Analysis of fractional integro-differential equations of thermistor type

Abstract

We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results on fractional integral and differential equations of thermistor type. Several nonlocal problems are considered: with Riemann-Liouville, Caputo, and time-scale fractional operators. Existence and uniqueness of positive solutions are obtained through suitable fixed-point theorems in proper Banach spaces. Additionally, existence and continuation theorems are given, ensuring global existence.

This is a preprint of a paper whose final and definite form is in 'Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory', De Gruyter. Submitted 28/March/2018; revised 03/May, 04/May and 04/July/2018; accepted 04/July/2018

References in corpus (10)

• Fractional Calculus: Integral and Differential Equations of Fractional Order
• Discrete-Time Fractional Variational Problems
• The Variable-Order Fractional Calculus of Variations
• A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
• Calculus of Variations on Time Scales with Nabla Derivatives
• A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration
• Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform
• Chain rules and inequalities for the BHT fractional calculus on arbitrary time scales
• Existence and uniqueness results for a fractional Riemann-Liouville nonlocal thermistor problem on arbitrary time scales
• Optimal Control of Nonlocal Thermistor Equations

Cited by in corpus (4)

• Cauchy's formula on nonempty closed sets and a new notion of Riemann--Liouville fractional integral on time scales
• Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets
• Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives
• Time-Fractional Optimal Control of Initial Value Problems on Time Scales