Nonlinear Dynamic in Engineering by Akbari-Ganji’S Method

Copyright © 2015 by Alireza Ahmadi; Mohammadreza Akbari;

                        Davood Domairry Ganji.

Library of Congress Control Number:   2015913937

ISBN:      Hardcover          978-1-5144-0169-9

                Softcover          978-1-5144-0170-5

                eBook               978-1-5144-0171-2

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.

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Rev. date: 11/06/2015

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Contents

Preface

1.   Nonlinear Differential Equations

1.1. A Brief Explanation to Nonlinear Differential Equations

1.2. The Application of Nonlinear Differential Equations

in Different Sciences

1.3. Analytic Methods

1.3.1. The First-Order Nonlinear Differential Equations

1.3.2. The Second- and Higher-Order Differential Equations (Nonvibrational)

1.3.3. Vibrational Differential Equations

1.4. Applied Commands in AGM by Maple Software

1.4.1. Analyzing the Second- and Higher-Order Differential Equations (Nonvibrational)

1.4.2. Applied Instructions for Vibrational Equations

without Any External Force

1.4.3. Applied Commands for Solving Vibrational Equations

with External Force

1.5. References

2.   The First-Order Nonlinear Differential Equations

2.1. Investigation of the First-Order Differential Equations

2.2. Applied Examples

2.3. Problems

2.4. References

3.   The Second- and Higher-Order Nonvibrational Differential Equations

3.1. Physical Examples

3.2. Problems

4.   Nonlinear Vibrational Differential Equations

4.1. Applied examples

4.2. Problems

5.   Nonlinear Integrodifferential Equations

5.1. A Short Description of Integrodifferential Equations

5.2. Miscellaneous Examples

5.3. Problems

5.4. References

6.   Application of AGM in Partial Differential Equations

6.1. Partial Differential Equations

6.2. A Broad Range of Applied Examples

6.3. Problems

6.4. References

Appendices

Maple Codes

Preface

Being nonlinear is the challenge of most users dealing with differential equations in mathematical and engineering sciences. In fact, so far, there has not been any especial method to solve various forms of nonlinear differential equations by an easy approach in different fields of study. It is necessary to mention that in recent years, researchers have proposed some semianalytical methods for solving a few groups of nonlinear differential equations, but none of them can be useful for a broad range of nonlinear equations. To understand more, it is better to indicate that some of the existing methods for solving nonlinear equations are HAM, HPM, VIM, DTM, ADM, and so on—in which each method is well applied only for a small category of nonlinear differential equations. Furthermore, in some cases, application of every method has really made significant errors in calculations. Therefore, the aforementioned methods cannot apply for the whole nonlinear problems, and each of them can just answer certain problems. To explain more, the long and complex process is the other weak point of these methods.

In this book, a convenient approach is introduced for all the different kinds of nonlinear differential equations and also for various sets of nonlinear equations, which can analytically solve all the differential equations very easily and obtain the final solution of every differential equation as an algebraic function. This method is named Akbari-Ganji’s method (AGM) and is aimed to solve the whole nonlinear differential equations algebraically by choosing an answer function for a differential equation with constant coefficients, which can be computed by applying initial or boundary conditions in a particular manner.

Eventually, the authors are grateful to the ancient Persian mathematician, astronomer, and geographer Muḥammad ibn Mūsā al-Khwārizmī, who—in his Compendious Book on Calculation by Completion and Balancing—was the one that presented the first systematic solution of linear and quadratic equations. Furthermore, he has been considered as the original inventor of algebra, and Europeans have derived the term algebra from his book. And also, the expression algorithm has been taken from his name (the Latin form of his name).

Chapter 1

Nonlinear Differential Equations

1.1. A Brief Explanation to Nonlinear Differential Equations

This book has been published in order to solve nonlinear differential equations governing the procedures of engineering and basic sciences. Definitely, solving nonlinear differential equations governing physical phenomena can be considered as the root of most researches and studies in the applied sciences. And the obtained results have been utilized in order to design, implement, and apply in industry. It is necessary to mention that the behavior of most phenomena in engineering and basic sciences is revealed as nonlinearity,¹, ² so to comprehend the relationships among the components of each phenomenon, having sufficient mathematical knowledge is very essential. Up to here, a big step has been taken in understanding the relations among the elements of physical phenomena. In the next step, it is important to solve nonlinear differential equations. Since every differential equation consists of a series of independent and dependent variables and also has some parameters with respect to the physical properties of the system, it is better to solve differential equations analytically in terms of their parameters.³ It is clear that researchers can easily solve these kinds of equations numerically,⁴, ⁵ but there are some weak points in the numerical methods. For example, it is obligatory in a numerical solution to determine the values of existing parameters in differential equations at the beginning of the solution procedure, but the answer of differential equations in analytic methods is always achieved with the existing parameters in their own differential equation. As a result, different answers can be gained by choosing various amounts for these parameters.

1.2. The Application of Nonlinear Differential Equations in Different Sciences

Obviously, the existent analytic methods have not been comprehensive up to now, and they are not suitable for all the nonlinear differential equations,⁶–¹³ and they do not give us the exact answers while comparing them with numerical methods. The results show that the entities of the differential equations governing the systems are different from each other, and to remove this difficulty, we divide all the differential equations governing the system (including engineering and basic sciences) into the following three cases and then solve them by AGM. These analytic methods are useful for a special class of differential equations in a certain domain and just give reliable solutions in this domain.

Analytic solution of nonlinear differential equations by AGM is divided into the following three classes in terms of whether the reaction of the system is harmonic or nonharmonic:

1. Solving equations and set of second- and higher-order nonlinear differential equations for nonharmonic systems (systems whose answers are not a function of time)

These systems are mainly applied in mechanical engineering (heat and fluids), chemistry, civil engineering, and nanomaterials.¹⁴–²⁰

2. Solving equations and set of first-order nonlinear equations for harmonic and nonharmonic systems (systems whose answers are a function of time)

It is necessary to mention that the basic application of these systems is in electrical engineering (electrical circuits and control), chemistry (kinetics of chemical reactions), and gas dynamics.

3. Solving equations and set of second-order nonlinear differential equations for harmonic systems well-known to vibrational equations.²¹–³⁰

These systems are considerably used when their answers are alternative and vibrational, such as the existent problems of solid mechanics, civil engineering (earthquake), and so on.

1.3. Analytic Methods

1.3.1. The First-Order Nonlinear Differential Equations

The first-order nonlinear differential equations can be considered as the following general form:

And the relevant initial condition is expressed as

In this step, a compound series consisting of exponential and trigonometric terms has been selected as a suitable answer for this kind of nonlinear differential equation as follows:

It is citable that equation 1.3 is applicable for solving all the first-order nonlinear differential equations.

Moreover, for analyzing electric circuits in electrical engineering, we can utilize the above relation.

To understand more, it is better to indicate that to facilitate the computational operations in AGM, if the physical system is not vibrational, such as differential equations governing the heat transfer systems (radiation), chemical reactions, or nonvibrational differential equations, then it is logical to omit the trigonometric term of equation 1.3. Therefore, equation 1.3 can be rewritten as follows:

As an example, some kinds of the first-order nonlinear differential equations governing the chemical, heating, and electric systems have been presented as follows:

In the aforementioned equations, equation 1.3 can be a suitable choice for the answer function. Based on the given explanations, the constant coefficients of equation 1.3, 10174.png to 10189.png  , can be obtained by applying the given initial condition.

1.3.2. The Second- and Higher-Order Differential Equations (Nonvibrational)

Boundary conditions and initial conditions are required for analytic methods of each linear and nonlinear differential equation according to the physic of the problem. Therefore, we can solve every differential equation with any degrees. In order to comprehend the given method, reading the following lines is recommended.

In accordance with the boundary conditions, the general manner of a differential equation is as follows:

The nonlinear differential equation p (which is a function of u), the parameter u (which is a function of x), and their derivatives are considered as follows:

To solve the first differential equation, with respect to the boundary conditions in x = L in equation 1.9, the series of letters in the nth order with constant coefficients, which is the answer of the first differential equation, is considered as follows:

The more precise the answer of equation 1.8, the more choices of series sentences from equation 1.10. It means by increasing the number of series sentences in equation 1.10, the obtained solution is the real answer, or the exact solution of the problem. In applied problems, approximately five or six sentences from the series are enough to solve nonlinear differential equations. In the answer of differential equation 1.10 regarding the series from the degree (n), there are (n+1) unknown coefficients that need (n+1) equations to be specified. The boundary conditions of equation 1.9 are used to solve a set of equations that consists of (n+1) ones.

The boundary conditions are applied to the functions as follows:

a) The application of the boundary conditions for the answer of differential equation 1.10 is in the form of

If x = 0

And when x = L

b) After substituting equation 1.12 into equation 1.8, the application of the boundary conditions on differential equation 1.8 is done according to the following procedure:

With regard to the choice of n; (n<m) sentences from equation 1.10 and in order to make a set of equations that consist of (n+1) equations and (n+1) unknowns, we confront with a number of additional unknowns that are indeed the same coefficients of equation 1.10. Therefore, to remove this problem, we should derive m times from equation 1.8, in accordance with the additional unknowns in the aforementioned set of differential equations, and then this is the time to apply the boundary conditions of equation 1.9 on them.

c) Application of the boundary conditions on the derivatives of the differential equation 10297.png in equation 1.14 is done in the form of

The 10345.png equations can be made from equation 1.11 to equation 1.16 so that 10358.png unknown coefficients of equation 1.10 for example, 10369.png can be computed. The answer of the nonlinear differential equation 1.8 will be gained by determining coefficients of equation 1.10.

1.3.3. Vibrational Differential Equations

In general, vibrational equations and their initial conditions are defined for different systems as follows:

1.3.3.1. Choosing the Answer Function of Vibrational Differential Equations by AGM

In AGM, a total answer with constant coefficients is required in order to solve differential equations in various fields of study such as vibrations, structures, fluids, and heat transfer. In vibrational systems, with respect to the kind of vibration, it is necessary to choose the mentioned answer in AGM. To clarify here, we divide vibrational systems into two general forms:

1.3.3.1.a. Vibrational Systems without Any External Force

Differential equations governing this kind of vibrational systems are introduced in the following form:

Now, the answer of this kind of vibrational system is chosen as

According to trigonometric relationships, equation 1.20 is rewritten as follows:

It is notable that in the above equation, 10450.png and 10463.png .

Sometimes, for increasing the precision of the considered answer of equation 1.19, we are able to add another term in the form of cosine by inspiration of Fourier cosine series expansion as

In the above equation, we are able to omit the term 10494.png to facilitate the computational operations in AGM