Error Analysis of the <i>L</i>1 Scheme on a Modified Graded Mesh for a Caputo–Hadamard Fractional Diffusion Equation

Error Analysis of the <i>L</i>1 Scheme on a Modified Graded Mesh for a Caputo–Hadamard Fractional Diffusion Equation

The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn></mrow></semantics></math></inline-formula> scheme on a modified graded mesh is propo...

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Bibliographic Details
Main Authors:	Dan Liu, Libin Liu, Hongbin Chen, Xiongfa Mai
Format:	Article
Language:	English
Published:	MDPI AG 2025-04-01
Series:	Fractal and Fractional
Subjects:	Caputo–Hadamard fractional differential equations <i>L</i>1 scheme modified graded mesh error analysis
Online Access:	https://www.mdpi.com/2504-3110/9/5/286
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Summary:	The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn></mrow></semantics></math></inline-formula> scheme on a modified graded mesh is proposed to solve a Caputo–Hadamard fractional diffusion equation with order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Firstly, an improved graded mesh frame is innovatively constructed, and its mathematical properties are verified. Subsequently, a new truncation error bound for the L1 discretisation format of Caputo–Hadamard fractional-order derivatives is established by means of a Taylor cosine expansion of the integral form, and a second-order central difference method is used to achieve high-precision discretisation of spatial derivatives. Furthermore, a rigorous analysis of stability and convergence under the maximum norm is conducted, with special attention devoted to validating that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn></mrow></semantics></math></inline-formula> approximation scheme manifests an optimal convergence order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>−</mo><mi>α</mi></mrow></semantics></math></inline-formula> when deployed on the modified graded mesh. Finally, the theoretical results are substantiated through a series of numerical experiments, which validate their accuracy and applicability.
ISSN:	2504-3110