Mathematical formulation of the problem

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Figure 1

the force acting on the ball are Drag force,(F_D), the lift force(F₁), and the gravitation force,(F_W) as follows

The equatation of motion takes the form:

After substituting Eqs.(1) in to Eqs.(2),We have as follows

We can write:

So, the equations can be written entirely as a system set of nonlinear differeential equations for the velocity ad follows:

and value

Initial conditions as:

By selecting the physical values at below:

AYM solution process (Akbari Yasna’s Method) Output of the solution process by new approach AYM (Akbari Yasna’s Method) for set of nonlinear differential equations Eqs(5), according to the initial conditions Eqs.(6) and physical values Eq.(7), the solutions. set of the non-defferential equations is obtained as follows:

Comparing the achieved solutions by Numerical Method order Runge-Kutta and AYM (Akbari Yasna’s Method)

Figure 2

Figure 3

In this article, we proved that with this new method, all kinds of complex practical problems related to nonlinear differential equations in the designs for motions of a solid in a fluid can be easily solved analytically. Obviously, most of the phenomena in dunamics and aerodynamics are nonlinear so it is quite difficult to study and analyze nonlinear mathematical equations in this area, also we wanted to demonstrate the strength, capability and flexibility of the new AYM methods(Akbari-Yasna Methods).This Methods is newly created and it can have high power in analytical solution of all kinds of industrial and practical problems in engineering fields and basic sciences for complex nonlinear differential equations

Acknowledgment, History of AGM , ASM , AYM , AKLM , MR.AM and IAM methods: AGM (Akbari-Ganji Methods), ASM (Akbari-Sara’s Method) , AYM (Akbari-Yasna’s Method) AKLM (Akbari Kalantari Leila Method), MR.AM (MohammadReza Akbari Method)and IAM ( Integral Akbari Methods), 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.

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

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

^*AKLMmethod (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.

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

References

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1. Book Nonlinear Dynamic in Engineering by Akbari-Ganji's Method ISBN-13: 978-1514401699, ISBN-10: 151440169X.
2. MR Akbari, Sara Akbari, Esmaeil Kalantari (2020) ‘Akbari Ganji s method “AGM” to chemical reactor design for non-isothernal and non-adiabatic of mixed flow reactors'Journal of Chemical Engineering and Materials Science 11: 1-9.
3. MR Akbati,DD Ganji,M Nimafar (2014)"Signification progress" in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach", Frontiers of Mechanical Engineering Journal.
4. AK Rostami, MR Akbari, DD Ganji, S Heydari (2014) Investigating Jeffery-Hamel flow with high maganetic field and nanoparticle by HPM and AGM, Cent. Eur. J. Eng 4: 357-70.
5. MR Akbari, DD Ganji, A Majidian, AR Ahmadi (2014) "Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM", Frontiers of Mechanical Engineering
6. DD Ganji, MR Akbari, AR Goltabar (2014) “Dynamic Vibration Analysis for Non-linear Partial Differential Equation of the Beam - columns with Shear Deformation and Rotary Inertia by AGM”, Development and Applications of Oceanic Engineering (DAOE), ISSN Online: 2325-3762.
7. MR Akbari, DD Ganji, AR Ahmadi, Sayyid H Hashemi kachapi (2014) "Analyzing the Nonlinear Vibrational wave differential equation for the simplified model of Tower cranes by (AGM)" Frontiers of Mechanical Engineering Volume 9: 58-70.
8. MR Akbari, M Nimafar, DD Ganji, MM Akbarzade (2014) "Scruiny of non-linear differential equations Euler Bernoulli" beam with large rotational deviation by AGM” Springer.
9. M. R. Akbari, Sara Akbari , Esmaeil Kalantari, “A Study about Exothermic Chemical Reactor by ASM Approach Strategy”Crimson Publishers Wings to the Research,Published, March 17, 2020.