It is hereby declared that the thesis entitled "Approximate Analytic Solutions of the Nonlinear Cubic Oscillator by Iterative Method" was carried out by Md. 34; Approximate Analytic Solutions of the Nonlinear Cubic Oscillator by Iterative Method” has been approved by the Board of Examiners for the partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics, Khulna University of Engineering & Technology, Khulna, Bangladesh on 23 November 2016. Ikramu1 Haque, Associate Professor, Department of Mathematics, Khulna University of Engineering & Technology, Khulna-9203, Bangladesh, who taught the subject of this dissertation and provided excellent guidance with his constant dedication, endless inspiration, valuable suggestions, scholastic criticism , continued encouragement and helpful discussion.
I wish to express my sincere and wholehearted respect and gratitude to all the teachers of the Department of Mathematics, Khulna University of Engineering & Technology, Khulna-9203, Bangladesh for their necessary advice and cordial cooperation during my studies. Based on the iteration method, a modified approximate analytical solution of the cubic nonlinear oscillator " + x3 = 0" was obtained. In general, differential equations apply to many physical phenomena such as spring-mass systems, resistor-capacitor-inductor circuits, bending of rays, chemical reactions, motion of a pendulum, motion of a rotating mass around another body, etc. came
A disadvantage of the method is that it is difficult to predict for a given nonlinear differential equation whether a first-order harmonic balance calculation will yield a sufficiently accurate approximation of a periodic solution. Chapter 1V describes the Iterative method for obtaining approximate analytical solutions of the Cubic Truly Nonlinear Oscillator.
Trajectory
If a plane is such that every point in this plane describes the position and velocity of a dynamical particle, then this plane is called a phase plane. A convenient way to treat equation (2.1) is to rewrite it as a system of two first-order ordinary differential equations. A point that satisfies F(x,y)= 0 and G(x,y) =0 is called an equilibrium point. The solution to (2.3) can be depicted as a curve in the x - y phase plane passing through the point of initial conditions (x0 , y0.
If both f and g are not constant functions, then equation (2.7) defines a curve in the phase plane, which is called a path or path or trajectory of the system. A closed path in the phase plane, such that other non-closed paths spiraled towards it, either from within or without, as t —* oo , is called a limit cycle. If all orbits starting near a closed orbit (both inside and outside) spiral toward the closed orbit as t —* oo, then the limit cycle is asymptotically stable.
If the orbits on either side of the closed orbit spiral away as t —>oo, then the closed orbit is unstable. Such a system in which the independent variable t does not explicitly appear in the function P and Q on the right is called an autonomous system.
The Non-autonomous system Consider the systems of the form
Critical Point
Isolated Critical Point
Classifications of Critical Point
In each of the four domains, between any two of the four directions in (i), there are infinitely many paths that are arbitrarily closed at (0, 0) but do not approach (0,0) either as t —*+ co or as t. Suppose that (0, 0) is an isolated critical point of the above system. 1) Asymptotically stable Let us consider the system (2.6).
Characteristic Equation Consider the linear system
Equation (2.10) is called the characteristic equation of (2.8) and its roots are called characteristic roots or eigenvalues of equation (2.8).
Nature
It follows that some aspects of the behavior of a nonlinear system usually appear chaotic, unpredictable, or counterintuitive. Initially, Van der Pol [41] paid attention to the new (self-excitation) oscillations and indicated that their existence is inherent to the nonlinearity of the differential systems that characterize the procedure. This nonlinearity thus seems to be the essence of these phenomena and by linearizing the differential equation in the sense of the small oscillation method one simply eliminates the possibility of investigating such problems.
By replacing equation (3.2) with equation (3.1) and equating the coefficients of the different powers of g, the following equations are obtained. This method starts by solving the linear equation, assuming that, in the nonlinear case, the amplitude and phase in the solution of the linear equation are a time-dependent function rather than constants [7]. Now to determine an approximate solution of equation (3.1) for e small but different from zero, Krylov and Bogoliubov assumed that the solution is still given by equation (3.4) where a and 0 vary depending on the conditions.
Later, this solution was used by Mitropolskii [45] to investigate similar system (i.e. equation (3.1)) where the coefficient very slowly with time. An important consequence of this property is that the corresponding Fourier expansions of the periodic solutions only contain odd harmonics, i.e. 3.14). Step-i: Substitute equation (3.18) into equation (3.16) and expand the resulting form into an expression having the following structure.
In this equation (3.19), we keep only as many harmonics in our expansion as occur initially in the assumed approximation to the periodic solution. In general, the amplitude and angular frequency will be expressed in terms of the parameters A1 and A2. Secular tensors will not appear in the solution for x1 (t) if the coefficient of the cosO term is zero, i.e.
We now present the full details on how to evaluate the right-hand side of equation (3.48). Secular terms can be eliminated in the solution for x2 (t) as the coefficient of the cos 0. Substituting equation (3.62) then into equation (3.61) and expanding the right side of equation (3.61) to the Fourier series yields.
In equation (3.65), the particular solution is chosen so that it does not contain secular terms [27], which requires that the coefficient. This scheme is computationally easier to work with, for k ~: 2, than the one given in equation (3.69).
Introduction
The method
The method can proceed in any order of approximation; but due to increasing algebraic complexity, the solution is limited to a lower order, usually the second [52]. A linear differential equation for Xk+I (t) allows C2k to be determined by requiring that secular terms are absent.
Solution Procedure
To check secular terms in the solution, we need to remove cos9 from the right side of equation (4.20) which we get. To check secular terms in the solution, we have to remove cosO from the right-hand side of equation (4.36), we get To check secular terms in the solution, we have to remove cosO from the right Ic of equation (4.43) which we get.
To test the accuracy of the modified iterative method approach, we compare our results with other existing results of different methods. It turns out that in most cases our solutions provide more meaningful results than other existing results. Here we have calculated the first, second and third approximate frequencies, which are marked as QØ, 01 and C22 respectively.
We have also given the existing results determined by Mickens iterative method [27] and Mickens HB method [27].
Table
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