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We propose a powerful iteration scheme for solving analytically a class of partial equations with mixed derivatives. Our approach is based upon the Lagrange multiplier in two-dimensional spaces. The local convergence and uniqueness of the proposed method are analyzed. In order to demonstrate the applicability of our method, we present an algorithm to compute the solution for two examples.

In the recent decade, several scholars in the fields of partial differential equations have paid attention in showing the existence and the solutions of the class of partial differential equations involving mixed and nonmixed derivatives. Several methods were proposed, for instance, the Laplace transform method [

While doing a search in the literature, we noticed that there is a class of partial differential equations for which no analytical method or iteration method has been proposed to get to the bottom of their solutions. Without loss of generality, the general form of this class of equation is given below as

In this paper, our approach will be based upon the Lagrange multiplier in two-dimensional spaces. The local convergence and uniqueness of the proposed method will be analyzed in detail.

We devote this section to the discussion underpinning the general method to derive the special solution of (

We will assume that

We will in the coming section illustrate this extension by solving some problems with mixed derivatives. However, we will first deal with the convergence and uniqueness analysis of a specific equation (

The purpose of this section is to show the local convergence of the proposed method for solving an example of nonlinear equation and the uniqueness of the special solution obtained via the proposed method; we will therefore consider the following equation:

It is possible for us to find a positive constant say

To the extent that all

Let us consider

We will present the proof of this theorem by just verifying the Hypotheses

Using the definition of our operator

However, we can find a positive constant

Consider

Taking into account the initial conditions for (

Assuming that we can find another special solution, say

We will present in this section the application of this method for (

Consider

Consider the following:

Input:

Output:

We will in this subsection make use of the proposed algorithm to present the special solution.

We assume that the initial guest is given by

Special solution for

Let us consider the following partial differential equation:

Attention has not been paid to the class of partial differential equations with mixed derivatives only. But this class of partial differential equations is used to describe several physical occurrences or real world problems. More importantly, the nonlinear partial differential equations with mixed derivatives only cannot be handled with the commonly used analytical methods. Even some numerical methods [

The authors declare that there is no conflict of interests for this paper.

Abdon Atangana wrote the first draft. Both authors revised and corrected the final version.

The authors would like to thank the editor and anonymous reviewers for their valuable suggestions toward the enhancement of this paper. Abdon Atangana would like to thank the Claude Leon Foundation for their scholarship.