Maximum principles for time-fractional Caputo-Katugampola diffusion equations

Abstract

Maximum and minimum principles for time-fractional Caputo-Katugampola diffusion operators are proposed in this paper. Several inequalities are proved at extreme points. Uniqueness and continuous dependence of solutions for fractional diffusion equations of initial-boundary value problems are considered.

Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives

“…Both Katugampola and Almeida defined and developed new nonlocal notions of fractional derivatives, known as the generalized Caputo fractional (GCpFr) derivative (see [42,43]). One of the main advantages of the GCpFr derivative is the ability to combine all traditional fractional derivatives, and it satisfies the semigroup property, hence, GCpFr derivative is considered a generalized form of fractional derivatives.…”

“…One of the main advantages of the GCpFr derivative is the ability to combine all traditional fractional derivatives, and it satisfies the semigroup property, hence, GCpFr derivative is considered a generalized form of fractional derivatives. There are a number of research studies that have been done in the form of GCpFr derivative [43][44][45]. However, according to the authors' knowledge, there are no existing research studies on GCpFr involving p-Laplacian operator.…”

An extension of maximum principle with some applications

“…Then the key lemma and the fixed point theorem, are employed to prove the maximum and comparison principles and some of their corollaries. In [9], Cao, Kong and Zeng proposed maximum and minimum principles for time-fractional Caputo-Katugampola diffusion operators. They proved several inequalities at extreme points, and considered uniqueness and continuous dependence of solutions for fractional diffusion equations of initial-boundary value problems.…”

Some comparison principles for fractional differential equations and systems

“…In Al‐Refai and Luchko, 5 extremum principles for time fractional diffusion equations in the Riemann‐Liouville sense were derived. In Cao et al, 6 time fractional Caputo‐Katugampola diffusion operators were studied. Kirane and Torebek 7 established extremum principles for Hadamard‐type fractional derivatives and presented several applications to time fractional diffusion equations in the Hadamard‐type sense.…”