Advanced Transform Techniques for the One-Dimensional Non-Homogeneous Heat Equation with Non-Homogeneous BCs and IC

  • Authors

    • RAPHEAL OLADIPO FIFELOLA DEPARTMENT OF MATHEMATICAL SCIENCES, FACULTY OF SCIENCE, NIGERIAN DEFENCE ACADEMY
    • OKAFOR UCHENWA LINUS DEPARTMENT OF MATHEMATICAL SCIENCES, FACULTY OF SCIENCES, NIGERIAN DEFENCE ACADEMY.
    • ADEDAPO KEHINDE FEMI DEPARTMENT OF PHYSICAL AND CHEMICAL SCIENCES,FEDERAL UNIVERSITY OF HEALTH SCIENCES
    • JOHNSON SUNDAY EGBEJA DEPARTMENT OF MATHEMATICAL SCIENCES, FACULTY OF SCIENCES, NIGERIAN DEFENCE ACADEMY.
    2024-09-22
    https://doi.org/10.14419/kyyb9f56
  • Heat Equation, Boundary Conditions, Fourier Series, Non-Homogeneous PDE, Transformation Method.

  • Abstract

    This study addresses the one-dimensional non-homogeneous heat equation with non-homogeneous boundary conditions using a transformation method. We introduce a new dependent variable V(x,t) and a function ψ(x) to simplify the PDE into a homogeneous form, solving it analytically. The solution involves separating variables and applying Fourier series, leading to:

    Numerical simulations confirm the theoretical results, illustrating the method’s robustness for modeling heat conduction problems.

  • References

    1. Agarwal, R. P., & O'Regan, M. (2009). Generalized Integral Transformations. Springer.
    2. Berger, M. J., & Colella, J. O. (1989). Local adaptive mesh refinement for shock hydrodynamics. Journal of Computational Physics, 82(1), 64–84
    3. Çengel, Y. A. (2018). Heat Transfer: A Practical Approach. McGraw-Hill Education.
    4. Chung, T. J. (2002). Computational Fluid Dynamics. Cambridge University Press.
    5. d'Alembert, J. le Rond. (1747). Réflexions sur la cause générale des vents. Histoire de l'Académie Royale des Sciences.
    6. Duffy, D. G. (2015). Boundary Value Problems and Partial Differential Equations. Springer.
    7. Fourier, J. B. J. (1822). Théorie analytique de la chaleur. Firmin Didot.
    8. Ghosh, S., & Khanna, A. (2012). Mathematical Methods for Engineers and Scientists. Springer.
    9. Incropera, F. P., & DeWitt, D. P. (2002). Introduction to Heat Transfer. Wiley.
    10. Laplace, P. S. (1799). Mémoire sur la théorie des probabilités. Bachelier.
    11. Moin, P. (2010). Fundamentals of Computational Fluid Dynamics. Cambridge University Press.
    12. MIT OpenCourseWare. (2023). Linear Partial Differential Equations. Massachusetts Institute of Technology.
    13. Smith, B. A., Abel, T. W., & Howe, D. A. T. (2018). High-Performance Computing: An Introduction. CRC Press.
    14. Strauss, W. A. (2007). Partial Differential Equations: An Introduction. John Wiley & Sons.
    15. Wazwaz, A. M. (2011). Nonlinear Partial Differential Equations and Applications. Springer.
    16. Yagdjian, K. B. (2016). Integral Transforms and Special Functions. Springer.
    17. Zienkiewicz, O. C., Taylor, R. L., & Zhu, J. Z. (2013). The Finite Element Method: Its Basis and Fundamentals. Elsevier.

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  • How to Cite

    FIFELOLA, R. O., OKAFOR UCHENWA LINUS, ADEDAPO KEHINDE FEMI, & JOHNSON SUNDAY EGBEJA. (2024). Advanced Transform Techniques for the One-Dimensional Non-Homogeneous Heat Equation with Non-Homogeneous BCs and IC. International Journal of Applied Mathematical Research, 13(2), 96-102. https://doi.org/10.14419/kyyb9f56