Research Article
Solutions of One Dimensional Parabolic Partial Differential Equations: An Improved Finite Difference Approach
Omowo Babajide Johnson*,
Adeniran Adebayo Oludare,
Adetunkasi Taiwo Flora,
Ogunbanwo Samson Tolulope,
Olatunji Olakunle Henry
Issue:
Volume 13, Issue 4, August 2025
Pages:
237-244
Received:
7 May 2025
Accepted:
6 June 2025
Published:
14 July 2025
Abstract: This paper introduces a finite difference scheme derived from the classical Crank-Nicolson method. The proposed scheme offer an improved spatial accuracy while maintaining the second-order temporal accuracy of the original Crank-Nicolson scheme. The higher order of spatial accuracy leads to improved convergence properties. The consistency and stability of the new scheme are analyzed using Taylor series expansion and von Neumann stability analysis, respectively. To validate the efficiency of the proposed scheme, it is implemented in MATLAB to solve the one-dimensional heat equation. To explore the versatility of the scheme, it is further extended to solve the advection-diffusion equation. Numerical experiments demonstrated on diffusion equation show that the new scheme compares favorably with existing methods in terms of convergence and accuracy. The results of the numerical solutions are presented in tabular form to highlight the accuracy and rates of convergence of the method. In addition, graphical plots of the numerical solutions are provided at different time levels to visualize the behavior of the solution over time and to illustrate the consistency between the numerical and analytical results. These visual and numerical comparisons further emphasize the reliability and precision of the proposed scheme. The combination of improved spatial resolution, solid theoretical foundation, and practical implementation demonstrates the schemeˆ as potential for solving time-dependent partial differential equations efficiently and accurately. This makes the scheme a valuable contribution to the field of numerical methods for parabolic-type equations.
Abstract: This paper introduces a finite difference scheme derived from the classical Crank-Nicolson method. The proposed scheme offer an improved spatial accuracy while maintaining the second-order temporal accuracy of the original Crank-Nicolson scheme. The higher order of spatial accuracy leads to improved convergence properties. The consistency and stabi...
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Research Article
Homotopy Perturbation Technique for Solving Higher Dimensional Time Fractional Burgers-Huxley Equations
Umesh Kumari*
,
Inderdeep Singh
Issue:
Volume 13, Issue 4, August 2025
Pages:
245-255
Received:
11 June 2025
Accepted:
14 July 2025
Published:
31 July 2025
DOI:
10.11648/j.ajam.20251304.12
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Views:
Abstract: Many real-world phenomena such as nerve pulse transmission, fluid transport, and chemical kinetics are modeled using nonlinear partial differential equations with time-fractional derivatives. Among them, the time-fractional Burgers-Huxley equation plays a significant role due to its ability to capture both diffusion and reaction mechanisms with memory effects. Solving such equations in higher dimensions is highly challenging and calls for efficient analytical approaches. In this work we present a technique for handling the time-fractional Burgers-Huxley equation up to k-dimensions by employing Laplace transform based homotopy perturbation method (LT-HPM). The LT-HPM is adopted based on its advantage in dealing with nonlinear terms and simplify the solution process by converting fractional derivatives into more manageable expressions. Unlike other hybrid approaches, LT-HPM is less computational complex and provides rapid convergent series solutions without requiring any linearization or restrictive assumptions. To showcase the effectiveness of this approach, we solve a pair of examples: a 2-D case and a 3-D case. The obtained results confirm that LT-HPM is accurate and powerful in tackling complex nonlinear PDEs in higher dimensions.
Abstract: Many real-world phenomena such as nerve pulse transmission, fluid transport, and chemical kinetics are modeled using nonlinear partial differential equations with time-fractional derivatives. Among them, the time-fractional Burgers-Huxley equation plays a significant role due to its ability to capture both diffusion and reaction mechanisms with mem...
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