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Introduction
Professor Jafar Biazar teaches and researches at the department of applied mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht.
His research field is numerical analysis and numerical approaches to solve the applied equations.
He has received his undergraduate degree in numerical analysis from Brunel University in England.
Professor Biazar commenced his higher education from Tehran University (Education), continued the research at Kharazmi University, and the University of Dalhousie Canada. In 2002 received his Ph.D., from Kharazmi University in Iran.
Additional affiliations
Education
September 1996 - March 1997
September 1976 - March 2001
Kharazmi University
Field of study
- Numerical Analysis
Publications
Publications (318)
This paper applies a new modification of the Homotopy Perturbation Method that is called Rational Homotopy Perturbation Method (RHPM) to obtain an analytic approximation of stiff systems of ordinary differential equations. The procedure of the method will be explained briefly and some examples are presented to illustrate the method. The results of...
In the current study, a new numerical algorithm is presented to solve the fractional-order differential model of HIV-1 infection of CD4 + T-cells with the effect of drug therapy. Utilizing quasi-hat functions (QHFs) and their properties, the main problem is transformed into a number of trivariate polynomial equations. Ultimately, an example is prov...
In the current study, a one-step numerical algorithm is presented to solve strongly non-linear full fractional duffing equations. A new fractional-order operational matrix of integration via quasi-hat functions (QHFs) is introduced. Utilizing the operational matrices of QHFs, the main problem will be transformed into a number of univariate polynomi...
In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operation...
In this article, two numerical approaches are presented to solve a system of three
fractional differential equations that express the pollution of lakes. In our recent
study, a new class of hat functions, called quasi-hat functions (QHFs), are constructed. The proposed approaches utilize modified hat functions (MHFs) and
quasi-hat functions (QHFs)....
In this article, two numerical approaches are presented to solve a system of three fractional differential equations that express the pollution of lakes. In our recent study, a new class of hat functions, called quasi-hat functions (QHFs), are constructed. The proposed approaches utilize modified hat functions (MHFs) and quasi-hat functions (QHFs)....
In this study, a new numerical algorithm is proposed for solving a class of non-linear fractional Volterra integral equations of the second kind based on our newly constructed hat functions. New functions that are called cubic hat functions (CHFs) are applied and an operational matrix of fractional order integration of these functions. In a new num...
In the current study, a new numerical algorithm is presented to solve a class of non-linear fractional integral-differential equations with weakly singular kernels.
Cubic hat-functions (CHFs) and their properties will be introduced for the first time.
A new fractional-order operational matrix of integration via a CHFs is presented.
Utilizing the op...
In this study, a new numerical algorithm is proposed for solving a class of non-linear
fractional Volterra integral equations of the second kind based on our newly constructed hat functions.
New functions that are called cubic hat functions (CHFs) are applied and an operational matrix of
fractional order integration of these functions. In a new num...
This paper investigates the dynamics of a fractional-order model of the human liver. The proposed model is examined via quasi-hat functions (QHFs). Utilizing a method that incorporates the operational matrices of QHFs is used to reduce the problem to several systems of two equations with two unknowns. Finally, an illustrative example is provided to...
In this paper, perturbed Galerkin method is proposed to find numerical solution of an integro-differential equations using fourth kind shifted Chebyshev polynomials as basis functions which transform the integro-differential equation into a system of linear equations. The systems of linear equations are then solved to obtain the approximate solutio...
In this research, a new approach based on an alteration of usage of optimal homotopy asymptotic method (OHAM), called multistage optimal homotopy asymptotic method (MOHAM) is utilized to derive an approximate solution to system of Nonlinear Volterra integral equations of the first kind (SNVIEFK). One example is provided to show the efficiency and a...
In this article, a comparative study between optimal homotopy asymptotic method and multistage optimal homotopy asymptotic method is presented. These methods will be applied to obtain an approximate solution to the seventh-order Sawada-Kotera Ito equation. The results of optimal homotopy asymptotic method are compared with those of multistage optim...
In this paper, fractional Sturm–Liouville problems of high-order are studied. A simple and efficient approach
is presented to determine more eigenvalues and eigenfunctions than other approaches. Existence and uniqueness of
solutions of a fractional high-order differential equation with initial conditions is addressed as well as the convergence
of t...
As the main contribution of this article, we establish an option on a credit spread under a stochastic interest rate. The intense volatilities in financial markets cause interest rates to change greatly; thus, we consider a jump term in addition to a diffusion term in our interest rate model. However, this decision leads us to a partial integral di...
Radial Basis Function (RBF) is a real valued function whose value rests only on the distance from some other points called a center, so that a linear combination of radial basis functions are typically used to approximate given functions or differential equations. Radial Basis Function (RBF) approximation has the ability to give an accurate approxi...
In this paper. the optimal homotopy asymptotic method (OHAM) and multistage optimal
homotopy asymptotic method (MOHAM) is applied to obtain an analytic approximate solution
to a time-fractional Klein-Fock-Gordon (FKFG) equation. The FKFG equation plays an important
role in characterizing the relativistic electrons. The MOHAM relies on OHAM to obtai...
We use hat functions to solve a variety of integral equations
and find a new operational matrix for modification of hat functions.
In this paper, a fractional-ordered prey and predator population model is introduced and applied to obtain an approximate solution with help of optimal homotopy asymptotic method (OHAM). Some plots for populations of the prey and the predator versus time are presented to show the efficiency and the accuracy of the method and confirm that the method...
The main purpose of this work is to present an efficient approximate approach for solving linear systems of fractional integro-differential equations based on a new application of Taylor expansion. Using the mth-order Taylor polynomial for unknown functions and employing integration method the given system of fractional integro-differential equatio...
This paper is devoted to applying the sixth-order compact finite difference approach to the Helmholtz equation. Instead of using matrix inversion, a discrete sinusoidal transform is used as a quick solver to solve the discretized system resulted from the compact finite difference method. Through this way, the computational costs of the method with...
Nowadays, as the financial markets grow larger, to compare with the past, financial investments and their modeling become more complicated. One of the difficulties in these financial modeling is selecting an appropriate model for interest rate. The main reason is that, in real market some data goes under sudden changes in some points so, they may n...
This paper attempts to carry out a study on a specific type of time-fractional differential equation called Benjamin-Bona-Mahony-Burger (BBM-Burger). This equation describes the mathematical model of unidirectional transmission of low-amplitude long waves through frequency-dependent dispersive media. To go ahead the research, optimal homotopy asymp...
In this paper, we present a computational method for solving systems
of Volterra and Fredholm integral equations which is hybrid approach,
based on the third order Chebyshev polynomials and blockpulse
functions which we will refer to as (HBV), for short. By using
the HBV method and their operational matrix of integration, such systems
can be reduce...
determination of eigenvalues and eigenfunctions of a High-order Sturm-Liouville problem (HSLP) is considered. To this end, the Differential Transformation Method (DTM) is applied which is an efficient technique for solving differential equations. The results of the proposed approach are compared with those of some well-known methods reported in the...
In this note we extend the Adomian decomposition method to solve two concrete
problems, the population problem and a problem from polymer rheology.
The purpose of this research is to provide an effective numerical method for solving linear Volterra integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences leads to an integral equation. The proposed approach is based on the method of moments (Galerkin-Ritz) using orthonormal Bernstein pol...
The purpose of this research is to provide an effective numerical method for solving linear Volterra integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences lead into an integral equation. The proposed approach is based on the method of moments (Galerkin- Ritz) using orthonormal Bernstein p...
In this study, the asymptotic Adomian decomposition method (AADM) is implemented to solve fractional order Riccati differential equations. Product integration method is used to solve the singular integrals, resulted from fractional derivative. Some fractional order Riccati differential equations are presented as examples to illustrate the ability a...
The main purpose of this work is to present an efficient approximate approach for solving linear systems of fractional integro-differential equations based on a new application of Taylor expansion. Using the mth-order Taylor polynomial for unknown functions and employing integration method the given system of fractional integro-differential equatio...
This paper deals with the solution of a class of Volterra integral equations in the sense of the conformable fractional derivative. For this goal, the well-organized Neumann method is developed and some theorems related to existence, uniqueness, and sufficient condition of convergence are presented. Some illustrative examples are provided to demons...
In this paper, optimal homotopy asymptotic method (OHAM) and multistage optimal homotopy asymptotic (MOHAM) method are applied to find an approximate solution to Abel's integral equation, that is in fact a weakly singular Volterra integral equation. To illustrate these approaches one example is presented. The results confirm the efficiency and abil...
In this paper, the differential transform method is used for solving a system of fractional differential equations, which is resulted from the mathematical modeling of the diffusion of pollution in three connected lakes. The proposed approach will be applied to solve three systems with different pollution sources, i.e. constant pollution, a time-de...
In this work, a new explicit second-order �nite di�Terence scheme is proposed for
discretizing and also solving parabolic equations, peculiarly heat equation. Difference equations
are determined via a discretization approach which approximates spatial partial derivatives
together and simultaneously. Fourier stability analysis exhibits that the stab...
The purpose of this research is to provide an effective numerical method for solving linear Volterra-Fredholm integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences lead into an integral equation. The proposed approach is based on the collocation method using orthonormal Bernstein polynomi...
The one dimension Legendre Wavelet is a numerical method to solve one dimension
equation. In this paper Black-Scholes equation (B-S), that have applied via single asset American option
and Heston Cox- Ingersoll- Ross equation (HCIR), as partial differential equations have been studied
in the form of a stochastic model at first. The Black-Scholes an...
Fractional differential equations are often seeming perplexing to solve. Therefore, finding comprehensive methods for solving them sounds
of high importance. In this paper, a general method for solving second order fractional differential equations has been presented based on
conformable fractional derivative. This method realizes on determining a...
Absract The purpose of this research is to provide an effective numerical method for solving linear Volterra-Fredholm integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences lead into an integral equation. The proposed approach is based on the collocation method using orthonormal Bernstein...
This paper is aimed to find an approximate solution to linear Volterra integral equations of the first kind, by applying optimal homotopy asymptotic method (OHAM), which is powerful and efficient method and a new approach called multistage optimal homotopy asymptotic method (MOHAM). MOHAM is just a clear alteration of the standard optimal homotopy...
The present paper deals with the time-fractional unstable nonlinear Schrödinger equation
and the time fractional modified unstable nonlinear Schrödinger equation in the conformable
context. For this aim, the modified Kudryashov method and the sine–Gordon expansion
approach have been applied to retrieve a series of exact solutions for the previously...
In this research, an effective combination of the result of fractional partial differential equations
(PDEs) is envisioned. Decomposition coupled with modified integral transform (Elzaki transform) is applied to
solve partial differential equations, of a fractional order. It is observed that the proposed technique is extremely
useful. The effects o...
In this paper, the fractional-order differential model of HIV-1 infection of CD4 + T-cells with the effect of drug therapy has been introduced. ere are three components: uninfected CD4 + T-cells, x, infected CD4 + T-cells, y, and density of virions in plasma, z. e aim is to gain numerical solution of this fractional-order HIV-1 model by Laplace Ado...
In this paper, optimal homotopy asymptotic method (OHAM) and its implementation on subinterval, called multistage optimal homotopy asymptotic method (MOHAM), are presented for solving linear and nonlinear systems of Volterra integral equations of the second kind. To illustrate these approaches two examples are presented. The results confirm the eff...
In this paper, a modification of the finite integration method (FIM) is combined with the radial basis function (RBF) method to solve a time-fractional convection-diffusion equation with variable coefficients. The FIM transforms partial differential equations into integral equations and this creates some constants of integration. Unlike the usual F...
This paper is aimed to find an approximate solution to a weakly singular Volterra integral equation of the second kind, by applying a new approach called multistage optimal homotopy asymptotic method (MOHAM). MOHAM is just a clear alteration of the standard optimal homotopy asymptotic method (OHAM), in which in order to find a more accurate approxi...
This investigation intends to provide a new application of Tay-lor expansion approach for solving first kind Fredholm integral equations. The approach is based on employing the νth-degree Taylor polynomial of unknown function at an arbitrary point and integration method such that the first kind Fredholm integral equation is converted into a linear...
In this paper, a hybrid approach consisting of the third-order Chebyshev polynomials and block-pulse functions is used for solving systems of Volterra integral differential equations. Applying this approach transforms the system of integral differential equations into a system of algebraic equations. The existence and uniqueness of the solution, fo...
In this paper, optimal homotopy asymptotic and multistage optimal homotopy asymptotic methods are applied to find an approximate solution to Volterra integral equations. To illustrate these approaches two example are presented. The results confirm the efficiency and ability of these methods for such an equations. The results will be compared, to fi...
In this study, an operational approach, based on the shifted Jacobi polynomials, ispresentedtosolveaclassofweaklysingularfractionalintegro-differentialequations.Thefractional derivative operators are considered as the Caputo sense. In addition to findingthe operational matrices of integration and product, a new operational matrix is derivedto be ap...
The present paper deals with the time-fractional unstable nonlinear Schrödinger equation and the time fractional modified unstable nonlinear Schrödinger equation in the conformable context. For this aim, the modified Kudryashov method and the sine–Gordon expansion approach have been applied to retrieve a series of exact solutions for the previously...
Abstract This paper aims to obtain an approximate solution for fractional order Riccati differential equations (FRDEs). FRDEs are equivalent to nonlinear Volterra integral equations of the second kind. In order to solve nonlinear Volterra integral equations of the second kind, a class of Runge–Kutta methods has been applied. Runge–Kutta methods hav...
A simple presentation to learn about Homotopy Perturbation Method.
A wide range of fractional differential equations in applied sciences can be solved by integral transformations. In the present work, first some new theorems related to the Mellin transform and the conformable fractional operator are established, and then a few conformable fractional equations such as wave and heat equations are solved through the...
Our concern in the current article is studying conformable fractional Volterra integral equations (CFVIEs) of the second kind. For this, a method of calculation called the optimal homotopy asymp-totic method is developed to deal with this class of fractional integral equations. The results reveal the efficiency and accuracy of the method for Volter...
In this paper, a modification of finite integration method (FIM) is combined with the radial basis function (RBF) method to solve a time-fractional convection-diffusion equation with variable coefficients. The FIM transforms partial differential equations into integral equations and this creates some constants of integration. Unlike the usual FIM,...
In this paper, (2+1) dimensional parabolic equations are studied by compact Alternating Direction Implicit (ADI) method. In this method, second order spatial derivatives are approximated by fourth order compact finite differences. ADI scheme is applied to split the time step into two fractional steps. Stability of the method is addressed. Nu...
Fractional differential equations are often seeming perplexing to solve. Therefore, finding comprehensive methods for solving them sounds of high importance. In this paper, a general method for solving second order fractional differential equations has been presented based on conformable fractional derivative. This method realizes on determining a...
Obtaining analytical or numerical solution of fractional differential equations is one of the troublesome and challenging issues among mathematicians and engineers, specifically in recent years. The purpose of this paper is to solve linear and nonlinear fractional differential equations such as first order linear fractional equation, Bernoulli, and...
Solving fractional differential equations have a prominent function in different science such as physics and engineering. Therefore, are different definitions of the fractional derivative presented in recent years. The aim of the current paper is to solve the fractional differential equation by a semi-analytical method based on conformable fraction...
Analytic behavior of fractional differential equations are often seems confusing. Thus, finding comprehensive methods for solving those sounds of high importance. In the present study, Feng’s first integral method which is based on the ring theory of commutative algebra, is developed for analytic treatment fractional differential equations based on...
The radial basis functions (RBFs) depend on an auxiliary parameter, called the shape parameter. Great theoretical and numerical efforts have been made to find the relationship between the accuracy of the RBF-approximations and the value of the shape parameter. In many cases, the numerical approaches are based on minimization of an estimation of the...
Compact finite difference scheme is applied for a partial integro-differential equation with a weakly singular kernel. The product trapezoidal method is applied for discretization of the integral term. The order of accuracy in space and time is O(h⁴ , k2-a ) , where 0 <α<1. Stability and convergence in 2 L norm are discussed through energy method....
The traditional Adomian decomposition method
(ADM) usually is divergent to solve nonlinear systems of
ordinary differential equations of Emden–Fowler type. To
cover this deficiency, an effective modification of ADM is
formally adopted in the current study. The method overcomes
the singularity at the origin and provides the solution
of the problems...
In this paper, the Volterra’s population model is studied for population growth of a species within a
closed system. Modified Adomian decomposition method (MADM) in conjunction with Pade technique
is formally proposed to obtain an analytic approximation for the solution of the model, which is a
nonlinear intgro-differential equation. The results of...
This paper is aimed to develop two well-known nonlinear ordinary differential equations, Bernoulli and Riccati equations to fractional form. General solution to fractional differential equations are detected, based on conformable fractional derivative. For each equation, numerical examples are presented to illustrate the proposed approach.
This paper applies a new modification of the Homotopy Perturbation Method that is called Rational Homotopy Perturbation Method (RHPM) to obtain an analytic approximation of stiff systems of ordinary differential equations
In this paper, a new method is developed for approximating solution to the reaction-convection-diffusion equation, in which reaction rate and diffusion coefficient are small parameters. A compact finite difference scheme (CFD) is applied for discretizing spatial derivatives of linear reaction-convection-diffusion equation, which leads to a linear s...
This paper aimed to develop two well-known nonlinear ordinary different equations, Bernoulli and Riccati equations to fractional form. General solution to fractional differential equations are detected, based on conformable fractional derivative. For each equation, numerical examples are presented to illustrate the proposed approach.
In this work, the multi-step differential transformation method (MSDTM) is applied to approximate a solution of the hyperchaotic Rossler system. MSDTM is adapted from the differential transformation method (DTM). In this method, DTM is implemented in each subinterval. Results are compared with a fourth-order Runge Kutta method and a standard DTM. T...
Traditional Adomian decomposition method (ADM) usually fails to solve singular initial value problems of Emden-Fowler type. To overcome this shortcoming, a new and effective modification of ADM that only requires calculation of the first Adomian polynomial is formally proposed in the present paper. Three singular initial value problems of Emden-Fow...
In this paper, we propose an efficient implementation of the Chebyshev Galerkin method for first order Volterra and Fredholm integro-differential equations of the second kind. Some numerical examples are presented to show the accuracy of the method.
In radial basis function approximation, the shape parameter can be variable. The values of the variable shape parameter strategies are selected from an interval which is usually determined by trial and error. As yet there is not any algorithm for determining an appropriate interval, although there are some recipes for optimal values. In this paper,...
In this manuscript the implicit Runge-Kutta (IRK) method, with three slopes of order five has been explained, and is applied to Van der pol stiff differential equation. Truncation error, of order five, has been estimated. Stability of the
procedure for the Van der pol equation, is analyzed by the Lyapunov method. To illustrate the structure of the...
Meshless method of line is a powerful device to solve time-dependent partial differential equations. In integrating step, choosing a suitable set of points, such as adaptive nodes in spatial domain, can be useful, although in some cases this can cause ill-conditioning. In this paper, to produce smooth adaptive points in each step of the method, two...
Finite Difference Method and Radial Basis Functions are applied to solve partial integro-differential equations with a weakly singular kernel. The product trapezoidal method is used to compute singular integrals that appear in the discretization process. Different RBFs are implemented and satisfactory results are shown the ability and the usefulnes...
In this paper a numerical method, based on collocation method and radial basis functions (RBF) is proposed for solving integral equations with a weakly singular kernel. Integrals appeared in the procedure of the solution are approximated by adaptive Lobatto quadrature rule. Illustrative examples are included to demonstrate the validity and applicab...
We combine the Adomian decomposition method (ADM) and Adomian's asymptotic decomposition method (AADM