# Homotopy Analysis Technique for a Generalised (1+1)-Dimensional KdV Equation of Variable Coefficients

## TL;DR

This paper applies the homotopy analysis technique to find exact solutions for a generalized (1+1)-dimensional variable coefficient KdV equation, demonstrating convergence and solution behaviors.

## Contribution

It introduces a novel application of homotopy analysis to solve a generalized nonlinear KdV equation with variable coefficients.

## Key findings

- Solutions exhibit convergent behavior.
- Two cases of linear parts analyzed.
- Plots confirm solution convergence.

## Abstract

In this work, an exact solution to a new generalized nonlinear KdV partial differential equations has been investigated using homotopy analysis techniques. The mentioned partial differential equation has been solved using homotopy perturbation method (HPM). In details, the study is divided into two cases; the first case is that the linear part is the velocity (the first derivative with respect to time), and the initial guess was chosen at the initial time or the boundary values. Next, the other case was that the linear part consists of many linear terms from the equation, the chosen terms constructed a partial differential equation whose solution can be gained easily. Eventually, some plots of the found solutions have demonstrated the convergent behaviors of the solutions.