DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020042 DYNAMICAL SYSTEMS SERIES S EXTENSION OF TRIPLE LAPLACE TRANSFORM FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS Amir Khan ∗ and Asaf Khan Department of Mathematics and Statistics, University of Swat Khyber Pakhtunkhwa, Pakistan Tahir Khan and Gul Zaman Department of Mathematics, University of Malakand, Chakadara Lower Dir, Khyber Pakhtunkhwa, Pakistan Abstract. In this article, we extend the concept of triple Laplace transform to the solution of fractional order partial differential equations by using Caputo fractional derivative. The concerned transform is applicable to solve many classes of partial differential equations with fractional order derivatives and integrals. As a consequence, fractional order telegraph equation in two di- mensions is investigated in detail and the solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. The same problem is also solved by taking into account the Atangana-Baleanu fractional derivative. Numerical plots are provided for the comparison of Caputo and Atangana-Baleanu fractional derivatives. 1. Introduction. The applications of differential equation is one of the interesting and most essential area in the field of engineering, physics and other branches of applied disciplines. Although for the solution of such problems involving differential equations, there are no common techniques. The integral transform technique is one of the greatest known scheme used by numbers of researchers for the solution of ordinary and partial differential equations [7, 8]. The double Laplace transform and Sumudu transform were used in [15, 18] for the solution of wave equation and Poisson equation. The area of fractional calculus, which has been attracted much attention in last few decades due to its enormous numbers of applications in almost all disciplines of applied sciences and engineering. The fractional calculus became an aspirant to find out the solution of complex systems exist in numerous fields in sciences, (see for detail [14, 26, 6]). In the theoretical and applied point of view large numbers of sweeping problems are existed in this region [10, 11, 27], which needs solution, for illustration, the physical significance of the fractional order derivative, etc. In the field of mathematical modeling having, partial derivatives of fractional order naturally seem in dealing with the generality of the current traditional models. Consequently there is a requirement to improve the existing literature from the traditional to the fractional calculus. At the primary view this procedure looks very simple and straight. In fact this is a complex procedure, which plentiful courtesy is 2010 Mathematics Subject Classification. 34A08, 35R11. Key words and phrases. Fractional integral, fractional derivative, fractional telegraph equation, triple Laplace transform. * Corresponding author: Amir Khan. 755