In this paper, we combine the Elzaki transform method (ETM) with the new homotopy perturbation method (NHPM) for the first time. This hybrid approach can solve initial value problems numerically and analytically, such as nonlinear fractional differential equations of various normal orders. The Elzaki transform method (ETM) is used to solve nonlinear fractional differential equations, and then the homotopy is applied to the transformed equation, which includes the beginning conditions. To obtain the solution to an equation, we use the inverse transforms of the Elzaki transform method (ETM). The initial conditions have a big impact on the equation’s result. We give three beginning value issues that were solved as precise or approximation solutions with high rigor to demonstrate the method’s power and correctness. It is clear that solving nonlinear partial differential equations with the crossbred approach is the best alternative.

This study aims to obtain the approximate resolution of the fractional partial differential equation through the help of unprecedented new integral transform (NIT) called Elzaki Decomposition Method (EDM), applied to the new integral transform for fractional partial differential equations (PDEs). The technique converges to the right solution, fractional partial differential equations. Some examples were illustrated to confirm the accuracy of the method.

The objective of this paper is to compute the new modified method of variational iteration and the Laplace Adomian decomposition method for the solution of nonlinear fractional partial differential equations. We execute a comparatively newfangled analytical mechanism that is denoted by the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition method (LADM). The effect of the numerical results indicates that the double approximation is handy to execute and reliable when applied. It is shown that numerical solutions are gained in the form of approximately series which are facilely computable.

This paper intends to obtain accurate and convergent numerical solutions of linear space-time matching telegraph fractional equations by means of a double Sumudu matching transformation method. Moreover, the numerical model is equipped to explain the work, the accuracy of the work, and sobriety in its presentation method, and as a result, the proposed method shows an effective and convenient way, to employ proven problems in science and engineering.

In this research, we develop a new analytical technique based on the Elzaki transform (ET) to solve the fractional-order biological population model (FBPM) with initial and boundary conditions (ICs and BCs). This approach can be used to locate both the closed approximate solution and the exact solution of a differential equation. The usefulness and validity of this strategy for managing the solution of FBPM are demonstrated using a few real-world scenarios. The dependability of the suggested strategy is also shown using a table and a few graphs. The approximate solutions that were achieved and the convergence analysis are shown in numerical simulations in a range of fractional orders. From the numerical simulations, it can be seen that the population density increases with increasing fractional order, whereas the population density drops with decreasing fractional order.