NUMERICAL SOLUTION OF FRACTIONAL ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATION

Tejas Sharma; Shreekant Pathak; Gargi Trivedi; Vishant Shah; Bhavyata Patel; Trupti Shah

doi:10.11948/20250026

2026 Volume 16 Issue 2

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Tejas Sharma, Shreekant Pathak, Gargi Trivedi, Vishant Shah, Bhavyata Patel, Trupti Shah. NUMERICAL SOLUTION OF FRACTIONAL ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 779-793. doi: 10.11948/20250026

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Tejas Sharma, Shreekant Pathak, Gargi Trivedi, Vishant Shah, Bhavyata Patel, Trupti Shah. NUMERICAL SOLUTION OF FRACTIONAL ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 779-793. doi: 10.11948/20250026

NUMERICAL SOLUTION OF FRACTIONAL ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATION

1.
Department of Mathematics, N V Patel College of Pure and Applied Sciences, The Charutar Vidya Mandal University, Vallabh Vidyanagar-388120, India
2.
Department of Applied Science and Humanities, Madhuben & Bhanubhai Patel Institute of Technology (MBIT), Anand, Vallabh Vidyanagar, Gujarat-388120, India
3.
Department of Applied Mathematics, Faculty of Technology and Engineering, The Maharaja Sayajirao University of Baroda, Vadodara-390001, India
4.
Department of Mathematics, Marwadi University, Rajkot-360003, Gujarat, India

Author Bio: Email: ts29121983@gmail.com(T. Sharma); Email: shreekant.pathak@cvmu.edu.in(S. Pathak); Email: vishantmsu83@gmail.com(V. Shah); Email: bhavyatapatel11@gmail.com(B. Patel); Email: trupti.p.shah-appmath@msubaroda.ac.in(T. shah)
Corresponding author: Email: gargi1488@gmail.com(G. Trivedi)

Abstract

Abstract

This article develops a finite element method (FEM) for solving nonlinear fractional-order partial differential equations (PDEs) using the Caputo fractional derivative. The scheme achieves second-order convergence in L² and L^∞ norms and is supported by a rigorous qualitative analysis, including existence, uniqueness, and Ulam-Hyers stability, established via nonlinear functional analysis and fixed-point theorems. Applied to a fractionalorder Burgers' equation modeling longitudinal dispersion in homogeneous porous media, the method demonstrates superior accuracy compared to finite difference and B-spline Galerkin methods. Numerical results validate robustness across fractional orders α ∈ {0.5, 0.75, 1.0}, with applications in environmental engineering and fluid dynamics.
- Finite element method /
- fractional-order PDEs /
- Burgers' equation /
- nonlinear PDEs /
- longitudinal dispersion
MSC: 65M60, 76S05, 35R11

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References

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