Studying Different Numerical Methods in Solving First-Order Differential Equations
Chapter One
AIM AND OBJECTIVES OF THE STUDY
The main aim of the research work is to examine different numerical methods for solving first-order differential equations. Other specific objectives of the study are:
To determine the relationship between the numerical methods of solving first-order differential equations.
To investigate the factors affecting the numerical methods for solving differential equations.
To determine the convergence of these numerical methods.
CHAPTER TWO
SOLUTION OF DIFFERNTIAL EQUATIONS OF FIRST ORDER AND FIRST DRGREE BY NUMERICAL METHODS OF EARLY
STAGE
INTRODUCTION
The solution of ordinary differential equation means to find an explicit expression for the dependent variable in terms of a finite number of elementary functions of. Such a solution of differential equation is called closed or finite form of the solution. In most numerical methods we replace the differential equation by a difference equation and then solve it. The methods developed and applied to solve ordinary differential equations of first order and first degree will yield the solution [23] in one of the following forms
- A power series in for , from which the values of can be obtained by direct substitution.
- A set of tabulated values of and.
In single step methods such as, Taylor’s series method and Picard’s approximation method; the information about the curve represented by a differential equation at one point is utilized and the solution is not iterated. The methods of Euler, Milne, Adams-Moulton and Runge-Kutta belong to step by step method or marching method. In these methods the next point on the curve is evaluated in short steps ahead for equal intervals of width of the dependent variable, by performing iterations till the desired level of accuracy achieved.
In this chapter we will discuss the Taylor’s series method, Picard’s approximation and Euler’s method (with modified), whose are considered as the numerical methods of early stage.
TAYLOR’S SERIES METHOD
Derivation: Let us consider the initial value problem (2.2.1)
Let be the exact solution of (2.2.1) such that. Now expanding (2.2.1) by Taylor’s series [12] about the point we get (2.2.2)
In the expression (2.2.2), the derivatives are not explicitly known. However, if is differentiable several times, the following expression in terms of and its partial derivatives as the followings
By similar manner a derivative of any order of can be expressed in terms of and its partial derivatives.
As the equation of higher order total derivatives creates a hard stage of computation, to overcome the problem we are to truncate the Taylor’s expansion to a first few convenient terms of the series. This truncation in the series leads to a restriction on the value of for which the expansion is a reasonable approximation.
Now, for suitable small step length the function is evaluated at . Then the Tailor’s expansion (2.2.2) becomes (2.2.3)
The derivatives are evaluated at, and then substituted in (2.2.3) to obtain the value of at given by (2.2.4)
CHAPTER THREE
SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS BY PREDICTOR-CORRECTOR METHOD AND RUNGE-KUTTA METHOD.
INTRODUCTION
In the previous chapter, we have discussed three numerical methods of early stage for solving ordinary differential equations. Now, in this chapter we will discuss two modern numerical methods for solving ordinary differential equations. These are known as predictor-corrector method and Runge-Kutta method respectively. It to be noting that one of the predictor-corrector based method already have mentioned in previous chapter; namely, modified Euler’s method.
Now, we are describing the numerical methods mentioned above in detail with the applications, then will compare them.
DEFINITION OF PREDICTOR-CORRECTOR METHOD
In the methods so far described to solve an ordinary differential equation over an interval, only the value of at the beginning of the interval was required. Now in the predictor-corrector methods, four prior values are needed for finding the value of at given value of [2,6]. These methods though slightly complex, have the advantage of giving an estimate of error from successive approximations of, with.
CHAPTER FOUR
SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS.
INTRODUCTION
Partial differential equations occur in many branches of applied mathematics such as in hydrodynamics, electricity, quantum mechanics and electromagnetic theory. The analytical treatment of these equations is rather involved process and requires application of advanced mathematical methods. On the other hand, it is generally easier to produce sufficiently approximate solutions by simple and efficient numerical methods. Several numerical methods have been proposed for solution of partial differential equations. Among those methods we will only discuss the methods those are related to the solution of Elliptic, Parabolic and Hyperbolic partial differential equations. i.e. in this chapter we will solve Elliptic, Parabolic and Hyperbolic partial differential equations only.
CHAPTER FIVE
SOLUTION OF THE BOUNDARY VALUE PROBLEMS WITH APPLICATIONS.
INTRODUCTION
In the previous chapters we have discussed about some well-known methods for solving differential equations satisfying certain initial conditions, which are called initial value problems. In such problems initial conditions are given at a single point. In this chapter we will discuss the problems in which the conditions are satisfied at more than one point, which are known as boundary value problems. We will discuss some methods for solution of boundary value problems.
The simple examples of two-point linear boundary value problem [23] are (5.1.1)
With the boundary conditions
CHAPTER SIX
CONCLUSIONS
In this thesis paper we have discussed some numerical methods for solution of ordinary differential equations (in chapter-2 & chapter-3), partial differential equations (in chapter-4) and boundary value problems (in chapter-5). Also, we have proposed two modified numerical methods (in chapter-6) in this thesis paper.
The conclusions of these discussions are coming next here in brief.
In chapter-2, we get from section-2.3 and section-2.5, both of the Taylor’s series method and Picard’s method of successive approximations are correct to eight decimal places with the exact solution for the given initial value problem. But from the comparative discussion of them in section-2.6, we can conclude that Picard’s method of successive approximations is better than Taylor’s series method in this case.
Also, from section-2.8 it can be said that computed values of y deviate rapidly in Euler’s method and the disturbance have solved in section-2.9 at modified Euler’s method.
In chapter-3, from the comparison between predictor-corrector method and Runge-Kutta method in section-3.12, we have seen that finding local truncation error in Runge-Kutta method is more laborious than in predictor-corrector method, but the self-starting characteristic of Runge-Kutta method makes it favorable than predictorcorrector method. Also, Runge-Kutta method can be used for a wider range of the solution and it is stable for suitable step size.
Thus, we can conclude that for practical purposes Runge-Kutta method is to be chosen for better accuracy.
In chapter-4, from the comparison between iteration method and relaxation method in section-4.10, we have seen that iteration method is slow, sure and lengthy process whereas relaxation method is rapid, less certain and short process to get o solution of partial differential equations under certain conditions. Also, iteration method is self-correcting and has minimum error bound than relaxation method.
Moreover from section-4.12, we have seen that to solve a physical problem by iteration method and relaxation method, it needs to formulate as a partial differential equations whereas the Rayleigh-Ritz method will give an approximate solution without any formulation. Here it to be noted that Rayleigh-Ritz method is quite long and having complexity during the calculation.
Thus, we can choose the iteration method as the best of among three methods and Rayleigh-Ritz method would probably the third one in practice.
In chapter-5, from section-5.3 and section-5.5, we have seen that a two-point boundary value problem can be solved directly by finite-difference method and no other methods needs to its assistance, but the shooting method needs the help of one of other standard methods (i.e. Euler’s method, predictor-corrector method and Runge-Kutta method) after primary formulation. Thus, we can take finite-difference method as the better method between above two.
Also, from the section-5.7, we have seen that Green’s function is applicable to solve a two-point boundary value problem numerically.
Moreover, from section-5.8, we can conclude that multi-order (fourth order) two-point boundary value problem of various cases can be solved numerically by the help of the cubic B-spline method [25] with more accuracy.
Finally, in chapter-6, we have proposed a modified form of Milne’s predictorcorrector method for solving ordinary differential equation of first order and first degree. Also, a utilized formula of standard 5-point formula and diagonal 5-point formula for solving partial differential equation of elliptic type have offered here.
Now, the advantages, limitations and recommendations future research with aim of above two proposed methods are given below.
Advantages of the Milne’s (modified) predictor-corrector formulae:
- Milne’s (modified) predictor-corrector formulae estimate the value of y respecting the given value of x by means of five initial conditions, which is more contributive and logical.
- Milne’s (modified) predictor-corrector formulae need to calculate up-to fifth order Newton’s formula of forward interpolation, which will give better accuracy.
- At Milne’s (modified) predictor-corrector formulae the co-efficient of is zero, then the truncation error converging to zero, this will upgrade the level of accuracy of the method.
Advantages of the surrounding 9-point formula:
- Since surrounding 9-point formula depends upon all mesh points around it to determine any mesh point, it is more contributive and logical, which may give better accuracy.
- The initial zero substitution may enable us to solve a bigger domain at which most of mesh points are absent.
- Using of the Gauss-Seidal iteration formula may give the method a quick ending, this will save the estimation time.
Limitations of the Milne’s (modified) predictor-corrector formulae:
- In Milne’s (modified) predictor-corrector formulae it is require to use one more initial condition than the previous.
- It needs few more calculation time than previous formulae.
Limitations of the surrounding 9-point formula:
- Surrounding 9-point formula is not applicable to the domains having less than nine mesh (grid) points.
- It can used to solve partial differential equations of elliptic type only.
Recommendations future research:
We can proof the advantages mentioned above by substituting proper applications and comparisons. Due to the compression of the thesis paper we have omitted these proofs. But in section-6.3 and section-6.5 we have just shown some applications and comments about these methods comparing with exact solutions.
Therefore, in the future these proofs are to be tried.
Further work can be done:
- To measure efficiency of Milne’s (modified) predictor-corrector formulae and surrounding 9-point formula, compare them with previous all.
- To construct a generalized predictor-corrector formulae for solving ordinary differential equation of first order and first degree. Also. Similar formulae as surrounding 9-point formula for solving partial differential equations for parabolic and hyperbolic types are to be tried to construct.
- To implement Milne’s (modified) predictor-corrector formulae and surrounding 9-point formula to the real world problems.
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