Abstract
In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function on , when is locally integrable on , and the integral converges. We now discard the last condition that should converge, and discuss the same problem. In Appendices, polynomial form of particular solutions are given for the differential equations studied and Hermite’s differential equation with special inhomogeneous terms.
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
In the preceding paper [4], we discussed the solution of an fractional differential equation of the type of DE (1.1), that is given by a2t b2 0 DR2 u t a1t b1 0 DR u t a0t b0 u t f t, t 0, for 1 and 1 2
In [4], we adopt operational calculus in the framework of distribution theory developed for the solution of the fractional differential equation (fDE) with constant coefficients in [5,6]
Summary
We use x for x , to denote the least integer that is not less than x. In [4], we adopt the usual definition of the Riemann-Liouville fD, which defines 0 DR f t only for such a locally integrable function f t on >0 that 0 f t dt is finite. In a recent review [9], we discussed the analytic continuations of fD, where an analytic continuation of Riemann-Liouville fD, 0 DR f t , is such that the fD exists even for such a locally integrable function f t on. We adopt this analytic continuation of 0 DR f t. It is the purpose this paper to show how the presentation in [4] should be revised, with the change of definition of fD and the replacement of Condition IB with. In Appendix B, we show how the results presented in [10] are derived from those in Appendix A
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