Jalil Rashidinia

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• PhD
• Professor at Iran University of Science and Technology

The time-fractional Black-Scholes model (T-FBSM) is developed to assess price fluctuations in a correlated fractal transmission system. It is applied to price American and European call and put options on nondividend-paying stocks. This study focuses on numerically solving the T-FBSM for option pricing using a local compact integrated radial basis...

The time fractional Black-Scholes equation (TFBSE) is designed to evaluate price fluctuations within a correlated fractal transmission system. This model prices American or European put and call options on non-dividend-paying stocks. Reliable and efficient numerical techniques are essential for solving fractional differential models due to the glob...

This paper presents a novel numerical approach for approximating the solution of the model describing the infection of CD4+T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen...

This article presents a study on Fractional Anomalous Diffusion (FAD) and proposes a novel numerical algorithm for solving Caputo’s type fractional sub-diffusion equations to convert the fractional model into a set of nonlinear algebraic equations. These equations are efficiently solved using the Levenberg-Marquardt algorithm. The study provides th...

In this study, a spectral collocation method is proposed to solve a multi-term time fractional diffusion-wave equation. The solution is expanded by a series of generalized Laguerre polynomials, and then, by imposing the collocation nodes, the equation is reduced to a linear system of algebraic equations. The coefficients of the expansion can be det...

This research involves the development of the spectral collocation method based on orthogonalized Bernoulli polynomials to the solution of time-fractional convection-diffusion problems arising from groundwater pollution. The main aim is to develop the operational matrices for the fractional derivative and classical derivatives. The advantage of our...

This research involves in the development of the spectral collocation method using Bernoulli polynomials to the solution for time fractional convection-diffusion problems arising in groundwater pollution. The main aim is to develop the operational matrices as well as fractional derivative. The advantage of our approach is to orthogonalize the Berno...

This paper presents a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burger's equation, where the fractional derivative is considered in the Caputo sense. The proposed method is applied to two examples by using...

Fractional calculus (FC) is an important mathematical tool in modeling many dynamical processes. Therefore, some analytical and numerical methods have been proposed, namely, those based on symmetry and spline schemes. This paper proposed a numerical approach for finding the solution to the time-fractional modified equal-width wave (TFMEW) equation....

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a...

This paper presents a bilinear Chebyshev pseudo-spectral method to compute European and American option prices under the two-asset Black–Scholes and Heston models. We expand a function and its derivatives into their Chebyshev series, so the differentiation matrices that act on the Chebyshev coefficients are sparse and better conditioned. First, the...

In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials via the collocation me...

The primary purpose of this paper is the construction of the Green's function and Sinc approximation for a class of Caputo fractional boundary value problems (CFBVPs). By using the inverse derivative of the fractional order, we can derive the equivalent fractional order Volterra integral equations from CFBVPs, which is considered Green's function....

Regional bank branch management is the most important elements of a bank’s structure. Each regional bank branch manager (RBBM) manages a large group of branches. In this paper, we develop a bi-level structure for the evaluation of RBBMs. In the developed bi-level structure, RBBMs are positioned at the upper level, and each RBBM has a group of branc...

In this research paper, we provide sufficient conditions for the local and global existence of solution for a class of nonlinear distributed-order fractional differential equations in the time domain‎, ‎based on Schauder's and Tychonoff's fixed-point theorems‎. ‎Also‎, ‎we provide sufficient conditions for the uniqueness of the solution‎.

Two-dimensional First Boubaker polynomials (2D-FBPs) have been formulated and developed as the set of basis for the expansion of bivariate functions. These polynomials are an extension of one-dimensional (1D) FBPs that have previously been used to solve 1D nonlinear integral equations. The dual operational matrix, four operational matrices for inte...

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann-Liouville integral and u...

We devoted the numerical solution of one-dimensional Coupled Viscous Burgers’ (CVB) equations with RBF-QR method to show how it overcomes the ill-conditioning of coefficient matrix in comparison with GA and MQ RBFs for small values of shape parameters. Then we use a strategy to modify LOOCV for selecting optimal value of shape parameter. Accuracy o...

In this research study, we present an efficient method based on the generalized hat functions for solving nonlinear stochastic differential equations driven by the ‎multi-fractional‎ Gaussian noise‎. Based on the generalized hat functions, we derive a stochastic operational matrix of the integral operator with respect to the ‎variable order fractio...

In this paper, an efficient method for solving time-fractional sub-diffusion equations of distributed-order is presented‎. An ‎error bound for the new method is obtained. ‎Numerical experiments illustrate the efficiency of the proposed method‎.

A numerical framework based on fuzzy finite difference is presented for approximating fuzzy triangular solutions of fuzzy partial differential equations by considering the type of [gH-p]-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepa...

A numerical framework based on fuzzy finite difference is presented for approximating fuzzy triangular solutions of fuzzy partial differential equations by considering the type of $[gH-p]-$differentiability. The fuzzy triangle functions are expanded using full fuzzy Taylor expansion to develop a new fuzzy finite difference method. By considering th...

The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integr...

‎The main aim of this research study is to present a new and efficient numerical method based on the second kind Chebyshev wavelets for solving the general form of distributed order fractional differential equations in the time domain with the Caputo fractional derivatives‎. For the first time‎, based on the second kind Chebyshev wavelets, ‎an exac...

In this paper, we develop the numerical solution of nonlinear Klein-Gordon equation (NKGE) using the meshless methods. The finite difference scheme and the radial basis functions (RBFs) collocation methods are used to discretize time derivative and spatial derivatives, respectively. Numerical results are given to confirm the accuracy and efficiency...

‎The aim of this research is to present a new and efficient numerical method to approximate the solutions of some classes of two-dimensional nonlinear fractional integral equations using the operational matrices of two-variable shifted fractional-order Jacobi polynomials (SFOJPs)‎. ‎Discussion on the convergence analysis and error bound of the prop...

‎In this research study‎, ‎we present a new and efficient numerical method‎, ‎based on the second kind Chebyshev wavelets‎, ‎for solving the general form of distributed order time-fractional differential equations (DOFDEs) with the Caputo fractional derivatives‎. ‎Discussion on the error bound and convergence analysis for the proposed method is pre...

This paper studies a new computational method for the approximate solution of the space fractional advection–dispersion equation in sense of Caputo derivatives. In the first method, a time discretization is accomplished via the compact finite difference, while the fourth kind shifted Chebyshev polynomials are used to discretize the spatial derivati...

The aim of this paper is to present a new and efficient numerical method to approximate the solutions of two-dimensional nonlinear fractional Fredholm and Volterra integral equations‎. ‎For this aim‎, ‎the two-variable shifted fractional-order Jacobi polynomials are introduced and their operational matrices of fractional integration and product are...

We study combination of Sinc and B-Spline scaling functions to develop numerical method for the time-fractional convection-diffusion equations . Dou to exponential convergence of Sinc interpolation, the shifted Sinc functions based on double exponential transformation have been used for the spatial discretization , and the B-Spline scaling function...

In this paper, an efficient numerical method is presented to approximate the solutions of two-dimensional nonlinear fractional Volterra and Fredholm integral equations‎. We derive new operational matrices of ‎fractional-order integration and product based on two-variable shifted Jacobi polynomials. These operational matrices via shifted Jacobi coll...

Accuracy of radial basis functions (RBFs) is increased as the shape parameter decreases and produces an ill-conditioned system. To overcome such difficulty, the global stable computation with Gaussian radial basis function-QR (RBF-QR) method was introduced for a limited number of nodes. The main aim of this work is to develop the stable RBF-QR-FD m...

The time fractional Klein–Kramers model (TFKKM) is obtained by incorporating the subdiffusive mechanisms into the Klein–Kramers formalism. The TFKKM can efficiently express subdiffusion while an external force field is present in the phase space. The model describes the escape of a particle over a barrier and has a significant role in examining a v...

In this paper‎, ‎a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative‎. ‎The suggested method is based on the Müntz-Legendre wavelet approximation‎. ‎We derive a new operational vec...

The present paper is primarily aimed at obtaining the numerical solution of space fractional advection-diffusion equation including two fractional space derivatives of order. At the first stage, a difference scheme with the second-order accuracy is formulated to obtain a semi-discrete plan. Unconditional stability and convergence analysis have been...

The purpose of this research is to provide sufficient conditions for the local and global existence of solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations, based on the Schauder's and Tychonoff's fixed-point theorems. Also, we provide sufficient conditions for the uniqueness of the solutions. Moreover, we use...

We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergen...

Numerical solution of singular boundary value problems using Green’s function and Sinc-Collocation method

This study is the construction of the Green’s function and Sinc function for a class of nonhomogeneous singular boundary value problems (SBVPs). The equivalent Volterra-Fredholm integral equations can be derived from SBVPs by applying Green’s function. This can be approximated by Sinc-Collocation method. Convergence analysis is given. Our approach...

We developed three various classes of Sinc methodsSinc methods based on single exponential Single exponential (SE), double exponential Double-exponential (DE) and single-double exponential transformation Exponential transformation (SE-DE), to approximate the solution of linear and non-linear fractional differential equations Fractional differential...

We construct the two unique piecewise C3-splines of degree six and eight to approximate the solution of fractional integro-differential equations. Existence and uniqueness of the solution of second order fractional integro-differential equation has been proved. Convergence analysis of the presented methods have been discussed. Illustrative examples...

When the number of nodes increases more than thousands, the arising system of global radial basis functions (RBFs) method becomes dense and ill-conditioned. To solve this difficulty, local RBFs generated finite difference method (RBF-FD) were introduced. RBF-FD method is based on local stencil nodes and so it has a sparsity system. The main goal in...

‎In the present paper‎, ‎an attempt was made to develop a numerical method for solving a general form of two-dimensional nonlinear fractional integro-differential equations by using operational matrices‎. ‎Our approach is based on the hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials‎. ‎Error bound and co...

In this paper we develop a non polynomial cubic spline function which we called "TS spline", based on trigonometric functions. The convergence analysis of this spline is investigated in details. The definition of B-spline basis function for TS spline is extended and "TS B-spline" is introduced. This paper attempts to develop collocation method base...

In this paper, two classes of methods are developed for the solution of two-dimensional elliptic partial differential equations. We have used tension spline function approximation in both x and y spatial directions and a new scheme of order O(h4+k4) has been obtained. The convergence analysis of the methods has been carried out. Numerical examples...

‎In this paper‎, ‎we present a numerical method for solving the generalized two-dimensional nonlinear fractional integro-differential equations (2D-NFIDEs)‎. ‎Presented collocation method is based on the operational matrices of shifted Jacobi polynomials‎. ‎Convergence analysis and error bound in Jacobi-weighted Sobolev space is discussed‎. ‎Also‎,...

In our knowledge so far, the Non-polynomial Spline functions (NPS) have not been yet applied for approximating the integral equations. In this article, we want to use such functions for obtaining numerical solutions of Fredholm integral equations of the second kind. In our approach, the coefficients of the non polynomial spline are obtained by solv...

In this study, we approximate the solution of the fractional diffusion equations based on Gaussian radial basis function (GRBF). Our approach is based on the Caputo fractional derivative and the combination of GRBF and Sinc function, here the GRBF direct and GRBF-QR methods are developed. The Sinc quadrature rule combined with double exponential tr...

In this article, we consider the discretized classical Susceptible-Infected-Recovered (SIR) forced epidemic model to investigate the consequences of the introduction of different transmission rates and the effect of a constant vaccination strategy, providing new numerical and topological insights into the complex dynamics of recurrent diseases. Sta...

The well-known Hodgkin Huxley (HH) model employs the electrical circuit theory to describe the membrane potential of a cell in a giant squid axon. Hugh R. Wilson introduced human and mammalian neocortical neurons (HMNN) model which limits the potential nonlinearities to cubic polynomials while retains the same basic biophysics. This model has been...

In this paper, an efficient numerical scheme is presented in order to solve the time Caputo fractional reaction-diffusion equation. This method is based on the Legendre tau spectral method combined with the generalized shifted Legendre operational matrix, and our main purpose is to reduce the given equation to the solution of a system of algebraic...

A collocation method based on B-spline is developed for solving Convection-Reaction-Diffusion equation subjected to Dirichlet's boundary conditions. The method comprises an explicit finite difference associated with extended cubic B-spline collocation method. We analyze the convergence and stability of the presented method. The proposed method is a...

A new and efficient method is presented for solving three-dimensional Volterra–Fredholm integral equations of the second kind (3D-VFIEK2), first kind (3D-VFIEK1) and even singular type of these equations. Here, we discuss three-variable Bernstein polynomials and their properties. This method has several advantages in reducing computational burden w...

As we know the approximation solution of seventh order two points boundary value problems based on B-spline of degree eight has only 0(h ² ) accuracy and this approximation is non-optimal. In this work, we obtain an optimal spline collocation method for solving the general nonlinear seventh order two points boundary value problems. The O(h ⁸ ) conv...

In this paper, an efficient numerical method is proposed for solving a nonlinear system of Volterra–Fredholm integral equations of the second kind, using two-dimensional radial basis functions (RBFs). This method is based on interpolation by radial basis functions including multiquadric (MQ), using Legendre–Gauss–Radau nodes and weights. The propos...

The Sinc-collocation method using the single and double exponential transformations has been established to solve the second kind FIEs linear systems. The convergence analyses of the approach have been discussed and the exponential convergence rates have been obtained for both of the transformations. The method has been tested by examples and compa...

One of the most widely studied biological systems with excitable behavior is neural communication by nerve cells via electrical signaling. The Fitzhugh–Nagumo equation is a simplification of the Hodgin–Huxley model (Hodgin and Huxley, 1952) [24] for the membrane potential of a nerve axon. In this paper we developed a three time-level implicit metho...

Local RBF collocation method based on multiquadric and inverse multiquadric basis has been presented for solving time-dependent convection-diffusion equation. Our purpose is to reduce the computational cost by providing matrix form of local collocation method. The approach is based on square stencils with sizes of 3× 3 and 5× 5 around each interior...

In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to solve the nonlinear Volterra integral equation. When the iterations of the Quasilinear technique employed in nonlinear integral equation we obtain a linear integral equation then by using the Non-polynomial spline functions and collocation method the solu...

In this paper, an efficient numerical method is proposed for solving a class of two-dimensional nonlinear Volterra–Fredholm integral equations of the second kind based on two-dimensional radial basis functions (RBFs). This method is based on a hybrid of radial basis functions including the multiquadric and the Gaussian constructed on Legendre–Gauss...

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection-diffusion-reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avo...

We provide the numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions. Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc-collocation method is employed in space. A weakly singula...

Numerical schemes have been developed for solutions of systems of nonlinear mixed Volterra-Fredholm integral equations of the second kind based on the First Boubaker polynomials (FBPs). The classical operational matrices are derived. The unknown has been approximated by FBPs and the Newton-Cotes points were applied as the collocations points. Error...

In this work, a stable numerical scheme based on modified bi-cubic B-spline collocation method is developed for the valuation of Asian options. The option prices are governed with Black–Scholes equation. We use the \(\theta \)-method for temporal discretization and the modified bi-cubic B-spline collocation approach in spatial direction. This appro...

A collocation method is proposed to obtain an approximate solution of a system of multi pantograph type delay differential equations with variable coefficients subject to the initial conditions. The general approach is that, first of all the solution of the system has been expanded according to First Boubaker polynomials (FBPs) basis. Then, by empl...

In this paper, two classes of methods are developed for the solution of two space dimensional wave equations with a nonlinear source term. We have used non-polynomial cubic spline function approximations in both space directions. The methods involve some parameters, by suitable choices of the parameters, a new high accuracy three time level scheme...

In this work we extend the definition of nonic polynomial spline to non-polynomial spline function which depends on arbitrary parameter k. We derived and discussed the formulation and spline relations. Using such non-polynomial spline relations we developed the classes of numerical methods, for the solution of the problem in calculus of variations....

In the present paper, we obtain analytical-approximate solution of Abel Volterra integral equations by using Optimal Homotopy Asymptotic method (OHAM).This approach has been compared with some other powerful and efficient methods such as He's homotopy perturbation method (HPM) and Adomian decomposition method (ADM).This method uses simple computati...

In this paper, we study the mixed Volterra-Fredholm integral equations of the second kind by means of optimal homotopy asymptotic method (OHAM) and Homotopy Perturbation method (HPM).Our approach is independent of time and contains simple computations with quite acceptable approximate solutions in which approximate solutions obtained by these metho...

The Class of various order numerical methods based on non-polynomial spline have been developed for the solution of linear and non-linear sixth-order boundary value problems. We developed non-polynomial spline which contains a parameter ρ, act as the frequency of the trigonometric part of the spline function, when such parameter tends to zero the d...

In this paper, the Burgers’ equation which is two-dimensional in space, time dependent parabolic differential equation was solved by b-spline collocation algorithms for solving two-dimensional parabolic partial differential equation. At first b-spline interpolation is introduced moreover, the numerical solution is represented as a bi-variate piecew...

In this work, the numerical approximation of Convection-Reaction-Diffusion equation is investigated using the method based on tension spline function and finite difference approximation. For nonlinear term, nonstandard finite difference method by nonlocal approximation is utilized. We describe the mathematical formulation procedure in detail and al...

In this paper, a scheme based on Sinc and radial basis functions (RBF) is developed to approximate the solution of two-dimensional Rayleigh-Stokes problem for a heated generalized second-grade fluid with fractional derivatives‎. We use RBF and Sinc functions as basis functions to approximate spatial and time coordinates of the unknown function‎, re...

In this paper, the forced vibrations of the fractional viscoelastic beam with the Kelvin-Voigt fractional order constitutive relationship is studied. The equation of motion is derived from Newton’s second law and the Galerkin method is used to discretize the equation of motion in to a set of linear ordinary differential equations. For solving the d...

We reconstruct the variational iteration method that we call, parametric iteration method (PIM) to solve second kind weakly singular Volterra integral equations of Abel type. Convergence analysis of the proposed method is studied. The solution process is illustrated by some tested examples. Comparisons are made between PIM and exact solution. The r...

A collocation method based on modified cubic B-spline functions has been developed for the valuation of European, American and barrier options of a single asset. The new approach contains discretization of temporal derivative using finite difference approximation of and approximating the option price with the modified B-spline functions. Stability...

We present a collocation method based on redefined extended cubic B-spline basis functions to solve the second-order one-dimensional hyperbolic telegraph equation. Extended cubic B-spline is an extension of cubic B-spline consisting of a parameter. The convergence and Stability of the method are proved and shown that it is unconditionally stable an...

The main aim of this paper is to apply the polynomial wavelets for the numerical solution of nonlinear Klein-Gordon equation. Polynomial scaling and wavelet functions are rarely used in the contexts of numerical computation. A numerical technique for the solution of nonlinear Klein-Gordon equation is presented. Our approach consists of finite diffe...

The current paper proposes a technique for the numerical solution of generalized Kuramoto–Sivashinsky equation. The method is based on finite difference formula combined with the collocation method, which uses the polynomial scaling functions (PSF). Mentioned functions and their properties are employed to derive a general procedure for forming the...

We reconstruct the variational iteration method that we call, parametric iteration method (PIM) to solve second kind weakly singular Volterra integral equations of Abel type. Convergence analysis of the proposed method is studied. The solution process is illustrated by some tested examples. Comparisons are made between PIM and exact solution. The r...

A study of Sinc-Galerkin method based on double exponential transformation for solving a class of nonlinear weakly singular two point boundary value problems with nonhomogeneous boundary conditions is given. The properties of the Sinc-Galerkin approach are utilized to reduce the computation of nonlinear problem to nonlinear system of equations with...

We study the performance of the sinc methods based on single and double exponential transformations, for numerical solution of Lane–Emden type equations. the sinc-Galerkin and collocation methods based on single and double exponential transformations developed and for comparison, these methods are applied to three examples. By considering the maxim...

In this article, we develop the Sinc-Galerkin method based on double exponential transformation for solving a class of weakly singu- lar nonlinear two-point boundary value problems with nonhomogeneous boundary conditions. Also several examples are solved to show the ac- curacy efficiency of the presented method. We compare the obtained numerical re...