Abstract
A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.
Highlights
Numerous scientific and engineering problems can be modelled employing nonlocal partial differential equations
We concerned by the solvability of a semilinear pseudoparabolic equation accompanied by nonclassical boundary conditions
The first appearance of the term pseudoparabolic equation was in the works of Ting and Showalter [21, 26], since the pseudoparabolic equation is referred to a partial differential equation with a first-order derivate in time appearing in the highest-order term, in particular, such kind of problems forms a subclass of Sobolev-type equations [25]
Summary
Numerous scientific and engineering problems can be modelled employing nonlocal partial differential equations. Rothe–Legendre spectral method for a semilinear pseudoparabolic equation subject to Dirichlet-type nonlocal boundary condition u(x, t) = Ku(t) := K(z, x)u(z, t) dz, x ∈ ∂Ω, t ∈ IT ,. Due to its applications in several fields of applied sciences, initial value problems for pseudoparabolic equations subject to classical, as well as nonclassical, boundary conditions are of important practical and theoretical interest and have been investigated in theoretical and numerical frames by several authors. This work is aimed at extending and improving some existing results on the solvability of nonlocal boundary-values problems of pseudoparabolic type as well as presenting an efficient numerical schema based on coupling the Euler finite-difference method for time discretization with Legendre spectral method for space discretization to obtain approximate solutions for this kind of problems. Numerical tests are presented in the last section to illustrate the effectiveness of the presented numerical algorithm
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