Exact Solution of Local and Nonlocal BVPs for the Laplace Equation in a Rectangle
An operational method for obtaining explicit solutions of a class of local and nonlocal boundary value problems for the Laplace equation is proposed. This method is based on a direct two-dimensional operational calculus built by means of two non-classical convolutions for the operators ∂ xx and ∂ yy . The corresponding operational calculus uses multiplier fractions instead of convolution fractions. An extension of the Duhamel principle to the space variables is proposed. Thus explicit solutions of the boundary value problems considered are obtained. The general approach is specialized to the case when some of the boundary value conditions are of integral type.
KeywordsNon-local boundary value problem right-inverse operator extended Duhamel principle generalized solutions convolutions operational calculus multiplier multipliers fractions
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