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.
| Published in | American Journal of Applied Mathematics (Volume 13, Issue 4) |
| DOI | 10.11648/j.ajam.20251304.11 |
| Page(s) | 237-244 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Finite Difference Method, Convergence, Taylor’expansion, von-Newmanns Stability, Heat Equation, Advection-diffusion Equation
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APA Style
Johnson, O. B., Oluudare, A. A., Flora, A. T., Tolulope, O. S., Henry, O. O. (2025). Solutions of One Dimensional Parabolic Partial Differential Equations: An Improved Finite Difference Approach. American Journal of Applied Mathematics, 13(4), 237-244. https://doi.org/10.11648/j.ajam.20251304.11
ACS Style
Johnson, O. B.; Oluudare, A. A.; Flora, A. T.; Tolulope, O. S.; Henry, O. O. Solutions of One Dimensional Parabolic Partial Differential Equations: An Improved Finite Difference Approach. Am. J. Appl. Math. 2025, 13(4), 237-244. doi: 10.11648/j.ajam.20251304.11
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Download@article{10.11648/j.ajam.20251304.11,
author = {Omowo Babajide Johnson and Adeniran Adebayo Oluudare and Adetunkasi Taiwo Flora and Ogunbanwo Samson Tolulope and Olatunji Olakunle Henry},
title = {Solutions of One Dimensional Parabolic Partial Differential Equations: An Improved Finite Difference Approach
},
journal = {American Journal of Applied Mathematics},
volume = {13},
number = {4},
pages = {237-244},
doi = {10.11648/j.ajam.20251304.11},
url = {https://doi.org/10.11648/j.ajam.20251304.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251304.11},
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.
},
year = {2025}
}
TY - JOUR T1 - Solutions of One Dimensional Parabolic Partial Differential Equations: An Improved Finite Difference Approach AU - Omowo Babajide Johnson AU - Adeniran Adebayo Oluudare AU - Adetunkasi Taiwo Flora AU - Ogunbanwo Samson Tolulope AU - Olatunji Olakunle Henry Y1 - 2025/07/14 PY - 2025 N1 - https://doi.org/10.11648/j.ajam.20251304.11 DO - 10.11648/j.ajam.20251304.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 237 EP - 244 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20251304.11 AB - 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. VL - 13 IS - 4 ER -
Department of General Studies (Mathematics Unit), Federal Polytechnic, Ile-Oluji, Nigeria
Department of General Studies (Mathematics Unit), Federal Polytechnic, Ile-Oluji, Nigeria
Department of Statistics, Federal Polytechnic, Ile-Oluji, Nigeria
Department of General Studies (Mathematics Unit), Federal Polytechnic, Ile-Oluji, Nigeria
