Numerical solution of an inverse reaction–diffusion problem via collocation method based on radial basis functions

Abstract

Please cite this article as: F. Parzlivand, A.M. Shahrezaee, Numerical solution of an inverse reaction-diffusion problem via collocation method based on radial basis functions, Appl. Math. Modelling (2014), doi: http://dx. Abstract.In this paper, a numerical technique is presented for the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement. This method is a combination of collocation method and radial basis functions. The operational matrix of derivative for radial basis functions is introduced and the new computational technique is used for this purpose. The operational matrix of derivative is utilized to reduce the problem to a set of algebraic equations. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results.

Hermite Radial Basis Collocation Method for Unsaturated Soil Water Movement Equation

“…Two most common infiltration models used to calculate the unsaturated conductivity ( ) and diffusivity ( ) in practice are Brooks-Corey (B-C) model and van Genuchten (V-G) model [1,38,39]. According to the B-C model and V-G model, we can obtain the expressions of nonlinear terms (27) and (28).…”

“…However, the accuracy of the derivative of the interpolating function is usually very poor on the boundary when collocation method is used. In order to enhance the accuracy of the derivative of the approximation functions, Chu et al [24] used Hermite radial basis point interpolation method for the nonuniform material gradient plane plate vibration; a meshless method was developed based on the collocation method and ridge basis function interpolation by Wang et al [25]; Xing [9] constructed the governing equations of laminated plates with large deflection bending problem, approximated these field variables with RBF and HRBF, and discreted the control equations with least square collocation method; Wang et al [26] proposed an improved meshless method which neither needs the computation of integrals nor requires a partition of the region and its boundary, and this method is applied to elliptic equations for examining its appropriateness; Parzlivand and Shahrezaee [27] mentioned a numerical technique which is a combination of collocation method, radial basis functions, the operational matrix of derivative for radial basis functions, and the new computational technique. Then they got the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement; Krowiak [28] proposed a Hermite radial basis function differential quadrature method for higher order equations and so on [29][30][31][32].…”

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

“…For the case of the hyperbolic equation, we do not need any boundary condition on the part on the part of the space-time boundary characterized by Ω × ft = Tg. The problem is solved as an ill-posed one, and the algebraic system is square (see [1]). Then, the system has the new form:…”

“…Such initial boundary value problems (parabolic or hyperbolic) can be solved by coupling time-stepping algorithms with different numerical methods such as finite element, finite volume, boundary elements methods, meshless methods, fundamental solutions, and spectral and wavelet methods. In [1], Parzlivand and Shahrezaee presented a numerical technique based on a combination of collocation method and radial basis functions to solve parabolic partial differential equations with time-dependent coefficients. Dehghan and Shokri [2] have employed the thin-plate splines RBFs in the collocation RBF method to solve the two-space dimensional linear hyperbolic equation with variable coefficients, subject to appropriate initial and Dirichlet boundary conditions.…”

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

“…This type of problem is ill-posed [24,34,35]. For solving retrospective inverse problems for parabolic equations the following methods have been used: quasireversibility [36,37], optimal filtering [38], boundary element [39], mollification [40], group preserving [41], operator-splitting [42], Fourier regularization [43,44], modified Tikhonov regularization [45], sequential function specification [31], and collocation [46]. However, one of the possible formulations of the inverse problem for determining the initial condition is a statement with additional information about the dynamics of the reaction front movement, if it is available for experimental observation (position of the shock wave front, reaction or combustion front, etc.).…”