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The aim of this study is to give a good strategy for solving some linear and nonlinear partial differential equations in engineering and physics fields, by combining Laplace transform and the modified variational iteration method. This method is based on the variational iteration method, Laplace transforms, and convolution integral, introducing an alternative Laplace correction functional and expressing the integral as a convolution. Some examples in physical engineering are provided to illustrate the simplicity and reliability of this method. The solutions of these examples are contingent only on the initial conditions.

Nonlinear equations are of great importance to our contemporary world. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. Despite the importance of obtaining the exact solution of nonlinear partial differential equations in physics and applied mathematics, there is still the daunting problem of finding new methods to discover new exact or approximate solutions.

In the recent years, many authors have devoted their attention to study solutions of nonlinear partial differential equations using various methods. Among these attempts are the Adomian decomposition method, homotopy perturbation method, variational iteration method [

Many analytical and numerical methods have been proposed to obtain solutions for nonlinear PDEs with fractional derivatives, such as local fractional variational iteration method [

In this work, we will use the new method termed He’s Laplace variational iteration method, and it will be employed in a straightforward manner.

Also, the main result of this paper is to introduce an alternative Laplace correction functional and express the integral as a convolution. This approach can tackle functions with discontinuities and impulse functions effectively.

To illustrate the idea of new Laplace variational iteration method, we consider the following general differential equations in physics:

According to the variational iteration method, we can construct a correction function for (

Also we can find the Lagrange multipliers easily by using integration by parts of (

Take Laplace transform of (

To find the optimal value of

In this section, we apply the Laplace variational iteration method for solving some linear and nonlinear partial differential equations in physics.

Consider the initial linear partial differential equation:

Consider the nonlinear partial differential equation:

Consider the physics nonlinear boundary value problem

The method of combining Laplace transforms and variational iteration method is proposed for the solution of linear and nonlinear partial differential equations. This method is applied in a direct way without employing linearization and is successfully implemented by using the initial conditions and convolution integral.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was funded by the Deanship of Scientific Research (DSR), King AbdulazizUniversity, Jeddah, under Grant no. (363-008-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.