The Elzaki Transform Method for Addressing Cauchy Problems in Higher Order Nonlinear PDEs

In this work, the new integral transform called the Elzaki transform (ET) is used to investigate and solve nonlinear higher-order partial differential equations (NHOPDEs), which serve as mathematical models in a range of practically significant disciplines of applied research. The NHOPDE solutions converge to exact solutions rather easily, were derived in a simple and easy-to-understand manner using ET. In addition, examples are given to illustrate how this method can be applied and how valid it is for the problem-solving form. There is a strong correlation between the analytical and exact solutions for the tested problems. This paper also covers the convergence of the ET technique to the exact solution of NHOPDEs. Numerical problems involving fourth and sixth order nonlinear hyperbolic equations and nonlinear wave-like equations with variable coefficients are solved to illustrate how the ET technique may efficiently yield accurate solutions for nonlinear PDEs of higher order with initial conditions. The results demonstrate the remarkable accuracy, efficiency, and dependability of the ET technique, which can be applied to a broad variety of nonlinear higher-order PDEs. This method greatly simplifies numerical calculations. The two primary goals of using this approach are to establish a fair frequency relationship and select an appropriate starting estimate. The precise, analytical, and numerical solutions to the examined problems show a high association with one another, further validating the robustness of this approach. Its unique properties, including its ability to simplify convolution operations and its close connection to the Laplace transform, also contribute to its effectiveness.