Numerical investigation of non-linear differential equations, non-linear Schr?dinger equation, Corteweg DeVry equation

Abstract

    Differential equations that appear in physical problems are often non-linear and their exact solution is necessary to obtain the solution. Since most of the nonlinear differential equations do not have an analytical solution, the numerical solution methods of these equations seem useful for things like plasma physics. For this purpose, in this thesis, the numerical solution of some nonlinear differential equations with partial derivatives has been investigated. Familiarity with nonlinear differential equations and their application in the first chapter and introduction of soliton as an example solution for nonlinear differential equations is given in the second chapter of the thesis. In the third chapter, some numerical methods for this category of equations have been investigated, including Adamian decomposition methods, homotopy disorder and repetition of changes. In the following, in order to obtain and compare the exact solution with numerical solutions, for two sample nonlinear equations (Schr?dinger's nonlinear equation and Cortex-DeVry equation), coding has been done with the help of Fortran programming software, and at the end, the accuracy of the used methods has been measured by analyzing the graphs and comparing each set of solutions. The results show that the method of repeating the changes has more convergence and is more consistent with the exact solution. Keywords: numerical solution, differential equation, nonlinear, soliton. In addition, the progress of the theory of differential equations is linked with the general progress of mathematics and cannot be separated from it. Many differential equations whose solutions cannot be obtained by analytical methods have been investigated in numerical approximation methods. Before 1900, relatively effective numerical integration methods were invented, but their implementation was extremely limited due to the need to perform calculations by hand or with very basic calculation tools. In the last fifty years, the ever-increasing development of powerful multi-purpose computers has greatly expanded the range of problems that can be effectively investigated by numerical methods. Another important work in the field of differential equations in the 20th century is the creation of geometric or topological methods, especially for nonlinear equations. The goal is to understand at least the qualitative behavior of the solutions from a geometrical as well as an analytical point of view. If more detailed information is required, numerical approximation can usually be used. In the last few years, these two trends have joined together. Computers, and especially computer graphics, are a new driving force for the study of nonlinear differential equations. Unexpected phenomena have been discovered, which are referred to by terms such as strange attractions, chaos, and spots, and they have been seriously investigated, which have led to new and important knowledge in some applications. Although differential equations is an old subject, which has a lot of information, but at the dawn of the 21st century, this subject remains a rich source of important and interesting unsolved problems.

Computers can be a valuable tool in studying differential equations. For years, computers were used to implement numerical algorithms to obtain numerical approximations for the solutions of differential equations. Currently, these algorithms have evolved and have reached a very high level in generalization and work. A few lines of computer code, written and executed in a high-level language on a relatively inexpensive computer. (often within a few seconds) is sufficient to numerically solve a wide range of differential equations. More advanced routines are available in most computer centers. The capabilities of these routines are a combination of the ability to deal with very large and complex devices and several useful diagnostic features that alert the user to problems they may encounter. The typical output of a numerical algorithm is a table of numbers containing the chosen values ??of the independent variable and the corresponding values ??of the dependent variables. With suitable computer graphics facilities, it is possible to easily display the solution of a differential equation through a diagram, whether the solution is obtained numerically or through some kind of analytical method.. This type of graphical representation is often more useful and enlightening for understanding and interpreting the solution of a differential equation than a table of numbers or a complex analytical formula. [1 and 2]

1-1 Linear and Nonlinear Differential Equations

Differential equations are divided into two categories, linear and non-linear. In a very simple definition, it is a linear equation in which the order of all the quantities in it is the same. If for the nonlinear equation, orders other than one are also present in the equation. The general form of these equations is as follows: (1-1) This equation is called linear when it is a function of variables. Therefore, the general form of the linear ordinary differential equation of the order is as follows

(1-2)

The mathematical theory of linear equations and their solution methods have been developed and expanded a lot. On the contrary, in the case of nonlinear equations, this theory is more complicated and their solution methods are less satisfactory. From this point of view, fortunately, many important problems are adjusted to linear ordinary differential equations in the first approximation. 1-2 Differences between linear and nonlinear equations In examining the initial value problem, the most fundamental questions to be considered are whether there is an answer, whether This answer is unique, in what areas is this answer defined and how can you get a useful formula for the answer or draw its graph. If the differential equation is linear, there is a general formula for the solution. In addition, for linear equations there is a general solution (involving an arbitrary constant) that includes all solutions, and possible points of discontinuity of the solution can be easily identified by determining the points of discontinuity of the coefficients. However, there is no corresponding formula for nonlinear equations, so it is more difficult to determine similar general properties for the solutions.

Definition interval

Linear problem

With an initial condition, it has a solution throughout any interval around which the functions and are continuous. On the other hand, for a nonlinear initial value, it may be difficult to determine the spaces in which there is a solution.

General answer

Linear and nonlinear equations differ in another aspect, which is related to our derivation of the general solution. For the first-order equation, a solution can be found that includes an arbitrary constant, and from that all possible solutions can be obtained by specifying the values ??of this constant. This may not be the case for nonlinear equations; Even if a solution containing an arbitrary constant can be found, there may be solutions that cannot be obtained for any value of this constant. Therefore, the term "general answer" is used only in the discussion of linear equations.

Implicit answers

We remind you once again that the first-order linear equation has an exact solution for the answer. As long as the necessary primary functions can be obtained, at any point the value of the answer can be obtained just by putting the appropriate value in the mentioned command. It is rare to find such an explicit answer for a nonlinear equation. Usually, in most cases, at most, interfaces in the form (1-4) can be included and found that the answer applies to it. Even this work can be done only for certain types of differential equations, of which separable equations are the most important. The above relation is obtained by an integral (or the first integral) of the differential equation and determines the answer as an implicit function; It means that for each value we have to solve the above equation to get the corresponding value. If the above relation is relatively simple, it can be solved with analytical methods and from there an explicit formula for the answer can be obtained. But this is often not the case, one must turn to numerical calculations in order to obtain the value for the assumed value. When several pairs of and have been calculated, it is often useful to plot them and plot the integral curve passing through those points.