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Mathematics
Volume 13
Issue 21
10.3390/math13213549
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

A Space–Time Collocation Method Using the Method of Particular Solutions with Polynomial Basis Functions for Solving the Fisher–KPP Equation

by
Thir Dangal
1,*,
Balaram Khatri Ghimire
2 and
Anup Lamichhane
3
1
Department of Mathematics, Augusta University, Augusta, GA 30912, USA
2
Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 39104, USA
3
Department of Mathematics and Physics, Ohio Northern University, Ada, OH 45810, USA
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(21), 3549; https://doi.org/10.3390/math13213549
Submission received: 19 October 2025 / Revised: 2 November 2025 / Accepted: 3 November 2025 / Published: 5 November 2025
Versions Notes

Abstract

The method of particular solutions (MPS) has been widely applied for solving various types of partial differential equations. In this paper, the space–time collocation technique is implemented using MPS with polynomial basis functions (MPS-PBF) to solve the nonlinear Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation in both one and two dimensions. The Picard iteration method is used to deal with the nonlinearity of the problem. Four numerical examples are provided, and their results are compared with established methods to demonstrate the effectiveness of the proposed scheme.
Keywords: Fisher—KPP equation; polynomial basis functions; radial basis functions; collocation method; particular solutions Fisher—KPP equation; polynomial basis functions; radial basis functions; collocation method; particular solutions

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MDPI and ACS Style

Dangal, T.; Khatri Ghimire, B.; Lamichhane, A. A Space–Time Collocation Method Using the Method of Particular Solutions with Polynomial Basis Functions for Solving the Fisher–KPP Equation. Mathematics 2025, 13, 3549. https://doi.org/10.3390/math13213549

AMA Style

Dangal T, Khatri Ghimire B, Lamichhane A. A Space–Time Collocation Method Using the Method of Particular Solutions with Polynomial Basis Functions for Solving the Fisher–KPP Equation. Mathematics. 2025; 13(21):3549. https://doi.org/10.3390/math13213549

Chicago/Turabian Style

Dangal, Thir, Balaram Khatri Ghimire, and Anup Lamichhane. 2025. "A Space–Time Collocation Method Using the Method of Particular Solutions with Polynomial Basis Functions for Solving the Fisher–KPP Equation" Mathematics 13, no. 21: 3549. https://doi.org/10.3390/math13213549

APA Style

Dangal, T., Khatri Ghimire, B., & Lamichhane, A. (2025). A Space–Time Collocation Method Using the Method of Particular Solutions with Polynomial Basis Functions for Solving the Fisher–KPP Equation. Mathematics, 13(21), 3549. https://doi.org/10.3390/math13213549

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