Radial Basis Functions Method for determining of unknown coefficient in parabolic

In this paper, we consider an inverse problem of finding unknown source parameterp(t) and u(x,t) satisfy equation

u_{t}=u_{xx}+p(t)u+f(t,x),    0≤x≤1, 0<t≤T,

with the initial-boundary conditions

u(x,0)=ϕ(x),     0≤x≤1

(0,t)=μ₁(t),     0<t≤T

u(1,t)=μ₂(t),     0<t≤T

subject to the overspecification over the spatial domain

u(x^{∗},t)=E(t),    0<x^{∗}≤1, 0< t≤T

where f(x,t),ϕ(x),μ₁(t),μ₂(t) and E(t)≠0 are known functions, x^{∗} is a fixed prescribed interior point in (0,1). If p(t) is known then direct initial boundary value problem (1)-(4) has a unique smooth solution u(x,t) [1]. If u represent a temperature distribution, then (1)-(4) can be interpreted as a control problem with source parameter. Based on the idea of the radial basis functions (RBF) approximation , a fast and highly accurate meshless method is developed for solving an inverse problem with a control parameter [2].

Some numerical examples using the proposed algorithm are presented.