Solving Heat Equations via Double Mohand Transform
DOI:
https://doi.org/10.48165/gjs.2025.2106Keywords:
Heat Equation, Partial Differential, Equations, Double Mohand, Transform MethodAbstract
Partial Differential Equations are vital in applied mathematics, chemistry, and physics since they model a variety of phenomena including the dynamics of fluids, the conduction of heat, and the propagation of waves. Due to their wide-ranging applications, PDEs have received a lot of attention for the development of efficient and accurate solution methodologies. Most of these attempts stress on accomplishing symmetry that simplifies the equations and facilitates easier solutions, whether analytical or numerical, to be found. To offer a different and efficient method to solving PDEs, we introduce in this article the Double Mohand Transform Method, an innovative integral transform method. This innovative double transform technique builds upon former integral transforms by incorporating vital elements that enhance solution efficiency and applicability. In this work, we explain comprehensively the double Mohand transform with special focus on its defining features, advantages, and unique qualities. In a more theoretical context, we present new results regarding its foundations and the partial derivatives of the transform. To explore the importance as well as effectiveness of the Double Mohand Transform Technique, we apply it on some cases of the heat equation with different boundary and initial conditions. By exploiting the symmetry features of the heat equation, our method correctly deals with complex PDEs by providing elegant and efficient solutions. The outcomes illustrate how this novel transformation could further progress the mathematical methods for differential equations applicable in various branches of science and engineering.
References
[1] El-Ajou, A., Abu Arqub, O., & Momani, S. (2012). Homotopy analysis method for second-order boundary value problems of integro-differential equations. Discrete Dynamics in Nature and Society, 2012, 365792.
[2] Saadeh, A., Qazza, K., & Amawi, A. (2022). New approach using integral transform to solve cancer models. Fractal and Fractional, 6, 490.
[3] Powers, D. L. (2014). Boundary value problems. Amsterdam, The Netherlands: Elsevier.
[4] Burqan, A., Saadeh, R., Qazza, A., & Momani, S. (2022). ARA-residual power series method for solving partial fractional differential equations. Alexandria Engineering Journal, 62, 47–62.
[5] Griffiths, D. F., & Higham, D. J. (2010). Numerical methods for ordinary differential equations: Initial value problems. London, UK: Springer.
[6] Ahmad, H., & Khan, T. A. (2020). Variational iteration algorithm I with an auxiliary parameter for the solution of differential equations of motion for simple and damped mass–spring systems. Noise & Vibration Worldwide, 51(1–2), 12–20.
[7] Abouelregal, A. E., Ahmad, H., & Yao, S. W. (2020). Functionally graded piezoelectric medium exposed to a movable heat flow based on a heat equation with a memory-dependent derivative. Materials, 13(18), 3953.
[8] Moa'ath, N. O., El-Ajou, A., Al-Zhour, Z., Alkhasawneh, R., & Alrabaiah, H. (2020). Series solutions for nonlinear time-fractional Schrödinger equations: Comparisons between conformable and Caputo derivatives. Alexandria Engineering Journal, 59, 2101–2114.
[9] Wang, F., Ahmad, I., Ahmad, H., Alsulami, M. D., Alimgeer, K. S., Cesarano, C., & Nofal, T. A. (2021). Meshless method based on RBFs for solving three-dimensional multi-term time fractional PDEs arising in engineering phenomenons. Journal of King Saud University–Science, 33(8), 101604.
[10] Saadeh, R., Burqan, A., & El-Ajou, A. (2022). Reliable solutions to fractional Lane-Emden equations via Laplace transform and residual error function. Alexandria Engineering Journal, 61, 10551–10562.
[11] Ahmed, S. A., Qazza, A., & Saadeh, R. (2022). Exact solutions of nonlinear partial differential equations via the new double integral transform combined with iterative method. Axioms, 11, 247.
[12] Harjunkoski, I., Maravelias, C. T., Bongers, P., Castro, P. M., Engell, S., Grossmann, I. E., Hooker, J., Méndez, C., Sand, G., & Wassick, J. (2014). Scope for industrial applications of production scheduling models and solution methods. Computers and Chemical Engineering, 62, 161–193.
[13] Karbalaie, A., Muhammed, H. H., Shabani, M., & Montazeri, M. M. (2014). Exact solution of partial differential equation using homoseparation of variables. International Journal of Nonlinear Science, 17, 84–90.
[14] Larsson, S., & Thomée, V. (2003). Partial differential equations with numerical methods. Berlin/Heidelberg, Germany: Springer.
[15] Saadeh, R., Al-Smadi, M., Gumah, G., Khalil, H., & Khan, R. A. (2016). Numerical investigation for solving two-point fuzzy boundary value problems by reproducing kernel approach. Applied Mathematics and Information Science, 10, 2117–2129.
[16] You, Y. L., & Kaveh, M. (2000). Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing, 9, 1723–1730.
[17] Qazza, A., Burqan, A., Saadeh, R., & Khalil, R. (2022). Applications on double ARA-Sumudu transform in solving fractional partial differential equations. Symmetry, 14, 1817.
[18] Ahmed, S., Elzaki, T., Elbadri, M., & Mohamed, M. Z. (2021). Solution of partial differential equations by new double integral transform (Laplace—Sumudu transform). Ain Shams Engineering Journal, 12, 4045–4049.
[19] Saadeh, R., Qazza, A., & Burqan, A. (2022). On the double ARA-Sumudu transform and its applications. Mathematics, 10, 2581.
[20] Ahmed, S., Elzaki, T., & Hassan, A. A. (2020). Solution of integral differential equations by new double integral transform (Laplace-Sumudu transform). Journal of Abstract and Applied Analysis, 2020, 4725150.
[21] Elnaqeeb, T., Shah, N. A., & Vieru, D. (2020). Weber-type integral transform connected with Robin-type boundary conditions. Mathematics, 8, 1335.
[22] Butzer, P. L., & Jansche, S. (1997). A direct approach to the Mellin transform. Journal of Fourier Analysis and Applications, 3, 325–376.
[23] Sullivan, D. M. (1996). Z-transform theory and the FDTD method. IEEE Transactions on Antennas and Propagation, 44, 28–34.
[24] Layman, J. W. (2001). The Hankel transform and some of its properties. Journal of Integer Sequences, 4, 1–11.
[25] Chauhan, R., Kumar, N., & Aggarwal, S. (2019). Dualities between Laplace-Carson transform and some useful integral transforms. International Journal of Innovative Technology and Exploring Engineering, 8, 1654–1659.
[26] Belgacem, F. B., & Karabali, A. A. (2006). Sumudu transform fundamental properties investigations and applications. International Journal of Stochastic Analysis, 2006, 091083.
[27] Mohand, M., & Mahgoub, A. (2017). The new integral transform "Mohand transform". Advances in Theoretical and Applied Mathematics, 12(2), 113–120.
[28] Ullah, S., & Aggarwal, S. (2024). Properties and application of double Mohand transform. Journal of Advanced Research in Applied Mathematics and Statistics, 9, 1–12.
[29] Saadeh, R., Sedeq, A. K., Ghazal, B., & Gharib, G. (2023). Double formable integral transform for solving heat equations. Symmetry, 15, 218.
