A New Chebyshev Spectral-collocation Method for Solving a Class of One-dimensional Linear Parabolic Partial Integro-differential Equations
Galal I. El–Baghdady *
Department of Engineering Physics and Mathematics Faculty of Engineering, Mansoura University El–Gomheria St., Mansoura, Dakahlia 35516, Egypt.
M. S. El–Azab
Department of Engineering Physics and Mathematics Faculty of Engineering, Mansoura University El–Gomheria St., Mansoura, Dakahlia 35516, Egypt.
*Author to whom correspondence should be addressed.
Abstract
In this work, the Chebyshev spectral-collocation method is applied to obtain approximate solution for some types of linear parabolic partial integro–differential equations (PPIDEs).
In the first approach, we convert our equation into two coupled Volterra integral equations of the second kind by using a proper transformation.
In the second approach, the integration in the resulting equations are approximated by replacing the integrand by its interpolating polynomials in terms of the Chebyshev polynomials instead of using the approximation by Gauss quadrature rules.
After approximation a linear algebraic system were raised, then it tested by the conditional number.
Finally, some numerical examples are included to illustrate the validity and applicability of the proposed technique.
Keywords: Spectral collocation method, Chebyshev polynomials, Chebyshev differentiation matrices, Lagrange basis function, parabolic partial integro–differential equations of Volterra type, Conditional number.