In this article, we want to analytically solve two examples of representative nonlinear differential equations in the fields of engineering and basic sciences using the AGM method. These are two types of complex non-linear differential equations called Langmuir Blodgett and Langmuir Boguslavski equations, which so far few people in the research department have been able to solve analytically, or they have been able to solve them analytically with difficulty. We will be able to solve these two types. We solve the nonlinear hard equation analytically using the AGM method. Besides, the methodology behind these techniques is completely understandable and easy to use, and users with common knowledge of mathematics will be capable of solving the most complicated equations at a low calculation cost. As all experts know, most of the actual system behavior in engineering is a nonlinear process, and analytical scrutiny of these nonlinear problems is difficult or sometimes impossible. Our purpose is to enhance the ability to solve the mentioned nonlinear differential equations in chemical engineering and similar issues with a simple and innovative approach entitled “Akbari Ganji Method” or "AGM". He’s Amplitude Frequency Formulation Method [1-3], which was first presented by Ji-Huan He gives convergent successive approximations of the exact solution and the Homotopy perturbation technique HPM [4,5]. It is necessary to mention that the above methods do not have this ability to gain the solution of the presented problem with high precision and accuracy, so nonlinear differential equations such as the presented problem in this case study should be solved by utilizing new approaches like AGM [6–13] that were created by Mohammadreza Akbari. In recent years, analytical methods for solving nonlinear differential equations have been presented and created by Mohammadreza Akbari. These methods are called AKLM [14] (Akbari Kalantari Leila Method), ASM [15, 16] (Akbari Sara’s Method), AYM [17-19] (Akbari Yasna’s Method), and IAM [20] (Integral Akbari Method). These examples somehow can be considered complicated cases to deal with for all of the existing analytical methods, especially in design slide engineering, which means old methods cannot resolve them precisely or even solve them in a real domain. These examples somehow can be considered complicated cases to deal with for all of the existing analytical methods, especially in the design of heat engineering, which means old methods cannot resolve them precisely or even solve them in a real domain.

Boundary conditions and initial conditions are required for analytical methods of each linear and nonlinear differential equation according to the physics of the problem. Therefore, we can solve every differential equation with any degree. In order to comprehend the given method in this section, a differential equation governing engineering processes will be solved in this new manner. In accordance with the boundary conditions, the general manner of a nonlinear differential equation is as follows:

To solve the first differential equation Eq. (1) with respect to the boundary conditions in Eq. (2), the series of letters in the nth order with constant coefficients, which is the answer to the first differential equation, is considered as follows: 

The more precise the answer to differential equation Eq. (1), the more choice of series sentences from Eq. (3). In applied problems, approximately five or six sentences from the series are enough to solve nonlinear differential equations. In the answer to differential Eq. (3) regarding the series from degree (n), there are (n+1) unknown coefficients that need (n+1) equations to be specified. The boundary conditions of Eq. (2) are used to solve a set of equations consisting of (n+1) ones.

The boundary conditions are applied on the functions such as follows:

1. The application of the boundary conditions for the answer of differential Eq. (2) is at form of:

 For x = 0:

And when be end x = L:

      ii.   After substituting Eq. (3) into differential equation Eq. (1), the application of the boundary conditions on the resulting equation is done according to the following procedure.

With regard to the choice of n; (n>m) sentences from Eq.(3) and in order to make a set of equations which is consisted  of (n+1) equations and (n+1) unknowns, we confront a number of additional unknowns that are indeed the same coefficients of Eq. (3). Therefore, to remove this problem, we should derive m times from Eq. (1) according to the additional unknowns in the afore-mentioned set of differential equations, and then this is the time to apply the boundary conditions of Eq. (2) to them.

     iii.    Application of the boundary conditions on the derivatives of the differential equation in Eq. (7) is done in the form of.

Second derivation:

And so for higher derivative  

By (n+1) Equations can be made from Eq. (4) to Eq. (9), so that (n+1) unknown coefficients of Eq. (3), for example will be computed. The answer of the nonlinear differential Eq. (1) will be gained by determining coefficients of Eq. (3).

Example 1:

The differential equation governing on the system of Langmuir Boguslavski [21] according to the nonlinear differential equation as follows:

The boundary conditions are in the following:

And we have:

Solving the differential equation with AGM

In AGM, the answer of Eq. (10) is considered as a finite series of polynomials with constant coefficients in the following:

Applying boundary conditions

In AGM, the boundary conditions are applied in two ways:

1. Applying the boundary conditions on the answer of the differential equation which in this case is Eq. (12).

      b. Applying the boundary conditions on Eq. (10) and its derivatives.

The above procedure is done after substituting Eq. (13) into Eq. (10).

The constant coefficients  are obtained by solving the set of ordinary equations which is consisted of four equations and four unknowns from Eq. (14-17).

To simplify, the following new variables are considered as:

The answer of Eq. (10) is achieved by solving Eq. (14-17), the constant coefficients from Eq. (13) will be calculated and we substituting the values of the constant coefficients in to Eq. (13), it is obtained as follows, which is the solution of the differential Eq. (10), as.

By choosing the following physical values, we will have:

Therefore, the answer of nonlinear differential Eq. (10) is gained in the form of:

Comparing the achieved solutions by the numerical method order Runge-Kutta and AGM (Akbari Ganji Method).

Figure 1: A Comparison between AGM and Numerical Solution.

For different values of n, the graphs compare the solution of AGM method and numerical solution as follows:

Figure 2:  Comparison between AGM and Numerical Solution for Different Values of N.

Example2:

The differential equation governing on the system of Langmuir Blodgett [21] according to the nonlinear differential equation as follows:

The boundary conditions are in the following:

Solving the Differential Equation with AGM

In AGM, the answer of Eq. (23) is considered as a finite series of polynomials with constant coefficients in the following:

Applying Boundary Conditions

In AGM, the boundary conditions are applied in two ways:

1. Applying the boundary conditions on the answer of the differential equation which in this case is Eq. (25).

       b.  Applying the boundary conditions on Eq. (23) and its derivatives.

The above procedure is done after substituting Eq. (25) into Eq. (23).

The constant coefficients are obtained by solving the set of ordinary equations which is consisted of four equations and four unknowns from Eqs. (26-29).

The answer of Eq. (23) is achieved by solving Eq.s (26-29), the constant coefficients from Eq. (25) will be calculated and we substituting the values of the constant coefficients in to Eq. (25), it is obtained as follows, which is the solution of the differential Eq. (23), as.

By choosing the following physical values, we will have:

Therefore, the answer of Eq. (23) is gained in the form of:

Comparing the achieved solutions by Numerical Method order Runge-Kutta and AGM (Akbari Ganji Method).

Figure 3: A Comparison between AGM and Numerical Solution.

For different values of u1, u2, the graphs compare the solution of AGM method and numerical solution as follows:

Figure 4: Comparison between AGM and Numerical Solution for different values of u1, u2.

Analytical solution of these two non-linear differential equations by other innovative methods of Mohammadreza Akbari

1. Nonlinear differential equation of Langmuir Boguslavski as follows:

The boundary conditions as:

And we have:

1. MrAM method (Mohammadreza Akbari) and AGM (Akbari Ganji Method)

Analytical solution of the nonlinear differential equation Langmuir Boguslavski, the following result is obtained by the MrAM and AGM methods:

Comparing the achieved solutions by Numerical Method order Runge-Kutta and AGM and MrAM methods

Figure 5: A Comparison between AGM and MrAM Methods and Numerical Solution.

      ii.  Nonlinear differential equation of Langmuir Blodgett as follows:

The boundary conditions as:

And we have:

     2.  MrAM method (Mohammadreza Akbari) and AGM (Akbari Ganji Method)

Analytical solution of the nonlinear differential equation Langmuir Blodgett, the following result is obtained by the MrAM and AGM methods:

Comparing the achieved solutions by Numerical Method order Runge-Kutta and AGM and MrAM methods

Figure 6: A Comparison between AGM and MrAM Methods and Numerical Solution.

In this article,we proved that with this method, AGM, all kinds of complicated practical problems related to nonlinear differential equations can be easily solved analytically. Obviously, most of the phenomena in the surroundings of the engineering field are nonlinear, so it is quite difficult to study and analyse nonlinear mathematical equations in this area. We also wanted to demonstrate the strength, capability, and flexibility of the AGM method. This method can have high power in the analytical solution of all kinds of industrial and practical problems in engineering fields and basic sciences for complicated nonlinear differential equations.

History of AGM, ASM, AYM, AKLM, MR.AM and IAM, WoLF,a, SYM, MrAM  methods:

AGM (Akbari-Ganji Methods), ASM (Akbari-Sara Method) , AYM (Akbari-Yasna Method) AKLM (Akbari Kalantari Leila Method), MR.AM (MohammadReza Akbari Method), IAM ( Integral Akbari Methods),WoLF,a method (Women Life Freedom,akbari),SYM method (Sara Yasna Method) and MrAM method (Mohammadreza Akbari) have been invented mainly by Mohammadreza Akbari (M.R.Akbari) in order to provide a good service for researchers who are a pioneer in the field of nonlinear differential equations.

*AGM method Akbari Ganji method has been invented mainly by Mohammadreza Akbari in 2014. Noting that Prof. Davood Domairy Ganji co-operated in this project.

*ASM method (Akbari Sara's Method) has been created by Mohammadreza Akbari on 22 of August, in 2019.

*AYM method (Akbari Yasna's Method) has been created by Mohammadreza Akbari on 12 of April, in 2020.

*AKLM method (Akbari Kalantari Leila Method) has been created by Mohammadreza Akbari on 22 of August, in 2020.

*MR.AM method (MohammadReza Akbari Method) has been created by Mohammadreza Akbari on 10 of November, in 2020.

*IAM method (Integral Akbari Method) has been created by Mohammadreza Akbari on 5 of February, in 2021.

*WoLF,a method (Women Life Freedom,akbari)has been created by Mohammadreza Akbari on 5 of February, in 2022. 

*SYM method (Sara Yasna Method) has been created by Mohammadreza Akbari on 25 of May, in 2023.

*MrAM method (Mohammadreza Akbari) has been created by Mohammadreza Akbari on 25 of August, in 2023. 

Tel: 0049-17647181018

akbari_hamid46@yahoo.com

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