Reza Pourgholi

In this paper, a numerical method is proposed for the numerical solution of the coupled nonlinear reaction–diffusion equations with suitable initial and boundary conditions by using the Haar wavelet method to determine the unknown boundary conditions. More precisely, we apply the Haar wavelet method for discretizing the space derivatives and then use a quasilinearization technique to linearize the nonlinear term in the equations. This process generates an ill-posed linear system of equations. Hence, to regularize the resultant ill-posed linear system of equations, we employ the Tikhonov regularization method to obtain a stable numerical approximation to the solution. We also prove the convergence of order one (i.e. O(1/M)) and discuss the error estimation and stability computation for the proposed method. Finally, we report some numerical results which in compared with the finite difference method and the radial basis function method show the efficiency and capability of the proposed method. © 2017, © 2017 Informa UK Limited, trading as Taylor & Francis Group.

In this paper, based on the cubic B-spline finite element (CBSFE) and the radial basis functions (RBFs) methods, the inverse problems of finding the nonlinear source term for system of reaction–diffusion equations are studied. The approach of the proposed methods are to approximate unknown coefficients by a polynomial function whose coefficients are determined from the solution of minimization problem based on the overspecified data. In fact, this work considers a comparative study between the cubic B-spline finite element method and radial basis functions method. The stability and convergence analysis for these problems are investigated and some examples are given to illustrate the results. © 2017, Springer-Verlag London Ltd., part of Springer Nature.

In this paper, a numerical method is proposed for the numerical solution of a linear system with suitable initial and boundary conditions using the cubic B-spline collocation scheme to determine the unknown boundary condition. We apply the cubic B-spline for the spatial variable and the derivatives which, generate an ill-posed linear system of equations. In this regard, to overcome, to this drawback, we employ the Tikhonov regularization method for solving the resultant linear system. It is proved that the proposed method has the order of convergence O(Δt + h2). Also, the conditional stability using the Von-Neumann method is established under suitable assumptions. Finally, some numerical results are reported to show the efficiency and robustness of the proposed approach for solving the inverse problems. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

A numerical procedure based on a fully implicit finite difference method for an inverse problem of identification of an unknown source in a heat equation is presented. The approach of the proposed method is to approximate unknown function from the solution of the minimisation problem based on the overspecified data. This problem is ill-posed, in the sense that the solution (if it exist) does not depend continuously on the data. To regularise this ill-conditioned, we apply the Tikhonov regularisation 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. A stability analysis shows that this numerical scheme approximation is unconditionally stable. Numerical results for two inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient. Copyright © 2018 Inderscience Enterprises Ltd.

In this paper, a spectral Tau method based on Legendre Wavelet basis is proposed. For this purpose we present a stable operational Tau method based on Legendre Wavelet basis. This method provides an efficient approximate solution for weakly singular Volterra integral equations by using reduced set of matrix operations. An error estimation of the Tau method is also introduced. Finally we demonstrate the validity and applicability of the method by numerical examples. © 2016 Informa UK Limited, trading as Taylor & Francis Group.

In this paper, we discuss a numerical method for solving an inverse Rosenau equation with Dirichlet’s boundary conditions. The approach used is based on collocation of a quintic B-spline over finite elements so that we have continuity of dependent variable and it first four derivatives throughout the solution range. We apply quintic B-spline for spatial variable and derivatives which produce an ill-posed system. We solve this system using Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on a test problem. Figures and comparisons have been presented for clarity. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. © 2017, Springer-Verlag London.

In this article, we discuss a numerical method for solving some nonlinear inverse parabolic partial differential equations with Dirichlet's boundary conditions. The approach used, is based on collocation of cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B-splines for spatial variable and derivatives, which produce an ill-posed system. We solve this system using the Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on two test problems. The figures and comparisons have been presented for clarity. Also the stability of this method has been discussed. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 88–104, 2017. © 2016 Wiley Periodicals, Inc.

Determination of an unknown time-dependent function in parabolic differential equations plays a very important role in many branches of science and engineering. In this paper, we present an approach based on ADM to solve inverse non-local initial-boundary value problems, since this problem is mildly ill-posed, the Tikhonov regularisation method is applied to deal with noisy input data and obtain a stable approximate solution. This method provides a reliable algorithm that requires less work if compared with the traditional techniques. Numerical results obtained from this method, indicating high accuracy and speed of the method. © 2015 Inderscience Enterprises Ltd.

In this paper, we will first study the existence and uniqueness of the solution for a one dimensional inverse heat conduction problem (IHCP) via an auxiliary problem. Then the present work is motivated by desire to obtain numerical approach for solving this IHCP. Our method begins with the utilisation of some transformations. These transformations allow us to eliminate an unknown term from parabolic equation to obtain an inverse parabolic problem with two unknown boundary conditions. To solve this inverse problem, we use the fundamental solution method. The effectiveness of the algorithm is illustrated by numerical example. Copyright 2015 Inderscience Enterprises Ltd.

The problem of identifying the solution (k(x, t),U(x, t)) in an inverse semilinear wave problem is considered. It is shown that under certain conditions of data φ, ψ, there exists a unique solution (k(x, t),U(x, t)) of this problem. Furthermore a numerical algorithm for solving the inverse semilinear wave problem is proposed. The approach for this inverse problem is given by using the semi-discretisation method. A polynomial function is proposed to approximate U(x, t) then the finite difference method is applied to approximate unknown k(x, t). Numerical results show efficiency of our method. Copyright © 2014 Inderscience Enterprises Ltd.

In this paper, we study the ADMB-KdV equation. We show that the associated initial value problem is locally well posed in Sobolev spaces Hs (ℝ2) for s > -1/2. We also prove that our result is sharp in some sense. © 2013 Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer.

In this paper we study the solitary waves of the Rosenau-RLW equation. By using some trigonometric function methods, a family of stable solitary wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency and wave speed, for what should be an equation relevant to modeling in a number of fields. © 2013 Foundation for Scientific Research and Technological Innovation.

In this paper a numerical approach combining the least squares method and the genetic algorithm (sequential and multi-core parallelization approach) is proposed for the determination of temperature in an inverse heat conduction problem (IHCP). Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data. Results show that an excellent estimation can be obtained by implementation sequential genetic algorithm within a CPU with clock speed 2.7. GHz, and parallel genetic algorithm within a 16-core CPU with clock speed 2.7. GHz for each core. © 2013 Elsevier Inc.

This paper studies the ADM-BKdV equation. The associated initial value problem will be proved to be locally well-posed in anisotropic Sobolev spaces Hs1,s2 (ℝ2) for s1>-3/2, s2>-1/2 and 2s1+4s2>-1. © 2013 Università degli Studi di Ferrara.

In this paper, two numerical methods are presented to solve an ill-posed inverse problem for Fisher's equation using noisy data. These two methods are the Haar basis and the Legendre wavelet methods combined with the Tikhonov regularization method. A sensor located at a point inside the body is used and u(x, t) at a point x = a, 0 < a < 1 is measured and these methods are applied to the inverse problem. We also show that an exponential rate of convergence of these methods. In fact, this work considers a comparative study between the Haar basis and the Legendre wavelet methods to solve some ill-posed inverse problems. Results show that an excellent estimation of the unknown function of the inverse problem which have been obtained within a couple of minutes CPU time at Pentium(R) Dual-Core CPU 2.20 GHz. © World Scientific Publishing Company.[/accordion]
AUTHOR KEYWORDS: error analysis; Haar basis method; Ill-posed inverse problems; Legendre wavelet method; noisy data
INDEX KEYWORDS: Comparative studies; Exponential rates; Fisher's equation; Haar basis method; ILL-posed inverse problem; Legendre wavelet methods; Noisy data; Tikhonov regularization method, Differential equations; Error analysis, Inverse problems

Pourgholi, R., Esfahani, A. An efficient numerical method for solving an inverse wave problem (2013) International Journal of Computational Methods, 10 (3), art. no. 1350009, .

DOI: 10.1142/S0219876213500096

In this paper, we will first study the existence and uniqueness of the solution of a one-dimensional inverse problem for an inhomogeneous linear wave equation with initial and boundary conditions via an auxiliary problem. Then a stable numerical method consisting of zeroth-, first-, and second-order Tikhonov regularization to the matrix form of Duhamel's principle for solving this inverse problem is presented. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition method. Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data. © 2013 World Scientific Publishing Company.

In this paper, we will first study the existence and uniqueness of the solution of an inverse initial-boundary-value problem, via an auxiliary problem. Furthermore, we propose a stable numerical approach based on the use of the solution to the auxiliary problem as a basis function for solving this problem in the presence of noisy data. Also note that the inverse problem has a unique solution, but this solution is unstable and hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method. The effectiveness of the algorithm is illustrated by numerical example. © 2013 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.

In this paper, we will first study the existence and uniqueness of the solution of a two-dimensional inverse heat conduction problem (IHCP) which is severely ill-posed, i.e.; the solution does not depend continuously on the data. We propose a stable numerical approach based on the finite-difference method and the least-squares scheme to solve this problem in the presence of noisy data. We prove the convergence of the numerical solution, then to regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition (SVD) method. © 2012 Korean Society for Computational and Applied Mathematics.

In this paper, a numerical approach combining the use of the least squares method and the genetic algorithm (sequential and multi-core parallelisation approach) is proposed for the determination of temperature in an inverse hyperbolic heat conduction problem (IHHCP). Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data. Results show that an excellent estimation can be obtained by implementation sequential genetic algorithm within a CPU with clock speed 2.4 GHz, and parallel genetic algorithm within a 16-core CPU with clock speed 2.4 GHz for each core. © 2013 Inderscience Enterprises Ltd.

In this paper a numerical method consists of combining Haar basis method and Tikhonov regularization method for solving some ill-posed inverse problems using noisy data is presented. By using a sensor located at a point inside the body and measuring the u(x, t) at a point x = a, 0 < a < 1, and applying Haar basis method to the inverse problem, we determine a stable numerical solution to this problem. Results show that an excellent estimation on the unknown functions of the inverse problem can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC. © Springer Science+Business Media, LLC 2012.[/accordion]
AUTHOR KEYWORDS: Haar basis method; Ill-posed inverse problems; Noisy data; Tikhonov regularization method
PUBLISHER: Springer International Publishing

Abtahi, M., Pourgholi, R., Shidfar, A. Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem (2011) Nonlinear Analysis, Theory, Methods and Applications, 74 (7), pp. 2462-2467.

DOI: 10.1016/j.na.2010.12.001

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f,g, and for a properly specified class A of admissible coefficients, there exists at least one a∈A such that (a,u) is a solution of the corresponding inverse problem. © 2011 Elsevier Ltd. All rights reserved.

This study is motivated by a desire to obtain a numerical approach combining the use of the finite difference method with the solution of ordinary differential equation proposed for the determination of unknown coefficient in an inverse heat conduction problem. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution. © 2010 Taylor & Francis.

This study is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions, and the initial condition are presented in a dimensionless form. The numerical approach is developed based on the use of the solution to the auxiliary problem as a basis function. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution. © 2009 Elsevier Inc.

In this paper, we consider an inverse heat conduction problem (IHCP). A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a transformation, the ill-posed IHCP becomes a Cauchy problem. Since the solution of Cauchy problem, exists and is unique but not always stable, the ill-posed problem is closely approximated by a well-posed problem. For this new well-posed problem, the existence, uniqueness, and stability of the solution are proved. © 2008 Elsevier B.V. All rights reserved.

This research is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions and the initial condition are presented in a dimensionless form. A numerical approach combining the use of the finite difference method with the solution of Ordinary Differential Equation (ODE) is proposed for the determination of temperature distribution in an Inverse Heat Conduction Problem (IHCP). The least-squares method is adopted to find the solution. Results show that an excellent estimation can be obtained within a couple of minute's CPU time at pentium IV-2.4 GHz PC. © 2008 Science Publications.

This research is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions and the initial condition are presented in a dimensionless form. A numerical approach combining the use of the finite difference method with the solution of Ordinary Differential Equation (ODE) is proposed for the determination of temperature distribution in an Inverse Heat Conduction Problem (IHCP). The least-squares method is adopted to find the solution. Results show that an excellent estimation can be obtained within a couple of minute's CPU time at pentium IV-2.4 GHz PC. © 2008 Science Publications.