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Abstract We begin with a treatment of the Caputo time‐fractional diffusion equation, by using the Laplace transform, to obtain a Volterra integro‐differential equation. We derive and utilize a numerical scheme that is derived in parallel to the L1‐method for the time variable and a standard fourth‐order approximation in the spatial variable. The main method derived in this article has a rate of convergence ofO(kα + h4)foru(x,t) ∈ Cα([0,T];C6(Ω)),0 < α < 1, which improves previous regularity assumptions that requireC2[0,T]regularity in the time variable. We also present a novel alternative method for a first‐order approximation in time, under a regularity assumption ofu(x,t) ∈ C1([0,T];C6(Ω)), while exhibiting order of convergence slightly more thanO(k)in time. This allows for a much wider class of functions to be analyzed which was previously not possible under the L1‐method. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques.
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The gradient descent method from the perspective of fractional calculus
Motivated by gradient methods in optimization theory, we give methods based onψ‐fractional derivatives of orderαin order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams–Bashforth–Moulton (ABM) method for the estimation of solutions to equations involvingψ‐fractional derivatives. Numerical examples using the ABM method show that the fractional orderαand weightψare tunable parameters, which can be helpful for improving the performance of gradient descent methods.
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- Award ID(s):
- 2027976
- PAR ID:
- 10452785
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematical Methods in the Applied Sciences
- Volume:
- 44
- Issue:
- 7
- ISSN:
- 0170-4214
- Page Range / eLocation ID:
- p. 5520-5547
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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