Abstract
In this paper we obtain some new explicit results for nonlinear equations involving Laguerre derivatives in space and/or in time. In particular, by using the invariant subspace method, we have many interesting cases admitting exact solutions in terms of Laguerre functions. Nonlinear diffusive-type and telegraph-type equations are considered and also the space and time-fractional counterpart are analyzed, involving Caputo or Prabhakar-type derivatives. The main aim of this paper is to point out that it is possible to construct many new interesting examples of nonlinear diffusive equations with variable coefficients admitting exact solutions in terms of Laguerre and Mittag-Leffler functions.
Highlights
Introduction and preliminariesIn this note we apply the invariant subspace method [5] to solve nonlinear partial differential equations involving space and/or time Laguerre derivatives of first order dd DL :=x dx dx Copyright c 2021 The Author(s)
N-order dd d that is the first order derivative applied to n-times the operator x d dx. It is well-known that a wide class of nonlinear partial differential equations admits a solution by separating variables in spaces generated by exponential, trigoniometric, hyperbolic and polynomial functions, according to the general theory developed by V
We consider some nonlinear equations with variable coefficients that admit solutions in terms of Laguerre functions and we will try to explain their possible meaning and utility in physics
Summary
In this note we apply the invariant subspace method [5] to solve nonlinear partial differential equations involving space and/or time Laguerre derivatives of first order dd DL. It is well-known that a wide class of nonlinear partial differential equations admits a solution by separating variables in spaces generated by exponential, trigoniometric, hyperbolic and polynomial functions, according to the general theory developed by V. Nonlinear equations involving Laguerre derivatives was not considered previously and the aim of this paper is to show that there are wide classes of nonlinear PDEs ( involging fractional derivatives) involving Laguerre derivatives that admit exact solutions Many of these equations can be seen as a generalization of classical nonlinear models. We obtain some exact results for the space or time-fractional counterpart of these interesting equations involving Caputo or Prabhakar derivatives (see e.g. [6, 12] and the references therein)
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