Abstract
We discuss the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We here study the solution of that differential Equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation.
Highlights
Yosida [1,2] discussed the solution of Laplace’s differential equation (DE), which is a linear DE with coefficients which are linear functions of the variable
The formulation is given in the style of primitive operational calculus, solving a Volterra integral equation with the aid of Neumann series
One of the purposes of the present paper is to give the recipe of obtaining the solution of the inhomogeneous equation as well as the homogeneous one, in the style of operational calculus in the framework of distribution theory
Summary
In our preceding papers [4,5], we discuss the initial-value problem of linear fractional differential equation (fDE) with constant coefficients, in terms of distribution theory. The formulation is given in the style of primitive operational calculus, solving a Volterra integral equation with the aid of Neumann series. One of the purposes of the present paper is to give the recipe of obtaining the solution of the inhomogeneous equation as well as the homogeneous one, in the style of operational calculus in the framework of distribution theory. A proof of a lemma in Section 2 is given in Appendix B The discussion is done in the style of our preceding papers [4,5]
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