Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function
Abstract
1. Introduction
2. Optimization of Entropy
3. Connection of the Pathway Model to Fractional Integrals
3.1. Fractional Integrals of the Second Kind
3.2. Fractional Integral of the First Kind or Left-Sided Integral
4. Mellin Convolutions of Products and Ratios for Other Functions
5. Connection of the Pathway Parameter to Fractional Indices in Fractional Differential Equations
6. The H-Function Thread
7. Diffusion Equation
- (i)
- For , the corresponding solution of (36), denoted by , can be expressed in terms of the H-function as provided below, and can be defined for as follows:Non-diffusion:
- (ii)
- When , then (36) reduces to the space-fractional diffusion equation, which is the fundamental solution of the following space–time fractional diffusion model:
- (iii)
- Next, if we take , then we obtain the time-fractional diffusion, which is governed by the following time-fractional diffusion model:
- (iv)
- If we set and , then for the fundamental solution of the standard diffusion equation
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Mathai, A.M.; Haubold, H.J. Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function. Axioms 2024, 13, 546. https://doi.org/10.3390/axioms13080546
Mathai AM, Haubold HJ. Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function. Axioms. 2024; 13(8):546. https://doi.org/10.3390/axioms13080546
Chicago/Turabian StyleMathai, Arak M., and Hans J. Haubold. 2024. "Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function" Axioms 13, no. 8: 546. https://doi.org/10.3390/axioms13080546
APA StyleMathai, A. M., & Haubold, H. J. (2024). Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function. Axioms, 13(8), 546. https://doi.org/10.3390/axioms13080546