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Related Experiment Video

Updated: Jul 14, 2025

An Experimental and Finite Element Protocol to Investigate the Transport of Neutral and Charged Solutes across Articular Cartilage

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An efficient spline technique for solving time-fractional integro-differential equations.

Muhammad Abbas¹, Sadia Aslam¹, Farah Aini Abdullah²

¹Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan.

|October 9, 2023

Summary

This summary is machine-generated.

This study introduces an extended cubic B-spline (ExCuBS) method for solving fractional partial integro-differential equations. The novel approach provides a stable and convergent numerical solution for complex mathematical models.

Keywords:

Approximation Caputo's fractional derivative ExCuBS Finite difference scheme Fractional order partial integro-differential equation Stability and convergence

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Area of Science:

Numerical Analysis
Applied Mathematics
Computational Science

Background:

Spline curves are essential in mathematics for approximating complex structures.
Fractional partial integro-differential equations (FPIDE) model various phenomena but are challenging to solve.
Weakly singular kernels (SK) present additional complexity in these equations.

Purpose of the Study:

To develop a numerical method for solving non-linear fractional partial integro-differential equations (FPIDE) with weakly singular kernels (SK).
To utilize extended cubic B-spline (ExCuBS) functions and a novel second-order derivative approximation.

Main Methods:

Discretization of spatial fractional derivatives using extended cubic B-spline (ExCuBS) functions.
Discretization of temporal fractional derivatives employing the Caputo finite difference scheme.
Implementation of a new second-order derivative approximation for enhanced accuracy.

Main Results:

The proposed ExCuBS method demonstrates stability and convergence for solving FPIDE.
Numerical solutions obtained through the ExCuBS method are validated against existing literature.
The approach effectively approximates complex structures within curved designs.

Conclusions:

The extended cubic B-spline (ExCuBS) method offers a robust and accurate technique for solving non-linear FPIDE with SK.
The study confirms the stability and convergence properties of the proposed numerical scheme.
This method provides a valuable tool for researchers working with fractional calculus and its applications.