The Exact Solutions of the Space and Time Fractional Telegraph Equations by the Double Sadik Transform Method

Abstract

The double integral transform is a robust implementation that is important in handling scientific and engineering problems. Besides its simplicity of use and straightforward application to the issue, the ability to reduce the problems to an algebraic equation that can be easily solved is a substantial advantage of the tool. Among the several integral transforms, the double Sadik transform is acknowledged to be one of the most frequently used in solving differential and integral equations. This work deals with investigating a generalized double integral transform called the double Sadik transform. The proof of the double Sadik transforms for partial fractional derivatives in the Caputo sense is displayed, and the double Sadik transforms method is introduced. The method has been applied to solve the initial boundary value problems for linear space and timefractional telegraph equations. Moreover, the suggested strategy can be used on non-linear problems via an iterative method and a decomposition concept. Some known-solution questions are evaluated with relatively minimal computational cost. The results are represented by utilizing the Mittag-Leffler function and covering the solution of a classical telegraph equation. The obtained exact solutions not only show the accuracy and efficiency of the technique, but also reveal reliability when compared to those obtained using other methods.

Exploring the Remarkable Properties of the Double Sadik Transform and Its Applications to Fractional Caputo Partial Differential Equations

“…The proof of the theorem can be found in [26] Theorem 3.7. [26] Let S 2 [f (x, t)] = F (w , v : α, β) then the double Sadik transform for the partial fractional Caputo derivatives are…”

“…Furthermore, the Laplace-ARA transform, which combines a single Laplace transform with an ARA transform, has been extensively studied, particularly its application to partial differential equations [25]. The recently introduced double Sadik transform has shown promise in solving fractional partial differential equations arising in scientific models [26]. Unfortunately, certain important properties of double integral transforms have not yet received formal acknowledgment or recognition.…”