation with non-zero background using the Riemann-Hilbert approach. The irregular Riemann-Hilbert problem is first constructed by direct and inverse scattering transform, and then regularized by introducing a novel transformation. The residue theorem is applied to derive the multi-soliton solutions at the simple poles of the Riemann-Hilbert problem. In particular, the interaction dynamics of the two-soliton solution are illustrated by considering their evolutions at different time.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Danhua He, Liguang Xu
Summary: This paper investigates the stability of conformable fractional delay differential systems with impulses. By establishing a conformable fractional Halanay inequality, the paper provides sufficient criteria for the conformable exponential stability of the systems.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Fei Sun, Xiaoli Li, Hongxing Rui
Summary: This paper presents a high-order numerical scheme for solving the compressible wormhole propagation problem. The scheme utilizes the fourth-order implicit Runge-Kutta method and the block-centered finite difference method, along with high-order interpolation technique and cut-off approach to achieve high-order and bound-preserving.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Zhijie Du, Huoyuan Duan
Summary: This study analyzes a direct discretization method for computing the eigenvalues of the Maxwell eigenproblem. It utilizes a specific finite element space and the classical variational formulation, and proves the convergence of the obtained finite element solutions.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Hongliang Li, Pingbing Ming
Summary: This paper proposes an asymptotic-preserving finite element method for solving a fourth order singular perturbation problem, which preserves the asymptotic transition of the underlying partial differential equation. The NZT element is analyzed as a representative, and a linear convergence rate is proved for the solution with sharp boundary layer. Numerical examples in two and three dimensions are consistent with the theoretical prediction.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Shuyang Xue, Yongli Song
Summary: This paper investigates the spatiotemporal dynamics of the memory-based diffusion equation driven by memory delay and nonlocal interaction. The nonlocal interaction, characterized by the given Green function, leads to inhomogeneous steady states with any modes. The joint effect of nonlocal interaction and memory delay can result in spatially inhomogeneous Hopf bifurcation and Turing-Hopf bifurcation.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Baoquan Zhou, Ningzhong Shi
Summary: This paper develops a stochastic SEIS epidemic model perturbed by Black-Karasinski process and investigates the impact of random fluctuations on disease outbreak. The results show that random fluctuations facilitate disease outbreak, and a sufficient condition for disease persistence is established.
APPLIED MATHEMATICS LETTERS
(2024)
