Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extention method

Abstract

Introduction:Boundary value problems for differential and integro-differential equations with multipoint and non-local boundary conditions often arise in mechanics, physics, biology, biotechnology, chemical engineering, medical science, finances and other fields. Finding an exact solution of a boundary value problem with Fredholm integro-differential equations is a challenging problem. In most cases, solutions are obtained by numerical methods.Purpose:Search for necessary and sufficient solvability conditions for abstract operator equations and their exact solutions. Results: A direct method is proposed for the exact solution of a certain class of ordinary differential or Fredholm integro-differential equations with separable kernels and multlpolnt/lntegral boundary conditions. We study abstract equations of the formBu = Au -gF(Au) = fandB1u = A2u -qF(Au) -gF(A2u) = fwith non-local boundary conditionsΦ(u ) =NѰ(u )andΦ(u ) =NѰ(u ),Φ(Au) =DF(Au) +NѰ(Au), respectively, where A is a differential operator,qandgare vectors,DandNare matrices, andF,ΦandѰare functional vectors. This method is simple to use and can be easily incorporated into any Computer Algebra System (CAS). The upcoming Part 2 of this paper will be devoted to decomposition method for this problem where the operatorB1is quadratic factorable.