Abstract
An exact series solution for nonhomogeneous parabolic coupled systems of the typeut-Auxx=Gx, t, A1u0, t+B1ux0, t=0, A2ul, t+B2uxl, t=0, 0<x<1, t>0, ux, 0=fx, whereA1, A2, B1, andB2are arbitrary matrices for which the block matrix is nonsingular, andAis a positive stable matrix, is constructed.
Highlights
Coupled partial differential systems with coupled boundaryvalue conditions are frequent in different areas of science and technology, as in scattering problems in quantum mechanics [1,2,3], in chemical physics [4,5,6], coupled diffusion problems [7,8,9], thermo-elastoplastic modelling [10], and so forth
The solution of these problems has motivated the study of vector and matrix Sturm-Liouville problems; see [11,12,13,14], for example
(2) Matrices Ai, Bi, i = 1, 2, are m × m complex matrices, and we assume that the block matrix
Summary
Coupled partial differential systems with coupled boundaryvalue conditions are frequent in different areas of science and technology, as in scattering problems in quantum mechanics [1,2,3], in chemical physics [4,5,6], coupled diffusion problems [7,8,9], thermo-elastoplastic modelling [10], and so forth. We will find the solution of nonhomogeneous problem with homogeneous boundary conditions (13)–(16) where we will suppose that the vector valued function G(x, t) satisfies the conditions that we will indicate to ensure the convergence of the solution proposal. Λn∈F 0 where, for a fixed value of t ∈ [c, d], the positive terms series bn has the partial sum bounded if we suppose that vector valued function G(x, t) satisfies the following condition: sup ∫ ‖G (x, t)‖2dx = M < ∞. Suppose that hypotheses of Theorem 3.1 of [16] hold; we can construct a solution V(x, t) of homogeneous problem with homogeneous boundary values conditions (34).
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