Analytical Solution of One-Dimensional Nonlinear Conformable Fractional Telegraph Equation by Reduced Differential Transform Method

Abstract

In this paper, the reduced differential transform method (RDTM) is successfully implemented to obtain the analytical solution of the space-time conformable fractional telegraph equation subject to the appropriate initial conditions. The fractional-order derivative will be in the conformable (CFD) sense. Some properties which help us to solve the governing problem using the suggested approach are proven. The proposed method yields an approximate solution in the form of an infinite series that converges to a closed-form solution, which is in many cases the exact solution. This method has the advantage of producing an analytical solution by only using the appropriate initial conditions without requiring any discretization, transformation, or restrictive assumptions. Four test modeling problems from mathematical physics, conformable fractional telegraph equations in which we already knew their exact solution using other numerical methods, were taken to show the liability, accuracy, convergence, and efficiency of the proposed method, and the solution behavior of each illustrative example is presented using tabulated numerical values and two- and three-dimensional graphs. The results show that the RDTM gave solutions that coincide with the exact solutions and the numerical solutions that are available in the literature. Also, the obtained results reveal that the introduced method is easily applicable and it saves a lot of computational work in solving conformable fractional telegraph equations, and it may also find wide application in other complicated fractional partial differential equations that originate in the areas of engineering and science.

Approximate Analytical Solution of Two-Dimensional Nonlinear Time-Fractional Damped Wave Equation in the Caputo Fractional Derivative Operator

“…From Equation (11), the principle of the CFRDTM can be determined to be derived from the power series expansion.…”

“…Linear and nonlinear fractional diferential equations can successfully simulate fractional derivatives in a range of scientifc and technical domains, including electrical networks, chemical physics, control theory of dynamical systems, reaction-difusion, signal processing, and heat transform [1][2][3][4][5][6][7]. Because fractional diferential equations (FDEs) often exist in several felds of engineering and science, many researchers focus their eforts on obtaining exact/approximate solutions to these dynamic fractional diferential equations utilizing a variety of powerful established approaches, including the fnite diference method [8], Caputo fractional-reduced diferential transform method [9][10][11], Padé-Sumudu-Adomian decomposition method [12], triple Laplace transform method [13][14][15], double Sumudu transform iterative method [16], shifted Chebyshev polynomial-based method [17], Laplace decomposition method [18,19], homotopy analysis method [20], double Laplace transform method [21], homotopy perturbation method [22,23], conformable reduced differential transform method [24], conformable fractionalmodifed homotopy perturbation, Adomian decomposition method [25], diferential transform method [26][27][28], and the new function method based on the Jacobi elliptic functions [29].…”

Diverse soliton solutions to the nonlinear partial differential equations related to electrical transmission line

“…Widely applied in the analysis of electric communications, wave propagation phenomena, and cable transmission systems, this equation offers superior performance compared to the heat equation when predicting parabolic physical events. Its versatile applications span various sectors, including radio frequency, wireless signals, telephone lines, chemical kinetics, biological population dispersion, random walk, and microwave transmission [33,34]. In the realm of physics, nonlinear electrical transmission lines (NTLs) function as nonlinear dispersive media through which electrical solitons can propagate in the form of voltage waves.…”

Modified conformable double Laplace–Sumudu approach with applications

“…Numerous mathematicians and authors have created new methods to solve conformal differential problems, including the simplest method [26], Kudryashov method [27], double Shehu transform [28], Tanh method [29,30], reliable methods [31], double Sumudu transform [32,33], conformable Laplace transform (CLT) method [34,35], conformable double Laplace transform method [36], and conformable Sumudu transform (CST) method and others [37][38][39][40][41][42][43][44].…”