A Fourier wavelet series solution of partial differential equation through the separation of variables method

• Published in: Applied mathematics and computation Vol. 388; p. 125480
• Main Authors: Sokhal, Simran, Ram Verma, Sag
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2021
• Subjects: Compact support; Fourier series; Integral transform; Partial differential equations; Wavelets and its transform; Compact support; Wavelets and its transform; Partial differential equations; Integral transform; Fourier series
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2020.125480

•Consideration of Partial Differential Equations.•Introduction of an analytical method to get the Fourier-wavelet series solution for PDE’s.•Extensive discussion of error bound and convergence analysis of the method.•Analysis of the existence and boundedness of the solution.•Comparison of proposed method with existing method through illustrative examples. In the present study, a new approach such as Fourier wavelet series solution of partial differential equation through the method of separation of variables has been discussed. This approach includes the process by which the Fourier-wavelet coefficients are calculated, and how these coefficients are used in place of Fourier coefficients to attain the solution. Also, the bounds of these coefficients have been estimated. Convergence analysis and the existence of the Fourier-wavelet series are discussed here. Moreover, it is clearly shown that if the proposed series is exactly convergent, then the Fourier-wavelet and Fourier coefficients coincide. Additionally, the existence of the difference of two equivalent Fourier-wavelet series has been computed. Four illustrative examples have been included to certify the proposed method, which shows incredible performance.