{"title":"具有记忆核的变阶分数扩散过程的高效数值模拟","authors":"Sabita Bera , Mausumi Sen , Sujit Nath","doi":"10.1016/j.jocs.2025.102705","DOIUrl":null,"url":null,"abstract":"<div><div>Diffusion equations are fundamental in modeling the transport of heat, mass, or contaminants in porous media. However, classical models often fail to capture the anomalous diffusion behavior inherent in heterogeneous and memory-dependent materials. To address this, we investigate a fractional diffusion integro-differential equation involving variable-order derivatives in both time and space, subject to suitable conditions. The solutions are shown to exist and be unique through the rigorous application of fixed-point theorems. A finite difference-based numerical scheme is formulated to handle the variable-order fractional operators and convolution-type integral terms efficiently. Stability analysis confirms the accuracy and robustness of the method. In addition, approximate solutions are computed for three representative cases:(i) constant-order fractional diffusion (<span><math><mrow><mi>α</mi><mo>=</mo><mtext>constant</mtext></mrow></math></span>), (ii) time-dependent order <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>, and (iii) fully variable-order <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>. By incorporating variable order dynamics and integro-differential structures, this work extends conventional models and provides a unified framework for simulating complex transport processes in porous media.</div></div>","PeriodicalId":48907,"journal":{"name":"Journal of Computational Science","volume":"92 ","pages":"Article 102705"},"PeriodicalIF":3.7000,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Efficient numerical simulation of variable-order fractional diffusion processes with a memory kernel\",\"authors\":\"Sabita Bera , Mausumi Sen , Sujit Nath\",\"doi\":\"10.1016/j.jocs.2025.102705\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Diffusion equations are fundamental in modeling the transport of heat, mass, or contaminants in porous media. However, classical models often fail to capture the anomalous diffusion behavior inherent in heterogeneous and memory-dependent materials. To address this, we investigate a fractional diffusion integro-differential equation involving variable-order derivatives in both time and space, subject to suitable conditions. The solutions are shown to exist and be unique through the rigorous application of fixed-point theorems. A finite difference-based numerical scheme is formulated to handle the variable-order fractional operators and convolution-type integral terms efficiently. Stability analysis confirms the accuracy and robustness of the method. In addition, approximate solutions are computed for three representative cases:(i) constant-order fractional diffusion (<span><math><mrow><mi>α</mi><mo>=</mo><mtext>constant</mtext></mrow></math></span>), (ii) time-dependent order <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>, and (iii) fully variable-order <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>. By incorporating variable order dynamics and integro-differential structures, this work extends conventional models and provides a unified framework for simulating complex transport processes in porous media.</div></div>\",\"PeriodicalId\":48907,\"journal\":{\"name\":\"Journal of Computational Science\",\"volume\":\"92 \",\"pages\":\"Article 102705\"},\"PeriodicalIF\":3.7000,\"publicationDate\":\"2025-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Computational Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1877750325001826\",\"RegionNum\":3,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1877750325001826","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Efficient numerical simulation of variable-order fractional diffusion processes with a memory kernel
Diffusion equations are fundamental in modeling the transport of heat, mass, or contaminants in porous media. However, classical models often fail to capture the anomalous diffusion behavior inherent in heterogeneous and memory-dependent materials. To address this, we investigate a fractional diffusion integro-differential equation involving variable-order derivatives in both time and space, subject to suitable conditions. The solutions are shown to exist and be unique through the rigorous application of fixed-point theorems. A finite difference-based numerical scheme is formulated to handle the variable-order fractional operators and convolution-type integral terms efficiently. Stability analysis confirms the accuracy and robustness of the method. In addition, approximate solutions are computed for three representative cases:(i) constant-order fractional diffusion (), (ii) time-dependent order , and (iii) fully variable-order . By incorporating variable order dynamics and integro-differential structures, this work extends conventional models and provides a unified framework for simulating complex transport processes in porous media.
期刊介绍:
Computational Science is a rapidly growing multi- and interdisciplinary field that uses advanced computing and data analysis to understand and solve complex problems. It has reached a level of predictive capability that now firmly complements the traditional pillars of experimentation and theory.
The recent advances in experimental techniques such as detectors, on-line sensor networks and high-resolution imaging techniques, have opened up new windows into physical and biological processes at many levels of detail. The resulting data explosion allows for detailed data driven modeling and simulation.
This new discipline in science combines computational thinking, modern computational methods, devices and collateral technologies to address problems far beyond the scope of traditional numerical methods.
Computational science typically unifies three distinct elements:
• Modeling, Algorithms and Simulations (e.g. numerical and non-numerical, discrete and continuous);
• Software developed to solve science (e.g., biological, physical, and social), engineering, medicine, and humanities problems;
• Computer and information science that develops and optimizes the advanced system hardware, software, networking, and data management components (e.g. problem solving environments).