Akbari-Ganji's Method

2017 ◽  
pp. 246-273
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Chemical Engineering ◽  
Solid Mechanics ◽  
Nonlinear Partial Differential Equation ◽  
The Other ◽  
Partial Differential ◽  
New Approach

The nonlinear partial differential equation governing on the mentioned system has been investigated by a simple and innovative method which we have named it Akbari-Ganji's Method or AGM. It is notable that this method has been compounded by Laplace transform theorem in order to covert the partial differential equation governing on the afore-mentioned system to an ODE and then the yielded equation has been solved conveniently by this new approach (AGM). One of the most important reasons of selecting the mentioned method for solving differential equations in a wide variety of fields not only in heat transfer science but also in different fields of study such as solid mechanics, fluid mechanics, chemical engineering, etc. in comparison with the other methods is as follows: Obviously, according to the order of differential equations, we need boundary conditions so in the case of the number of boundary conditions is less than the order of the differential equation, this method can create additional new boundary conditions in regard to the own differential equation and its derivatives. Therefore, a solution with high precision will be acquired. With regard to the afore-mentioned explanations, the process of solving nonlinear equation(s) will be very easy and convenient in comparison with the other methods.

Nonlinear Vibrations of Non-Uniform Beams Including Inertia Effects

2005 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Ion Stroe
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Effects ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Inertia Effects

This papers deals with nonlinear vibrations of non-uniform beams with geometrical nonlinearities such as moderately large curvatures, and inertia nonlinearities such as longitudinal and rotary inertia forces. The nonlinear fourth-order partial-differential equation describing the above nonlinear effects is presented. Using the method of multiple scales, each effect is found by reducing the nonlinear partial-differential equation of motion to two simpler linear partial-differential equations, homogeneous and nonhomogeneous. These equations along with given boundary conditions are analytically solved obtaining so-called zero-and first-order approximations of the beam’s nonlinear frequencies. Since the effect of mid-plane stretching is ignored, any boundary conditions could be considered as long as the supports are not fixed a constant distance apart. Analytical expressions showing the influence of these three nonlinearities on beam’s frequencies are presented up to some constant coefficients. These coefficients depend on the geometry of the beam. This paper can be used to study these influences on frequencies of different classes of beams. However, numerical results are presented for uniform beams. These results show that as beam slenderness increases the effect of these nonlinearities decreases. Also, they show that the most important nonlinear effect is due to moderately large curvature for slender beams.

Laplace Decomposition Method to Study Solitary Wave Solutions of Coupled Nonlinear Partial Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Arun Kumar ◽  
Ram Dayal Pankaj
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
Weak Nonlinearity ◽  
Partial Differential ◽  
New Approach ◽  
Laplace Decomposition Method

Analytical and numerical solutions are obtained for coupled nonlinear partial differential equation by the well-known Laplace decomposition method. We combined Laplace transform and Adomain decomposition method and present a new approach for solving coupled Schrödinger-Korteweg-de Vries (Sch-KdV) equation. The method does not need linearization, weak nonlinearity assumptions, or perturbation theory. We compared the numerical solutions with corresponding analytical solutions.

scholarly journals Transient-False Method for Solving System of Nonlinear Partial Differential Equations

2018 ◽  
Vol 1 (25) ◽  
pp. 509-522
Author(s):  
. Ali Khalaf Hussain
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations

          In this paper we study the false transient method  to  solve and transform a system of non-linear partial differential equations which can be solved using finite-difference method and give some problems which have a good results compared with the exact solution, whereas this method was used to transform the nonlinear partial differential equation to a linear partial differential equation which can be solved by using the alternating-direction implicit method after using the ADI method. The system of linear algebraic equations could be obtained and can be solved by using MATLAB.

Finite Difference, Finite Element and Boundary Element Formulae as a Language of Applied Physics

1991 ◽  
Vol 02 (01) ◽  
pp. 383-386
Author(s):  
JIŘÍ KAFKA ◽  
NGUYEN VAN NHAC
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Integral Form ◽  
The Other ◽  
Applied Physics ◽  
Partial Differential

When deducing the finite difference formulae, one has to discretize partial differential equations. On the other hand, those equations have been previously derived having started from laws of physics in their integral form. So, a question arises, why not avoid the approach to the limit (necessary to deduce the partial differential equation) and why not deduce the finite difference formulae directly on the base of laws of physics in their integral form.

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