# Ahmad GolbabaiIran University of Science and Technology · School of Mathematics

Ahmad Golbabai

PhD. applied math. London England

## About

115

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2,282

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Citations since 2017

## Publications

Publications (115)

This paper introduces a direct method derived from the global radial basis function (RBF) interpolation over arbitrary collocation nodes occurring in variational problems involving functionals that depend on functions of a number of independent variables. This technique parameterizes solutions with an arbitrary RBF and transforms the two-dimensiona...

The cable equation is one useful description for modeling phenomena such as neuronal dynamics and electrophysiology. The time-fractional cable model (TFCM) generalizes the classical cable equation by considering the anomalous diffusion that occurs in the ionic motion present for example in the neuronal system. This paper proposes a novel meshless n...

This paper addresses the solution of the Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid using fractional derivatives and the radial basis function-generated finite difference (RBF-FD) method. The time discretization is accomplished via the finite difference approach, while the spatial derivative terms are discretized using the...

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...

The fractional reactiondiffusion equation has an important physical and theoretical meaning, but its analytical solution poses considerable problems. This paper develops an efficient numerical process, the local radial basis function generated by finite difference (named LRBF-FD) method, for finding the approximation solution of the time-fractional...

This paper develops the solution of the two-dimensional time fractional evolution model using finite difference scheme derived from radial basis function (RBF-FD) method. In this discretization process, a finite difference formula is implemented to discrete the temporal variable, while the local RBF-FD formulation is utilized to approximate the spa...

In this paper, a projection-based recurrent neural network is proposed to solve convex quadratic bilevel programming problems (CQBPP). The Karush–Kuhn–Tucker optimal conditions (KKT) of the lower level problem are used to obtain identical one-level optimization problem. A projected dynamical system which its equilibrium point coincides with the glo...

The fractal mobile/immobile model of the solute transport is based on the assumption that the waiting times in the immobile region follow a power-law, and this leads to the application of fractional time derivatives. The model covers a wide family of systems that include heat diffusion and ocean acoustic propagation. This paper develops an efficien...

The mathematical modeling in trade and finance issues is the key purpose in the computation of the value and considering option during preferences in contract. This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. Due to the outstanding memory effect pre...

The price variation of the correlated fractal transmission system is used to deduce the fractional Black–Scholes model that has an \(\alpha \)-order time fractional derivative. The fractional Black–Scholes model is employed to price American or European call and put options on a stock paying on a non-dividend basis. Upon encountering fractional dif...

The nonlinear modified anomalous sub-diffusion model characterizes processes that become less anomalous as time progresses by including a second fractional time derivative acting on the term of diffusion. This paper introduces a radial basis function-generated finite difference (RBF-FD) method for solving the governing problem. The Grünwald–Letniko...

A nonlinear wave phenomenon is one of the significant fields of scientific research, which a lot of researchers in the past have deliberated about mathematical models clarifying the treatment. The nonlinear Korteweg-de Vries-Benjamin-Bona-Mahony-Burgers (KdV-BBM-B) model plays an essential role in numerous subjects of engineering and science. This...

Evolution equations containing fractional derivatives can offer efficient mathematical models for determination of anomalous diffusion and transport dynamics in multi-faceted systems that cannot be precisely modeled by using normal integer order equations. In recent times, researches have found out that lots of physical processes illustrate fractio...

In this study, we propose an efficient and accurate numerical technique that is called the rational Chebyshev collocation (RCC) method to solve the two dimensional flow of a viscous fluid in the vicinity of a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow, are reduced to a third-order ordinary differential equat...

Ill-conditioning is drawback of global radial basis functions (RBFs) method. Local RBFs method has overcome this difficulty and has become popular as an alternative methodology. In this paper, we analyze local RBFs method to solve singularly perturbed linear convection–diffusion problems. In some articles, it is investigated numerically but our aim...

Ill-conditioning problem of Global Radial Basis Functions (GRBFs) is a fundamental limitation for approximation of differential equation using this method. Most recently, some researchers have applied Local Radial Basis Functions (LRBFs) to approximate singularly perturbed convention diffusion problems. In these kinds of problems, mostly appear bou...

In this paper, a new local meshless approach based on radial basis functions (RBFs) is presented to price the options under the Black–Scholes model. The global RBF approximations derived from the conventional global collocation method usually lead to ill-conditioned matrices. Employing the idea of local approximants of the finite difference (FD) me...

In this paper, a feedback neural network model is proposed to compute the solution of the mathematical programs with equilibrium constraints (MPEC). The MPEC problem is altered into an identical one-level non-smooth optimization problem, then a sequential dynamic scheme that progressively approximates the non-smooth problem is presented. Besides as...

In this paper, we develop a new local meshless approach based on radial basis functions (RBFs) to solve the Black–Scholes equation. The global RBF approximations derived from conventional global collocation method usually lead to ill-conditioned matrices. The new scheme employs the idea of the finite difference method to localize them. It removes t...

In this paper, two-dimensional Schrödinger equations are solved by differential quadrature method. Key point in this method is the determination of the weight coefficients for approximation of spatial derivatives. Multiquadric (MQ) radial basis function is applied as test functions to compute these weight coefficients. Unlike traditional DQ methods...

The paper considers the ill-conditioning issue of the systems induced from RBF method for solving optimal control problems. One important aspects of RBF methods is that while the method has good accuracy its related matrix has high condition number, which is something strange. In this paper authors employ effective condition number idea and use rel...

In this note, the solution of nonlinear integral equations was discussed using radial basis functions (RBFs) method. This method will represent the solution of nonlinear integral equation by interpolating the RBFs based on Legendre-Gauss-Lobatto (LGL) nodes and weights. Zeros of the shifted Legendre polynomials are used as the collocation points. T...

The main contribution of the current paper is to propose a new effective numerical method for solving the first-order linear matrix differential equations. Properties of the Legendre basis operational matrix of integration together with a collocation method are applied to reduce the problem to a coupled linear matrix equations. Afterwards, an itera...

Stability, convergence and application of radial basis function finite difference (RBF-FD) scheme is studied for solving the reaction-diffusion equations (RDEs). We show that the explicit RBF-FD method is stable, and stability condition depends on the shape parameter of related radial basis function.The generalized multiquadric (GMQ) is applied as...

In this paper, a technique generally known as meshless numerical scheme for solving fractional differential equations is considered. We approximate the exact solution by use of Radial Basis Function (RBF) collocation method. This technique plays an important role to reduce a fractional differential equation to a system of equations.The numerical re...

In this paper we consider a Dynamic investment model. In the model, firm's objective is maximizaing discounted sum of profits over an interval of time. The model assumes that firm's capital in time t increases with investment and decreases with depreciation rate that can be expressed by means of differential equation.
We propose a direct method fo...

The current paper contributes a novel framework for solving a class of linear matrix differential equations. To do so, the operational matrix of the derivative based on the shifted Bernoulli polynomials together with the collocation method are exploited to reduce the main problem to system of linear matrix equations. An error estimation of presente...

The present study is an attempt to investigate some features of Radial Basis Functions (RBFs) approximation methods related to variational problems. Thereby authors applied some properties of RBFs to develop a direct method which reduces constrained variational problem to a static optimization problem. To assess the applicability and effectiveness...

The current paper contributes a novel framework for solving a class of linear matrix differential equations. To do so, the operational matrix of the derivative based on the shifted Bernoulli polynomials together with the collocation method are exploited to reduce the main problem to system of linear matrix equations. An error estimation of presente...

This paper developes a direct method for solving variational problems via a set of Radial Basis Functions (RBFs) . Operational matrices of differentiation, the product of two RBF vectors and some other formulas are derived and are utilized to propose a method which essentially reduces a variational problem to the linear system of algebraic equation...

The current paper deals with elaborating a novel framework for solving a class of linear matrix differential equations. To this end, the operational matrices of integration and the product based on the shifted Bernoulli polynomials are presented and a general procedure for forming this matrices is given. The properties of this matrices are exploite...

One of the most popular meshless methods is constructed by radial kernels as basis called radial basis function method. It has a unique feature which affects significantly on accuracy and stability of approximation: existence of a free parameter known as shape parameter that can be chosen constantly or variably. Several techniques for selecting a v...

In this paper, the optimal homotopy asymptotic method (OHAM) and the traditional homotopy analysis method (HAM) are used to obtain analytical solution for a strongly nonlinear oscillation. Moreover, the homotopy-pade technique is employed to accelerate the convergence of solution series of traditional HAM. Results show that the second-order approxi...

We develop a highly accurate numerical method for pricing discrete double barrier options under the Black–Scholes (BS) model. To this aim, the BS partial differential equation is discretized in space by the parabolic finite element method, which is based on a variational formulation and thus is well-suited for dealing with the non-smoothness of the...

Conventionally, artificial neural network technique is used to predict the chaotic time series. This paper proposes an approach for the construction of width factor using genetic algorithm to optimize the Gaussian function in RBF networks. Our experimental results show that the developed evolving RBF networks are able to predict the chaotic time se...

We investigate the performances of the finite element method in solving the Black–Scholes option pricing model. Such an analysis highlights that, if the finite element method is carried out properly, then the solutions obtained are superconvergent at the boundaries of the finite elements. In particular, this is shown to happen for quadratic and cub...

Shape parameters play an important role in radial basis function (RBF) approximations. Therefore, the choice of them is an active field in numerical analysis research. In this paper, first we review the available strategies in the literature for selecting shape parameters. Then, we introduce an alternative approach called hybrid strategy for scalin...

In this paper, a meshfree method based on radial basis functions (RBFs) is developed to approximate the eigenvalues of Stokes equations in primitive variables in a square domain. To avoid the inaccuracy near the boundaries, the collocation on boundary technique is applied. This approach leads to more accurate solutions in comparisons with finite el...

This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed t...

The fractional Zakharov–Kuznetsov equations are increasingly used in modeling various kinds of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. This has led to a significant interest in the study of these equations. In this work, solitary pattern solutions...

In this paper, we consider a partial integro-differential equation (PIDE) problem with a free boundary, arising in an American option model when the stock price follows a diffusion process with jump components. We use a front-fixing transformation of the underlying asset variable to fix the free boundary conditions and approximate the integral term...

This paper presents radial basis function (RBF) collocation methods for the coupled Klein-Gordon-Schrödinger equations. Unlike traditional mesh oriented methods, RBF collocation methods require only a scattered set of nodes in the domain where the solution is approximated. For the RBF collocation method in finite difference mode (RBF-FD), weights f...

Conventionally, in radial basis function (RBF) network width factor is constructed by obtaining r-nearest neighbor rule or taking equal to a constant for all Gaussian functions. This paper proposes an approach for the construction of width factor using genetic algorithm to optimize the Gaussian function. Our experimental results show that our propo...

In this paper, we are giving analytic approximate solutions
to nonlinear PDEs using the Homotopy Analysis Method
(HAM) and Homotopy Pad´e Method(HPad´eM). The HAM contains
the auxiliary parameter h, which provides us with a simple
way to adjust and control the convergence regions of solution series.
It is illustrated that HPad´eM accelerates the co...

This paper outlines a reliable strategy to use the homotopy perturbation method based on Jumarie’s derivative for solving fractional differential equations. In this framework, compact structures of fourth-order fractional diffusion-wave equations are considered as prototype examples. Moreover, convergence of the proposed approach for these types of...

This paper presents meshless method using RBF collocation scheme for the coupled Schrödinger-KdV equations. Instead of traditional mesh oriented methods such as finite element method (FEM) or finite difference method (FDM), this method requires only a scattered set of nodes in the domain. For this scheme, error estimates and stability analysis are...

The fractional advection–dispersion equation (FADE) known as its non-local dispersion, is used in groundwater hydrology and has been proven to be a reliable tool to model the transport of passive tracers carried by fluid flow in a porous media. In this paper, compact structures of FADE are investigated by means of the homotopy perturbation method w...

The homotopy perturbation method is applied to the generalized fourth-order fractional diffusion–wave equations. The problem is formulated in the Caputo sense. Moreover, a reliable scheme for calculating nonlinear operators is proposed. The results reveal that the present method is very effective and convenient.

In this paper, we apply the homotopy analysis method (HAM)
to obtain approximate solution for the Ratio-dependent predator-prey system with
constant effort harvesting. We optimize the values of h1 and h2 by an Euclidean
residual for the system of equations. The validity of this method is verified, because it
agrees with Runge-Kutta (RKF78) in figur...

In this paper, a family of high-order compact finite difference methods in combination with Krylov subspace methods is used for solution of the nonlinear sine–Gordon equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite-difference eq...

In this paper, we apply the homotopy analysis method (HAM) to obtain approximate solution for the Ratio-dependent predator-prey system with constant effort harvesting. We optimize the values of h 1 and h 2 by an Euclidean residual for the system of equations. The validity of this method is verified, because it agrees with Runge-Kutta (RKF78) in fig...

In this paper, we propose a simple general form of high-order approximation of O(c2+ch2+h4) to solve the two-dimensional parabolic equation αuxx+βuyy=F(x,y,t,u,ux,uy,ut), where α and β are positive constants. We apply the compact form for solving diffusion–convection equation. The results of numerical experiments are presented and compared with ana...

In this paper, the He's variational iteration method (VIM) based on a reliable modiflcation of Adomian algorithm has been used to obtain so- lutions of the nonlinear boundary value problems (BVP). Comparison of the result obtained by the present method with that obtained by Adomian method (A. M. Wazwaz, Found Phys. Lett. 13 (2000) 493 and G. L. Liu...

In this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.

This paper presents the application of the Homotopy Anal-ysis Method (HAM) and Homotopy Perturbation Method (HPM) for solving systems of integral equations. HAM and HPM are two ana-lytical methods to solve linear and nonlinear equations which can be used to obtain the numerical solution. The HAM contains the auxiliary parameter h, provide us with a...

A numerical solution for solving non-linear Fredholm integral equations is presented. The method is based upon homotopy perturbation theory. The result reveal that the modified homotopy perturbation method (MHPM) is very effective and convenient.

In this paper, a novel learning strategy for radial basis function networks (RBFN) is proposed. By adjusting the parameters of the hidden layer, including the RBF centers and widths, the weights of the output layer are adapted by local optimization methods. A new local optimization algorithm based on a combination of the gradient and Newton methods...

We present a modification to homotopy perturbation method for solving some non-linear Fredholm integral equations of the first kind. Solved problems reveal that the proposed method is very effective and simple and in some cases it gives the exact solution rather than the approximated one. Editorial remark: There are doubts about a proper peer-revie...

In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge–Kutta method for time integration to solve the generalized Burger’s–Huxley equation (GBHE). To reduce round-off error in spectral collocation (pseudospectral) method we use preconditioning. Firstly, theory of application...

In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge–Kutta method for time integration to solve the generalized Burger’s–Fisher equation (B–F). Firstly, theory of application of Chebyshev spectral collocation method (CSCM) and domain decomposition on the generalized Burger’...

This paper is about the construction of a simple method to approximate the solution of system of linear Fredholm integral equations of the second kind based on Adomian’s decomposition method. Easy computations rather than successive integrations are used with simple algorithm. Some solved problems are given to show the efficiency of the method. The...

In this paper we present a modification to homotopy perturbation method for solving linear Fredholm integral equations. Comparisons are made between the standard HPM and the modified one. The results reveal that the proposed method is very effective and simple and gives the exact solution.

A new learning strategy is proposed for training of radial basis functions (RBF) network. We apply two different local optimization methods to update the output weights in training process, the gradient method and a combination of the gradient and Newton methods. Numerical results obtained in solving nonlinear integral equations show the excellent...

Linear integral and integro-differential equations of Fredholm and Volterra types have been successfully treated using radial basis function (RBF) networks in previous works. This paper deals with the case of a system of integral equations of Fredholm and Volterra types with a normalized radial basis function (NRBF) network. A novel learning algori...

In this paper, exact and numerical solutions are obtained for the generalized Zakharov equation (GZE) by the well known variational iteration method (VIM). This method is based on Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method creates a sequence which tends to the exact solution of the pro...

In this paper, we demonstrate the efficiency of the multiquadric radial basis functions (MQ-RBFs) collocation method for solving partial differ-ential equations (PDEs), as theoretically compared to the finite element method (FEM). The MQ-RBF has the property of exponential convergence with respect to the shape parameter. Although the optimal choice...

The variational iteration method (VIM) is employed to obtain approximate analytical solutions of the Stefan problem. The method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials.

This article presents a new method to solve linear weakly singular Volterra integral equations using radial basis function (RBF) networks. The approximate solution is written as a linear combination of radial basis functions. This method employs a growing RBF network, as the basic approximation element, whose parameters are obtained by optimization...

In this paper, the well-known He’s variational iteration method (VIM) is used to construct solitary wave solutions for the generalized Zakharov equation (GZE). The chosen initial solution (trial function) can be in soliton form with some unknown parameters, which can be determined in the solution procedure.

In this paper, He’s variational iteration method is employed successfully for solving parabolic partial differential equations with Dirichlet boundary conditions. In this method, the solution is calculated in the form of a convergent series with an easily computable component. This approach does not need linearization, weak nonlinearity assumptions...

In this study, we use the spectral collocation method with preconditioning to solve various nonlinear Schrödinger equations. To reduce round-off error in spectral collocation method we use preconditioning. We study the numerical accuracy of the method. The numerical results obtained by this way have been compared with the exact solution to show the...

Homotopy perturbation method (HPM) is applied to construct a new family of Newton-like iterative methods for solving system of non-linear equations. Comparison of the result obtained by the present method with Newton–Raphson method reveals that the accuracy and fast convergence of the new method.

In this paper, we present a new iterative method for solving nonlinear algebric equations by using modified homotopy perturbation method. We also discuss the convergence criteria of the present method. To assess its validity and accuracy, the method is applied to solve several test problems.

A numerical method based on modified homotopy perturbation method (HPM) is proposed for solving nonlinear algebric equations. It is shown that the proposed method has third-order convergence. To assess its validity and accuracy, the method is applied to solve several test problems.

Homotopy perturbation method is applied to the numerical solution for solving eighth-order boundary value problems. Comparison of the result obtained by the present method with that obtained by modified decomposition method [M. Mesrovic, The modified decomposition method for eighth-order boundary value problems, Appl. Math. Comput. (2006), doi:10.1...

A numerical method based on Chebyshev polynomials and local interpolating functions is proposed for solving the one-dimensional parabolic partial differential equation subject to non-classical conditions. To assess its validity and accuracy, the method is applied to solve several test problem.

In this paper, He’s homotopy perturbation method is proposed to solve nth-order integro-differential equations. The results reveal that the method is very effective and simple.

Homotopy perturbation method (HPM) is applied to construct a new iterative method for solving system of nonlinear algebric equations. Comparison of the result obtained by the present method with that obtained by revised Adomian decomposition method [Hossein Jafari, Varsha Daftardar-Gejji, Appl. Math. Comput. 175 (2006) 1–7] reveals that the accurac...

Homotopy perturbation method is applied to the numerical solution for solving system of Fredholm integral equations. Comparison of the result obtained by the present method with that obtained by Taylor-series expansion method [K. Maleknejad, N. Aghazade, M. Rabbani, Numerical solution of second kind Fredholm integral equations system by using a Tay...

The purpose of this paper is to present a new method to solve the linear integro-differential equations using radial basis function (RBF) networks. The approximate solution is represented by means of RBFs, whose coefficients are computed by training a RBF network, and is written as a sum of two parts. The first part employs a growing RBF network, w...

In this paper homotopy perturbation is applied to system of linear Fredholm integral equations of the first kind .For these systems, degenerate kernels are considered and in order to avoid successive integrations ,an easy matrix computation is derived for approximating the solution of the problem.

We present a new method for solving of the parabolic partial differential equation with Neumann boundary conditions by using the collocation formula for calculating spectral differentiation matrix for Chebyshev-Gauss-Lobatto point. Firstly, the theory of application of spectral collocation method on parabolic partial differential equation is presen...

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