Article
Mathematics, Applied
Harendra Singh
Summary: In this paper, we solve the fractional advection-dispersion equation (ADE) using the Jacobi collocation method. The proposed method is validated and its efficiency is demonstrated through convergence analysis and illustrative examples.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Multidisciplinary Sciences
Mohamed Abdalla, Mohamed Akel, Junesang Choi
Summary: This paper introduces a modified matrix of Riemann-Liouville fractional integrals and investigates its relationship with certain functions and polynomials. It also considers the matrix in connection with Jacobi polynomials and points out potential avenues for further research on fractional integrals.
Article
Mathematics, Interdisciplinary Applications
Youssri Hassan Youssri, Ahmed Gamal Atta
Summary: In this paper, we propose a spectral collocation method to solve a specific class of nonlinear singular Lane-Emden equations with generalized Caputo derivatives that appear in the study of astronomical objects. The method approximates the solution as a truncated series of shifted Jacobi polynomials and uses the spectral collocation method to obtain the unknown expansion coefficients. Our solutions can be easily generalized to the classical Lane-Emden equation, and numerical examples demonstrate the effectiveness and applicability of the method.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Interdisciplinary Applications
Mohamed A. Abdelkawy, Ahmed Z. M. Amin, Antonio M. Lopes, Ishak Hashim, Mohammed M. Babatin
Summary: In this paper, a fractional-order shifted Jacobi-Gauss collocation method is proposed to solve variable-order fractional integro-differential equations with weakly singular kernel. By solving systems of algebraic equations, the approximate solutions of the equations are obtained using Riemann-Liouville fractional integral and derivative as well as fractional-order shifted Jacobi polynomials. The method demonstrates superior accuracy through numerical examples.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Leila Moradi, Dajana Conte, Eslam Farsimadan, Francesco Palmieri, Beatrice Paternoster
Summary: This work introduces a novel formulation for solving optimal control problems related to nonlinear Volterra fractional integral equations systems. By studying the properties of Chelyshkov polynomials, a spectral approach is implemented, with the use of operational matrices and the Galerkin method for discretization. The proposed approach is illustrated through several examples, showing its accuracy and improved efficiency compared to other methods.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
P. Agarwal, A. A. El-Sayed, J. Tariboon
Summary: This paper presents a numerical method for finding approximate solutions to fractional integro-differential equations of variable order, by converting the problem into algebraic equations and using shifted Vieta-Fibonacci polynomials to construct operational matrices. The method is demonstrated to be applicable and accurate through numerical applications.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Zhendong Gu
Summary: The research investigates the spectral collocation method for Riemann-Liouville fractional terminal value problems, aiming to solve the corresponding nonlinear weakly singular Volterra-Fredholm integral equation by transforming it into a new integral equation with a linear kernel. The provided convergence analysis and numerical experiments show that the convergence rate of numerical errors depends on the regularity of the solution to the corresponding VFIE.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Lakhlifa Sadek
Summary: In this paper, a new type of fractional derivatives involving exponential cotangent function in their kernels called Riemann-Liouville Ds,?, and Caputo cotangent fractional derivatives CDs,? were introduced, along with their corresponding integral Is,?. These new fractional derivatives possess a semi-group property, and they have special cases such as the Riemann-Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann-Liouville fractional integral (RL-FI) when ?=1. The paper also provided theorems, lemmas, and solutions to linear cotangent fractional differential equations using the Laplace transform of Ds,?, CDs,?, and Is,?. Lastly, the application of this new type of fractional calculus on the SIR model was discussed. This new approach can be useful for researchers working on related subjects.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Yong Zhou, Jing Na Wang
Summary: The paper investigates the existence of solutions for the nonlinear Rayleigh-Stokes problem for a generalized second grade fluid with Riemann-Liouville fractional derivative. It demonstrates that the solution operator of the problem is compact and continuous in the uniform operator topology, and provides an existence result of mild solutions for the nonlinear problem.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
M. Pourbabaee, A. Saadatmandi
Summary: In this article, a general technique for forming a new operational matrix of the distributed-order fractional derivative is proposed using orthonormal Bernoulli polynomials. The tau approach and the obtained operational matrix are then applied to solve several distributed-order time-fractional partial differential equations. The study confirms the validity and accuracy of the proposed technique through error analysis and comparison with exact solutions and numerical results from relevant studies.
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION
(2023)
Article
Mathematics
Tahereh Eftekhari, Jalil Rashidinia
Summary: In this research, we establish sufficient conditions for the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations and prove their uniqueness. We then utilize operational matrices of two-variable shifted Jacobi polynomials via the collocation method to transform the equations into a system of equations. We also obtain error bounds of the proposed method and successfully apply it to solve five test problems, demonstrating its accuracy, efficiency, and applicability.
Article
Mathematics, Applied
Caiyu Jiao, Abdul Khaliq, Changpin Li, Hexiang Wang
Summary: In this study, it was found that the Riesz derivative and the fractional Laplacian have differences on a proper subset of the real line.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2021)
Article
Multidisciplinary Sciences
Jalil Rashidinia, Tahereh Eftekhari, Khosrow Maleknejad
Summary: This paper presents an efficient numerical method for solving two-dimensional nonlinear fractional integral equations, by deriving new operational matrices and utilizing shifted Jacobi collocation method. The resulting systems are solved using the Newton method, with discussion on error bound and convergence analysis provided. The method's efficiency, accuracy, and validity are demonstrated through application to test examples and comparison with existing methods.
JOURNAL OF KING SAUD UNIVERSITY SCIENCE
(2021)
Article
Thermodynamics
Eduardo De-la-Vega, Anthony Torres-Hernandez, Pedro M. Rodrigo, Fernando Brambila-Paz
Summary: Concentrator photovoltaic systems have the highest conversion efficiency among solar applications, but need to increase it to compete with traditional photovoltaic systems. Hybridization with thermoelectric generators can enhance global efficiency by recovering waste heat. Two innovative methods using fractional derivatives were developed in this paper, marking their first application in the photovoltaic field. These methods can be used for energy prediction and design optimization.
APPLIED THERMAL ENGINEERING
(2021)
Article
Mathematics
Milan Medved, Eva Brestovanska
Summary: In this paper, a new type of fractional derivative called the tempered Psi-Caputo fractional derivative is defined, which generalizes the tempered Caputo fractional derivative and the Psi-Caputo fractional derivative. The Cauchy problem for fractional differential equations with this derivative is discussed, and some existence and uniqueness results are proven. Additionally, a Henry-Gronwall type inequality for an integral inequality with the tempered Psi-fractional integral is presented and applied in the proof of an existence theorem. Furthermore, a result on a representation of solutions of linear systems of Psi-Caputo fractional differential equations is proved, and an example is provided in the last section.
MATHEMATICAL MODELLING AND ANALYSIS
(2021)
Article
Mathematics, Applied
Behzad Nemati Saray, Mehrdad Lakestani, Mehdi Dehghan
Summary: The paper presents the design, analysis, and testing of the multiwavelets Galerkin method for solving the two-dimensional Burgers equation. By discretizing time using the Crank-Nicolson scheme, a PDE is obtained for each time step and then solved using the multiwavelets Galerkin method. The results demonstrate the effectiveness of the method by reducing the number of nonzero coefficients while maintaining the error within a certain range.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Majid Haghi, Mohammad Ilati, Mehdi Dehghan
Summary: In this paper, a high-order compact scheme is proposed for solving two-dimensional nonlinear time-fractional fourth-order reaction-diffusion equations. The unique solvability of the numerical method is proved in detail, and the convergence of the proposed algorithm is proved using the energy method. Numerical examples are given to verify the theoretical analysis and efficiency of the developed scheme.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics, Applied
Hadi Mohammadi-Firouzjaei, Hojatollah Adibi, Mehdi Dehghan
Summary: In this paper, backward difference and local discontinuous Galerkin (BDLDG) methods are used for temporal and spatial discretization of fourth-order partial integrodifferential equations (PIDEs) with memory terms containing weakly singular kernels. The stability analysis of the proposed method is provided, and numerical experiments demonstrate the stability of the resulting scheme and numerically show that the optimal convergence rate is O(h(k+1)) in the discrete L(2)norm.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Mohammad Shirzadi, Mohammadreza Rostami, Mehdi Dehghan, Xiaolin Li
Summary: In this paper, a valuation algorithm is developed for pricing American options under the regime-switching jump-diffusion processes, using a combination of moving least-squares approximation and an operator splitting method. The numerical experiments with American options under different regimes demonstrate the efficiency and effectiveness of the proposed computational scheme.
CHAOS SOLITONS & FRACTALS
(2023)
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
Mathematics, Applied
Mostafa Abbaszadeh, Mahmoud A. Zaky, Ahmed S. Hendy, Mehdi Dehghan
Summary: In this paper, a numerical formulation with second-order accuracy in the time direction and spectral accuracy in the space variable is proposed for solving a nonlinear high-dimensional Rosenau-Burgers equation. The spectral element method and the two-grid idea are combined to simulate the equation, and a three-level algorithm is used for the proposed technique. The existence and uniqueness of the solutions to Steps 1, 2, and 3 are investigated, and error analysis is also discussed.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Biology
Niusha Narimani, Mehdi Dehghan
Summary: This paper numerically studies the therapies of prostate cancer in a two-dimensional space. The proposed model describes the tumor growth driven by a nutrient and the effects of cytotoxic chemotherapy and antiangiogenic therapy. The results obtained without using any adaptive algorithm show the response of the prostate tumor growth to different therapies.
COMPUTERS IN BIOLOGY AND MEDICINE
(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
Mahboubeh Najafi, Mehdi Dehghan
Summary: In this work, two-dimensional dendritic solidification is simulated using the meshless Diffuse Approximate Method (DAM). The Stefan problem is studied through the phase-field model, considering both isotropic and anisotropic materials for comparisons. The effects of changing some constants on the obtained patterns are investigated.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Computer Science, Interdisciplinary Applications
Hasan Zamani-Gharaghoshi, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: This paper presents a local meshless collocation method for solving reaction-diffusion systems on surfaces. The proposed numerical procedure utilizes Pascal polynomial approximation and closest point method. This method is geometrically flexible and can be used to solve partial differential equations on unstructured point clouds. It only requires a set of arbitrarily scattered mesh-free points representing the underlying surface.
ENGINEERING WITH COMPUTERS
(2023)
Article
Computer Science, Interdisciplinary Applications
Reza MohammadiArani, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: Different coupled systems for the shallow water equation, bed elevation, and suspended load equation have been proposed. The main goal of this paper is to utilize an advanced lattice Boltzmann method (LBM) to solve this system of equations. In addition, a practical approach is developed for applying open boundary conditions in order to relax the solution onto a prescribed equilibrium flow.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics, Applied
Mehdi Dehghan, Zeinab Gharibi
Summary: This paper discusses the incompressible miscible displacement of two-dimensional Darcy-Forchheimer flow and formulates a mathematical model with two partial differential equations: a Darcy-Forchheimer flow equation for the pressure and a convection-diffusion equation for the concentration. The model is discretized using a fully mixed virtual element method (VEM) and stability, existence, and uniqueness of the associated mixed VEM solution are proved under smallness data assumption. Optimal error estimates are obtained for concentration, auxiliary flux variables, and velocity, and several numerical experiments are presented to support the theoretical analysis and illustrate the applicability for solving actual problems.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Fatemeh Asadi-Mehregan, Pouria Assari, Mehdi Dehghan
Summary: In this research, we present a numerical approach for solving a specific type of nonlinear integro-differential equations derived from Volterra's population model. This model captures the growth of a biological species in a closed system and includes an integral term to account for toxin accumulation. The proposed technique utilizes the discrete Galerkin scheme with the moving least squares (MLS) algorithm to estimate the solution of the integro-differential equations.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2023)
Article
Mathematics, Applied
Mehdi Dehghan, Zeinab Gharibi, Ricardo Ruiz-Baier
Summary: In this article, a fully coupled, nonlinear, and energy-stable virtual element method (VEM) is proposed and analyzed for solving the coupled Poisson-Nernst-Planck (PNP) and Navier-Stokes (NS) equations. The stability, existence, and uniqueness of solution of the associated VEM are proved, and optimal error estimates for both the electrostatic potential and ionic concentrations of PNP equations, as well as for the velocity and pressure of NS equations, are derived. Numerical experiments are presented to support the theoretical analysis and demonstrate the method's performance in simulating electrokinetic instabilities in ionic fluids and studying their dependence on ion concentration and applied voltage.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Rooholah Abedian, Mehdi Dehghan
Summary: This paper presents a new formulation of conservative finite difference radial basis function weighted essentially non-oscillatory (WENO-RBF) schemes to solve conservation laws. Unlike previous methods, the flux function is generated directly with the conservative variables, and arbitrary monotone fluxes can be employed. Numerical simulations of several benchmark problems are conducted to demonstrate the good performance of the new scheme.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2023)
