Abstract
In this study, a hybrid technique for improving the differential transform method (DTM), namely the modified differential transform method (MDTM) expressed as a combination of the differential transform method, Laplace transforms, and the Padé approximant (LPDTM) is employed for the first time to ascertain exact solutions of linear and nonlinear pantograph type of differential and Volterra integro-differential equations (DEs and VIDEs) with proportional delays. The advantage of this method is its simple and trusty procedure, it solves the equations straightforward and directly without requiring large computational work, perturbations or linearization, and enlarges the domain of convergence, and leads to the exact solution. Also, to validate the reliability and efficiency of the method, some examples and numerical results are provided.
Highlights
Mathematical modeling of various phenomena in science and engineering such as biological population management, chemistry, physics, physiological and pharmaceutical kinetics and chemical kinetics, medicine, infectious diseases, economy, nonlinear dynamical system, communication networks, number theory, electrodynamics, the navigational control of ships and aircraft and control problems and electronic systems leads to one of the most important kinds of delay differential equations (DDEs), namely pantograph equation [1,2,3,4,5,6,7]
We present the application of the modified differential transform method (MDTM) as a hybrid approach, for improving DTM’s truncated series solutions in convergence rate combining DTM, Laplace transforms, and Padé approximant
Using the inverse Laplace transform on the Padé approximants (5.22), we arrive at an improved solution that corresponds to the exact solution u(t) = e–t cos(t)
Summary
Mathematical modeling of various phenomena in science and engineering such as biological population management, chemistry, physics, physiological and pharmaceutical kinetics and chemical kinetics, medicine, infectious diseases, economy, nonlinear dynamical system, communication networks, number theory, electrodynamics, the navigational control of ships and aircraft and control problems and electronic systems leads to one of the most important kinds of delay differential equations (DDEs), namely pantograph equation [1,2,3,4,5,6,7]. We present the application of the modified differential transform method (MDTM) as a hybrid approach, for improving DTM’s truncated series solutions in convergence rate combining DTM, Laplace transforms, and Padé approximant.
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