A Computational Analysis of Nonlinear Fractional Partial Integro-differential Equation Using Meshfree Multiquadric Radial Basis Function Method

Arshed Ali; Fasiha Shaheen; Imtiaz Ahmad; Hadia Atta

Authors

Arshed Ali Department of Mathematics, Islamia College Peshawar, Pakistan
Fasiha Shaheen Department of Mathematics, Islamia College Peshawar, Pakistan
Imtiaz Ahmad Institute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional (UNITEN), Kajang, Selangor, Malaysia
Hadia Atta Department of Mathematics, Islamia College Peshawar, Pakistan

Keywords:

Fractional Partial Integro-Differential Equation, Caputo Derivative, Weakly Singular Kernel, Meshfree Method, Radial Basis Functions, Integral Operator

Abstract

Fractional partial integro-differential equations play an important role in describing physical and engineering systems that exhibit memory and nonlocal effects. Their nonlinear structure and the presence of weakly singular kernels make analytical solutions difficult to obtain, which highlights the need for accurate and flexible numerical strategies. This study develops a meshfree computational method based on multiquadric radial basis functions for solving a nonlinear fractional partial integro-differential equation involving the Caputo derivative. The temporal discretization is carried out using a backward difference formula, and the spatial operators are approximated through radial basis function interpolation. The resulting scheme avoids mesh generation and is suitable for irregular or scattered spatial nodes. Numerical experiments are presented to illustrate the accuracy, reliability, and efficiency of the method for representative test problems. The results indicate that the proposed meshfree approach provides a robust tool for nonlinear fractional models with weakly singular kernels.

References

I. Podlubny, “Fractional Differential Equation,” Math. Sci. Eng., vol. 198, 1999, [Online]. Available: https://www.researchgate.net/publication/249993249_Fractional_Differential_Equations_and_Their_Applications

J. L. Suzuki, M. Gulian, M. Zayernouri, and M. D’Elia, “Fractional Modeling in Action: a Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials,” J. Peridynamics Nonlocal Model. 2022 53, vol. 5, no. 3, pp. 392–459, Oct. 2022, doi: 10.1007/S42102-022-00085-2.

I. A. Imtiaz Ahmad, “Solutions of a three-dimensional multi-term fractional anomalous solute transport model for contamination in groundwater,” plos one, 2023, doi: https://doi.org/10.1371/journal.pone.0294348.

D. B. Zaid Odibat, “On a New Modification of the Erdélyi–Kober Fractional Derivative,” Fractal Fract, vol. 5, no. 3, p. 121, 2021, doi: https://doi.org/10.3390/fractalfract5030121.

A. A. B. Imtiaz Ahmad, “Investigating Virus Spread Analysis In Computer Networks With Atangana–Baleanu Fractional Derivative Models,” Fractals, vol. 32, 2024, doi: https://doi.org/10.1142/S0218348X24400437.

R. A. K. Arshed Ali, “A Computational Simulation of Fractional Advection-Diffusion Model Using Differential Quadrature and Local Radial Basis Functions,” Int. J. Agric. Sustain. Dev., vol. 7, no. 2, 2025, doi: 10.33411/ijist/202572926938.

Imtiaz Ahmad, Arshed Ali, “An Efficient RBF-Based Collocation Approach for Second-Order Fractional Partial Integro-Differential Problems with Singular Kernel,” J. Appl. Comput. Mech., 2025, doi: 10.22055/jacm.2025.48574.5336.

S. Zaeri, H. Saeedi, and M. Izadi, “Fractional integration operator for numerical solution of the integro-partial time fractional diffusion heat equation with weakly singular kernel,” https://doi.org/10.1142/S1793557117500711, vol. 10, no. 4, Oct. 2017, doi: 10.1142/S1793557117500711.

S. Arshed, “B-spline solution of fractional integro partial differential equation with a weakly singular kernel,” Numer. Methods Partial Differ. Equ., vol. 33, no. 5, pp. 1565–1581, Sep. 2017, doi: 10.1002/NUM.22153;JOURNAL:JOURNAL:10982426;WGROUP:STRING:PUBLICATION.

H. K. Mehwish Saleem, Arshed Ali, Fazal-i-Haq, “Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods,” Partial Differ. Equations Appl. Math., vol. 15, p. 101223, 2025, doi: https://doi.org/10.1016/j.padiff.2025.101223.

Z. A. Tayyaba Akram, “A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel,” Fractal Fract, vol. 5, no. 3, p. 85, 2021, doi: https://doi.org/10.3390/fractalfract5030085.

J. Guo, D. Xu, and W. Qiu, “A finite difference scheme for the nonlinear time-fractional partial integro-differential equation,” Math. Methods Appl. Sci., vol. 43, no. 6, pp. 3392–3412, Apr. 2020, doi: 10.1002/MMA.6128;SUBPAGE:STRING:ABSTRACT;WEBSITE:WEBSITE:PERICLES;ISSUE:ISSUE:DOI.

T. A. Imtiaz Ahmad, Mehwish Saleem, Arshed Ali, Aziz Khan, “A Computational Analysis of Convection-Diffusion Model with Memory using Caputo-Fabrizio Derivative and Cubic Trigonometric B-Spline Functions,” Eur. J. Pure Appl. Math., vol. 18, no. 2, 2025, [Online]. Available: https://www.ejpam.com/index.php/ejpam/article/view/6186

E. S. M. Aslefallah, “A Nonlinear Partial Integro-differential Equation Arising in Population Dynamic Via Radial Basis Functions and Theta-method,” J. Math. Comput. Sci., vol. 13, no. 1, pp. 14–25, 2014, doi: http://dx.doi.org/10.22436/jmcs.013.01.02.

“Numerical simulation of nonlinear parabolic type Volterra partial integro-differential equations using quartic B-spline collocation method,” Nonlinear Stud., vol. 27, no. 3, 2020, [Online]. Available: https://nonlinearstudies.com/index.php/nonlinear/article/view/1688

F. Mirzaee and S. Alipour, “Fractional-order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro-differential equations,” Math. Methods Appl. Sci., vol. 42, no. 6, pp. 1870–1893, Apr. 2019, doi: 10.1002/MMA.5481.

S. K. Hayman Thabet, “Numerical Analysis of Iterative Fractional Partial Integro-Differential Equations,” J. Math., 2022, doi: https://doi.org/10.1155/2022/8781186.

G. R. Liu, “Meshfree Methods: Moving Beyond the Finite Element Method, Second Edition,” CRC Press, p. 792, 2009, [Online]. Available: https://books.google.com.pk/books/about/Meshfree_Methods.html?id=KsaNEQAAQBAJ&redir_esc=y

T. Belytschko, Y. Y. Lu, and L. Gu, “Element‐free Galerkin methods,” Int. J. Numer. Methods Eng., vol. 37, no. 2, pp. 229–256, Jan. 1994, doi: 10.1002/NME.1620370205;REQUESTEDJOURNAL:JOURNAL:10970207;PAGE:STRING:ARTICLE/CHAPTER.

K. S. P. Lancaster, “Surfaces generated by moving least squares methods,” Math. Comput., vol. 37, no. 155, pp. 141–158, 1981, [Online]. Available: https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616367-1/

Greg Fasshauer, “Meshfree Approximation Methods with Matlab, Lecture 3: Dealing with ill-conditioned RBF systems,” Dolomites Res. Notes Approx., 2008, [Online]. Available: https://doaj.org/article/427ab33f8fde422b953a6e15c2df49af

M. A. Imtiaz Ahmad, “Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method,” Symmetry (Basel)., vol. 11, no. 3, p. 394, 2019, doi: https://doi.org/10.3390/sym11030394.

M. N. K. Phatiphat Thounthong, “Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations,” Mathematics, vol. 6, no. 12, p. 327, 2018, doi: https://doi.org/10.3390/math6120327.

I. A. Jun Feng Li, “Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method,” Open Phys., vol. 18, no. 1, 20201, [Online]. Available: https://www.degruyterbrill.com/document/doi/10.1515/phys-2020-0222/html?lang=en&srsltid=AfmBOoqgKblEVke-X6PUlp8bi091uAVHk4WcFuClatX9Y_OQ7PQ8FkmS

I. Ahmad, H. Ahmad, A. E. Abouelregal, P. Thounthong, and M. Abdel-Aty, “Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences,” Eur. Phys. J. Plus 2020 1359, vol. 135, no. 9, pp. 759-, Sep. 2020, doi: 10.1140/EPJP/S13360-020-00784-Z.

J. Z. Fuzhang Wang, “A Novel Meshfree Strategy for a Viscous Wave Equation With Variable Coefficients,” Front. Phys., vol. 9, 2021, doi: https://doi.org/10.3389/fphy.2021.701512.

A. H. A. Imtiaz Ahmad, Asmidar Abu Bakar, Ihteram Ali, Sirajul Haq, Salman Yussof, “Computational analysis of time-fractional models in energy infrastructure applications,” Alexandria Eng. J., vol. 82, pp. 426–436, 2023, doi: https://doi.org/10.1016/j.aej.2023.09.057.

H. Ahmad et al., “A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models,” AIMS Math. 2023 819677, vol. 8, no. 8, pp. 19677–19698, 2023, doi: 10.3934/MATH.20231003.

null S.-I. Imtiaz Ahmad, “Local meshless differential quadrature collocation method for time-fractional PDEs,” Discret. Contin. Dyn. Syst. - S, vol. 13, no. 10, 2020, [Online]. Available: https://www.aimsciences.org/article/doi/10.3934/dcdss.2020223

E.J. Kansa, “Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates,” Comput. Math. with Appl., vol. 19, no. 8–9, pp. 127–145, 1990, doi: https://doi.org/10.1016/0898-1221(90)90270-T.

Martin D. Buhmann, “Radial Basis Functions: Theory and Implementations,” Cambridge Univ. Press, p. 259, 2003, [Online]. Available: https://books.google.com.pk/books/about/Radial_Basis_Functions.html?id=TRMf53opzlsC&redir_esc=y