Abstract
In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields of science and engineering. Many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The purpose of this paper is to present a certain class of the explicit particular solutions of the associated Cauchy-Euler fractional partial differential equation of arbitrary real or complex orders and their applications as follows: where u=u(x,t); A, B, C, M, N, α and β are arbitrary constants.MSC: 26A33, 33C10, 34A05.
Highlights
1 Introduction, definitions and preliminaries The subject of fractional calculus (that is, derivatives and integrals of any real or complex order) has gained importance and popularity during the past two decades or so, due mainly to its demonstrated applications in numerous seemingly diverse fields of science and engineering (cf. [ – ])
1 Introduction, definitions and preliminaries The subject of fractional calculus has gained importance and popularity during the past two decades or so, due mainly to its demonstrated applications in numerous seemingly diverse fields of science and engineering
We present a direct way to obtain explicit solutions of such types of the associated Cauchy-Euler fractional partial differential equation with initial and boundary values
Summary
1 Introduction, definitions and preliminaries The subject of fractional calculus (that is, derivatives and integrals of any real or complex order) has gained importance and popularity during the past two decades or so, due mainly to its demonstrated applications in numerous seemingly diverse fields of science and engineering (cf. [ – ]). ) are repeated; that is, m = m = m; (c) u(x, t) = eλt k xa+bi + k xa–bi , when the discriminant (A – B) – A(C – Mλα – Nλβ) < , and a + bi, a – bi are the conjugate pair roots of Equation
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