Solutions to Differential-Differential Difference Equations with Variable Coefficients by Using Fourier Transform Method

Year 2023, Volume: 18 Issue: 3, 259 - 267, 23.11.2023

Murat Düz , Sunnet Avezov , Ahmad Issa

https://doi.org/10.29233/sdufeffd.1318890

Abstract

In this paper, differential-differential difference equations with variable coefficients have been solved using the Fourier Transform Method (FTM). In addition, new definitions and theorems are introduced. Besides, the efficiency of the proposed method is verified by solving five important examples. Furthermore, we have noted that the Fourier transform method is a powerful technique for solving ordinary differential difference equations (ODDEs) with variable coefficients. It involves transforming the ODDEs into the frequency domain using the Fourier transform, solving the transformed equation, and then applying the inverse Fourier transform to obtain the solution in the time domain.

Keywords

Linear differential equation, Variable coefficients, Fourier transform, Dirac delta function

References

• T. M. Elzaki and S. M. Elzaki, “On the Elzaki transform and ordinary differential equation with variable coefficients”, Advances in Theoretical and Applied Mathematics, 6(1), 41-46, 2011.
• S. Aggarwal, N. Sharma, R. Chauhan, A. R. Gupta and A. Khandelwal, “A new application of Mahgoub transform for solving linear ordinary differential equations with variable coefficients”, Journal of Computer and Mathematical Sciences, 9(6), 520-525, 2018.
• M. Düz, A. Issa and S. Avezov, “A new computational technique for Fourier transforms by using the Differential transformation method”, Bulletin of International Mathematical Virtual Institute, 12(2), 287-295, 2022.
• Osgood, “The Fourier transform and its applications”, Lecture notes for EE, 261, 2009, pp. 20.
• N. Wheeler, Simplified Production of Dirac Delta Function Identities, Reed College, 1997.
• M. Sezer and A. Akyüz-Daşcıoğlu, “Taylor polynomial solutions of general linear differential–difference equations with variable coefficients”, Applied Mathematics and Computation, 174(2), 1526-1538, 2006.
• K.L. Cooke, “Differential Difference Equations”, New York, Academic Press, 1963.
• A. Arikoglu and I. Ozkol, “Solution of differential–difference equations by using differential transform method”, Applied Mathematics and Computation, 181(1), 153-162, 2006.
• J. K. Zhou, “Differential Transformation and Its Applications for Electrical Circuits”, Wuhan, Huazhong University Press, 1986.
• L. Zou, Z. Wang and Z. Zong, “Generalized differential transform method to differential-difference equation”, Physics Letters A, 373(45), 4142-4151, 2009.