Decomposition of abstract linear operators on Banach spaces

Abstract

Introduction: The majority of the known decomposition methods for solving boundary value problems (Adomian decomposition method, natural transform decomposition method, modified Adomian decomposition method, combined Laplace transform - Adomian decomposition method, and Domain decomposition method) use so-called Adomian polynomials or iterations to get approXimate solutions. To our knowledge, a direct method for obtaining an eXact analytical solution is not yet proposed. Purpose: Developing, in an arbitrary Banach space, a new universal decomposition method for the class of ordinary or partial integro-differential equations with non-local and initial boundary conditions in terms of the abstract operator equation B1X = f. Results: A class of integro-differential equations in a Banach space with non-local and initial boundary conditions in terms of an abstract operator equation B1X = AX - S0F(AX) - G0Φ(AX) = f, X = D(B1)has been studied, where A, A are linear abstract operators, S0, G0are vectors and Φ, F the functional vectors. Usually, A, A are linear ordinary or partial differential operators, and F(AX), Φ(AX) are Fredholm integrals. The eXistence and uniqueness are proved under the assumption that the operator B1 has a decomposition of the form B1= B0B with B and B0 being different abstract linear operators of special forms. The proposed decomposition method is universal and essentially different from other decomposition methods in the relevant literature. This method can be applied to either ordinary integro-differential or partial integro-differential equations, providing a unique eXact solution in closed analytical form in a Banach space. The stages of the method are illustrated by numerical eXamples corresponding to specific problems. Computer algebra system Mathematica is used to demonstrate the solution outcomes and to assess the effectiveness of the analysis. Practical relevance: The main advantage of the proposed solution method is that it can be integrated in the interface of any CAS software in an easy, programing-free way. © 2021 Saint Petersburg State University of Aerospace Instrumentation. All rights reserved.