Some Recent Approximate Solution to Non Linear Boundary Value Problems
Chapter One
AIM/OBJECTIVE OF THE WORK
In this research work, we use Galerkin method, Variational Iteration Method and Homotopy Perturbation Method comparing with the exact solution of the classical buckling and post or primary buckling problem from the secondary bifurcation and imperfection sensitivity of columns of nonlinear foundation to check which is the better than others.
CHAPTER TWO
CLASSICAL METHODS OF SOLVING BOUNDARY VALUE PROBLEMS
BOUNDARY VALUE PROBLEMS (BVPs)
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problem is the Sturm–Liouville problem. The analyses of these problems involve the eigenfunctions of a differential operator.
To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.
Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace’s equation); the solution was given by the Dirichlet’s principle.
A more mathematical way to picture the difference between an initial value problem and a boundary value problem is that an initial value problem has all of the conditions specified at the same value of the independent variable in the equation (and that value is at the lower boundary of the domain, thus the term “initial” value). On the other hand, a boundary value problem has conditions specified at the extremes of the independent variable. For example, if the independent variable is time over the domain [0,1], an initial value problem would specify a value of y(t) and y‘(t) at time t = 0, while a boundary value problem would specify values for y(t) at both t = 0 and t = 1.
If the problem is dependent on both space and time, then instead of specifying the value of the problem at a given point for all time the data could be given at a given time for all space. For example, the temperature of an iron bar with one end kept at absolute zero and the other end at the freezing point of water would be a boundary value problem.
Concretely, an example of a boundary value (in one spatial dimension) is the problem (2.1)
to be solved for the unknown function y(x) with the boundary conditions 2.2)
Without the boundary conditions, the general solution to this equation is (2.3)
From the boundary condition y(0) = 0 one obtains
which implies that B = 0. From the boundary condition y(π / 2) = 2 one finds and so A = 2. One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is
TYPES OF BOUNDARY VALUE PROBLEM
Neumann Boundary Condition
If the boundary gives a value to the normal derivative of the problem then it is a Neumann boundary condition. For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.
That is are given.
Dirichlet Boundary Condition
If the boundary gives a value to the problem then it is a Dirichlet boundary condition. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.
That is are given.]
CHAPTER THREE
SOME RECENT METHODS OF SOLVING NON-LINEAR BOUNDARY VALUE PROBLEMS
NON-LINEAR BOUNDARY VALUE PROBLEMS
A non-linear differential equation is a differential equation which is not linear. A non-linear system does not satisfy the superposition principle, or whose output is not directly proportional to its input. Non-linear problems are of interest to Engineers, Physicists and Mathematicians because most physical systems are inherently non-linear in nature. Non-linear equations are difficult to solve and give rise to interesting phenomena such as chaos. The problems involving non-linear differential equations are extremely diverse and the methods of solution or analysis are problem dependent. When the behavior of a nonlinear system cannot be predicted, it is said to be indeterminism. It is multistability when the non-linear system is alternating between two or more exclusive states. The behavior is aperiodic oscillation when a non linear system functions cannot repeat values after some period.
A non-linear differential equation together with a set of additional restraints called boundary conditions is referred as non-linear boundary value problems. A boundary value problem (BVP) should be well posed so that it can be useful in application [18].
The simplest non-linear equation is the first order equation
CHAPTER FOUR
APPLICATION OF THESE METHODS TO BUCKLING PROBLEM
APPROXIMATE SOLUTION OF A CLASSICAL BUCKLING PROBLEM
The non-dimensional form of the boundary value problem for the lateral displacement w(x) of an imperfect elastic column supported laterally by a continuous non-linear foundation consists of
CHAPTER FIVE
RESULT AND CONCLUSION
NUMERICAL RESULT
In this work, some recent methods for the approximate solution of Non-linear Boundary Value Problems were applied. In doing this, Galerkin Methods, VIM and HPM were introduced and compared.
In the chapter one, the analysis of the Galerkin Methods, VIM and Perturbation Methods that is Classical Perturbation Method and Homotopy Perturbation Method were detailed.
In the chapter two, some of the previous methods of solving Non-Linear Boundary Value Problems were introduced such as Shooting Methods and Finite Difference Methods which reveal how tedious, rigorous and burdensome the methods are like reformulating the boundary conditions to initial conditions as in Shooting Methods. Also, applications of Galerkin Methods to some BVPs were provided. For example, figure 2.2 shows a very good approximation to the exact solution of differential equation.
In the chapter three, several applications of VIM and HPM to some Non-Linear BVPs were provided. For example, tables: 3.1, 3.2 and 3.3 show a good approximation to the exact solutions of the given Non-Linear BVPs.
In the chapter four, the Galerkin Methods, VIM and HPM were applied to solving the Classical Buckling and Post Buckling problems. The figure 4.1 shows a very accurate approximation to the analytic solution of Classical Buckling problem by using Galerkin Methods. Actually, the approximate solution is equal to the analytic solution. The figure 4.4 shows the approximate solution of the Classical Bucking using the VIM. The solution was obtained using the equation solver found in the MATLAB package and pylab, matplotlib found in python. The figure 4.6 shows the approximate solution of the Classical Buckling using HPM. The figure 4.8 shows the graph of the four solutions. The graph shows that the Galerkin solution and VIM behave better than HPM.
It is clear from the comparison between the figures that there is a very close agreement between the solutions with the boundary conditions.
The VIM and HPM were also applied to the Post Buckling state, the Galerkin Methods cannot solve the problem because it can only be apply to mildly Non-Linear BVPs.
The figure 4.9 shows the solution of the Post Buckling problems solved by Amazigo and Oyesanya [18] using Classical Perturbation Methods assuming ε = 1. The figure 4.10 shows the approximate solution of the system using VIM. The figure 4.11 shows the approximate solution of the same system using HPM. The figure 4.12 shows the graph of the three solutions. At, the classical perturbation and VIM intercept. It is observed from the comparison between the figures that there is a very good agreement with the solutions.
CONCLUSION
In this work, considering the Classical Buckling problems, the Galerkin Method gives the same solution as the Analytic solution. This shows that the Galerkin method gives more accurate solution than VIM and HPM.
In the Post Buckling state, the numerical results are almost the same except for slight difference between the solutions. The HPM behaves better than VIM.
The VIM can be easily comprehended only with the basic knowledge of advance calculus and the knowledge of calculus of variation in pure Mathematics will be an advantage. One iterational is enough to obtain the accurate approximate solution.
It can be concluded that the methods are efficient techniques in finding exact solutions and approximate solutions for wide classes of problems.
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