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
N. H. Sweilam, S. M. Ahmed, M. Adel
Summary: A simple numerical technique was proposed in this paper to solve two important types of fractional anomalous sub-diffusion equations, and its stability was analyzed to demonstrate accuracy and effectiveness. Four numerical examples were presented to illustrate the method's performance.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Bo Yu
Summary: This paper investigates the multi-term time fractional mixed diffusion and diffusion-wave equation, deriving a high-accuracy compact finite difference scheme. The study rigorously discusses the stability and convergence of the method, with numerical experiments confirming its efficiency and convergence properties. A practical example demonstrates the applicability of the model and the efficiency of the derived high-order compact finite difference method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Lei Ren
Summary: In this paper, a high order compact finite difference method is proposed for the time multi-term fractional sub-diffusion equation. The method achieves high accuracy in both time and space, and a proof of stability and convergence is provided. Numerical results demonstrate the efficiency of the proposed method.
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
Ben Wongsaijai, Phakdi Charoensawan, Tanadon Chaobankoh, Kanyuta Poochinapan
Summary: This paper investigates the performance of a compact structure-preserving finite difference scheme, discussing the convergence, stability, and accuracy of the approximate solution with respect to grid refinement. The method, which maintains invariants precisely, is found to have fourth-order spatial accuracy and is verified through comparison with second-order finite difference schemes. The efficiency of the scheme is confirmed through simulations of the problem over a long time period.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Qing Li, Huanzhen Chen, Hong Wang
Summary: In this article, a proper orthogonal decomposition-compact difference scheme (POD-CDS) is proposed for the displacement-stress form of a simply supported plate vibration model. It is proven that the POD-CDS can maintain the same spatial and temporal convergence rates and unconditional stability as the compact difference solution, while significantly improving computing efficiency. Stability and convergence analysis for the corresponding compact difference scheme is also conducted. Numerical experiments verify the theoretical findings and demonstrate that the POD-CDS is nearly 10-30 times faster than the compact difference scheme.
NUMERICAL ALGORITHMS
(2023)
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
Computer Science, Interdisciplinary Applications
Kanyuta Poochinapan, Ben Wongsaijai
Summary: This paper introduces weighted parameters in standard compact difference operators and applies high-order compact finite difference operators and the Crank-Nicolson/Adams-Bashforth method to solve coupled BBM equations. The numerical results demonstrate that the proposed weighted compact finite difference scheme can improve computational efficiency and numerical accuracy.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Engineering, Multidisciplinary
Rahul Kumar Maurya, Vinita Devi, Vineet Kumar Singh
Summary: Stable multistep schemes based on Caputo fractional derivative approximation are proposed for solving 1D and 2D nonlinear fractional model arising from dielectric media. The schemes are numerically verified to be effective and stable through test functions.
APPLIED MATHEMATICAL MODELLING
(2021)
Article
Mathematics, Applied
Gengen Zhang, Chunmei Su
Summary: This paper introduces a highly accurate conservative method for solving the quantum Zakharov system, which is fourth-order accurate in space and second-order accurate in time according to detailed numerical analysis. The proposed scheme's conservation properties and high accuracy are confirmed through various numerical examples. Additionally, the compact scheme is used to study the convergence rate of the quantum Zakharov system to its limiting model in the semi-classical limit.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Materials Science, Multidisciplinary
M. H. Heydari
Summary: In this paper, a modified anomalous space-time fractional sub-diffusion equation in two dimensions is proposed and an effective computational technique based on Chebyshev cardinal polynomials is developed. The technique involves obtaining the fractional derivative matrices of these polynomials and approximating the unknown solution using these matrices and the collocation technique. The approach is validated with the successful solution of two test problems.
RESULTS IN PHYSICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Ziyang Luo, Xingdong Zhang, Shuo Wang, Lin Yao
Summary: In this paper, a new numerical scheme is proposed to solve time fractional partial integro-differential equations with a weakly singular kernel. The proposed scheme, based on weighted and shifted Grunwald formula and compact difference operate, ensures stability and convergence, with accuracy independent of the fractional parameter a.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Computer Science, Interdisciplinary Applications
Jianqiang Xie, Quanxiang Wang, Zhiyue Zhang
Summary: This paper develops and analyzes two stable and efficient time second-order difference schemes for handling space fractional KGZ systems. One scheme is based on multi-point weighted time second-order scheme to construct energy conservative linearized difference scheme, while the other scheme splits the original system into linear and nonlinear parts and advances the subproblems with three stages. The proposed schemes are linear decoupled and can be easily applied in parallel computing, especially in long time simulations.
JOURNAL OF COMPUTATIONAL SCIENCE
(2022)
Article
Mathematics, Applied
Emadidin Gahalla Mohmed Elmahdi, Sadia Arshad, Jianfei Huang
Summary: In this paper, a linearized compact difference scheme for one-dimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions is proposed. The initial singularity of the solution is considered and the Crank-Nicolson technique, combined with other formulas, is used for time and spatial discretization. The proposed scheme is shown to have unconditional stability, convergence, and high accuracy in both time and space through theoretical analysis and numerical experiments.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(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
Moh. Ivan Azis, Mostafa Abbaszadeh, Mehdi Dehghan
Summary: LT-BEM is applied to solve a diffusion-convection equation with variable coefficients for anisotropic functionally graded media. By transforming the variable coefficients equation into a constant coefficients equation, a boundary-only integral equation is derived to find numerical solutions. The study highlights the significant impact of anisotropy and inhomogeneity of materials on solutions in experimental and practical studies.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(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
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, Mehdi Dehghan, Ionel Michael Navon
Summary: This paper discusses the development of a fast and robust numerical method for simulating a system of fractional PDEs, using finite difference and spectral Galerkin methods with reduced-order technique. The stability and convergence properties of this new technique are analyzed, and examples are provided to validate the theoretical results.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Junfeng Cao, Ke Chen, Huan Han
Summary: This paper proposes a two-stage image segmentation model based on structure tensor and fractional-order regularization. In the first stage, fractional-order regularization is used to approximate the Hausdorff measure of the MS model. The solution is found using the ADI scheme. In the second stage, thresholding is used for target segmentation. The proposed model demonstrates superior performance compared to state-of-the-art methods.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Dylan J. Oliver, Ian W. Turner, Elliot J. Carr
Summary: This paper discusses a projection-based framework for numerical computation of advection-diffusion-reaction (ADR) equations in heterogeneous media with multiple layers or complex geometric structures. By obtaining approximate solutions on a coarse grid and reconstructing solutions on a fine grid, the computational cost is significantly reduced while accurately approximating complex solutions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Nathan V. Roberts, Sean T. Miller, Stephen D. Bond, Eric C. Cyr
Summary: In this study, the time-marching discontinuous Petrov-Galerkin (DPG) method is applied to the Vlasov equation for the first time, using backward Euler for a Vlasov-Poisson discretization. Adaptive mesh refinement is demonstrated on two problems: the two-stream instability problem and a cold diode problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Yizhi Sun, Zhilin Sun
Summary: This work investigates the convexity of a specific class of positive definite probability measures and demonstrates the preservation of convexity under multiplication and intertwining product. The study reveals that any integrable function on an interval with a polynomial expansion of fast absolute convergence can be decomposed into a pair of positive convex interval probabilities, simplifying the study of interval distributions and discontinuous probabilistic Galerkin schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Bhagwan Singh, Komal Jangid, Santwana Mukhopadhyay
Summary: This paper examines the prediction of bending characteristics of nanoscale materials using the Moore-Gibson-Thompson thermoelasticity theory in conjunction with the nonlocal strain gradient theory. The study finds that the stiffness of the materials can be affected by nonlocal and length-scale parameters, and the aspect ratios of the beam structure play a significant role in bending simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Guoliang Wang, Bo Zheng, Yueqiang Shang
Summary: This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method, and the post-processing technique. The algorithm generates a global continuous approximate solution using the partition of unity method and improves the smoothness of the solution by adding an extra coarse grid correction step. It has good parallel performance and is validated through theoretical error estimates and numerical test examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Hao Xu, Zeng-Qi Wang
Summary: Fluid flow control problems are crucial in industrial applications, and solving the optimal control of Navier-Stokes equations is challenging. By using Oseen's approximation and matrix splitting preconditioners, we can efficiently solve the linear systems and improve convergence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhengya Yang, Xuejuan Chen, Yanping Chen, Jing Wang
Summary: This paper focuses on the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for spatial discretization and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. As a result, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Several numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed numerical scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
