Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels
Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations (FPIDEs) represent a signif...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/9/6/392 |
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| Summary: | Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations (FPIDEs) represent a significant class of such evolution equations and are widely used in diverse scientific and engineering fields. In this study, we use the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sinc</mi></semantics></math></inline-formula>-collocation and iterative Laplace transform methods to solve a specific FPIDE with a weakly singular kernel. Specifically, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sinc</mi></semantics></math></inline-formula>-collocation method is applied to discretize the spatial domain, while a combination of numerical techniques is utilized for temporal discretization. Then, we prove the convergence analytically. To compare the two methods, we provide two examples. We notice that both the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>sinc</mi></semantics></math></inline-formula>-collocation and iterative Laplace transform methods provide good approximations. Moreover, we find that the accuracy of the methods is influenced by fractional 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> and the memory-kernel parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We observe that the error decreases as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> increases, where the kernel becomes milder, which extends the single-value study of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> in the literature. |
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| ISSN: | 2504-3110 |