Premium
The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data
Author(s) -
Dehghan Mehdi,
Tatari Mehdi
Publication year - 2007
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20204
Subject(s) - mathematics , partial differential equation , domain (mathematical analysis) , mathematical analysis , boundary value problem , radial basis function , partial derivative , parabolic partial differential equation , basis (linear algebra) , inverse problem , work (physics) , geometry , physics , machine learning , artificial neural network , computer science , thermodynamics
Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p ( t ) in the inverse linear parabolic partial differential equation u t = u xx + p ( t ) u + φ, in [0,1] × (0, T ], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ∫ 0 1 k ( x ) u ( x , t ) dx = E ( t ), 0 ≤ t ≤ T , for known functions E ( t ), k ( x ), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ∫ 0 s ( t ) u ( x , t ) dx = E ( t ), 0 < t ≤ T , 0 < s ( t ) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007