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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2025-06-01
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| author | Kamel Al-Khaled Isam Al-Darabsah Amer Darweesh Amro Alshare |
| author_facet | Kamel Al-Khaled Isam Al-Darabsah Amer Darweesh Amro Alshare |
| author_sort | Kamel Al-Khaled |
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| description | 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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| spelling | doaj-art-7d36b68c20d44bfba4ce4f1a46d41a7f2025-06-25T13:52:14ZengMDPI AGFractal and Fractional2504-31102025-06-019639210.3390/fractalfract9060392Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory KernelsKamel Al-Khaled0Isam Al-Darabsah1Amer Darweesh2Amro Alshare3Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, JordanDepartment of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, JordanDepartment of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, JordanDepartment of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, JordanEvolution 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.https://www.mdpi.com/2504-3110/9/6/392fractional partial integro-differential equationsinc-collocation methoditerative Laplace transform methodweakly singular kernel |
| spellingShingle | Kamel Al-Khaled Isam Al-Darabsah Amer Darweesh Amro Alshare Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels Fractal and Fractional fractional partial integro-differential equation sinc-collocation method iterative Laplace transform method weakly singular kernel |
| title | Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels |
| title_full | Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels |
| title_fullStr | Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels |
| title_full_unstemmed | Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels |
| title_short | Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels |
| title_sort | analytical and numerical treatment of evolutionary time fractional partial integro differential equations with singular memory kernels |
| topic | fractional partial integro-differential equation sinc-collocation method iterative Laplace transform method weakly singular kernel |
| url | https://www.mdpi.com/2504-3110/9/6/392 |
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