Boundary Value Problems Research Papers (original) (raw)

This thesis paper is mainly analytic and comparative among various numerical methods for solving differential equations but Chapter-4 contains two proposed numerical methods based on (i) Predictor-Corrector formula for solving ordinary differential equation of first order and first degree (ii) Finite-difference approximation formula for solving partial differential equation of elliptic type.

Two types of problems are discussed in detail in this thesis work, namely the ordinary differential equation in Chapters-2 & Chapter-3 and partial differential equations in Chapter-4. Also, Chapter-5 highlights the boundary value problems.

The various chapters of this thesis paper are organized as follows:

The chapters of this thesis paper are organized as follows: Chapter-1 of the thesis is an overview of differential equations and a brief discussion of their solutions by numerical methods.

Chapter-2 deals with the solution of ordinary differential equations by Taylor’s series method, Picard’s method of successive approximation and Euler’s method. Derivation of Taylor’s series method with truncation error and application are discussed here. The solution of ordinary differential equations by Picard’s method of successive approximations and its application is discussed in detail. The definition of Euler’s method is mentioned, the simple pendulum problem is solved to demonstrate Euler’s method. Error estimations and geometrical representation of Euler’s method and the improved Euler’s method are mentioned as a Predictor-Corrector form, which forms being discussed in Chapter-3 next.

Also in it, the comparison between Taylor’s series method and Picard’s method of successive approximation has given. Moreover the advantages and disadvantages of these three methods narrated in it.

Chapter-3 provides a complete idea of the Predictor-Corrector method. Derivation of Milne’s predictor-corrector formula and Adams-Moulton Predictor-Corrector formula with their local truncation errors and applications are discussed here. Solutions of ordinary differential equations by the Runge-Kutta method with error estimation are studied in this chapter. Some improved extensions of the Runge-Kutta method are explained. Also, the general form of the Runge-Kutta method has given here.

The law of the rate of nuclear decay is solved in this chapter by means of standard fourth-order Runge-Kutta method and then the obtained solution is compared with the exact solution, which is an application of the numerical method to the nuclear physics. Comparison between the Predictor-Corrector method and the Runge-Kutta method discussed in detail. Also, the advantages and disadvantages of these two methods discussed in it.

Chapter-4 gives a review of the solution of partial differential equations. Three types of partial differential equations such as elliptic equations, parabolic equations and hyperbolic equations with methods of their solutions are discussed at length. To solve the method of the elliptic equation of iterations and relaxation are discussed. Schmidt's method and the Crank-Nicholson method are discussed to solve parabolic equations. The solution of vibrations of a stretched string is mentioned as a method of solution of hyperbolic equations.

The solution of vibrations of the rectangular membrane by the Rayleigh-Ritz method has given here. A comparison between the iterative method and relaxation method has highlighted and then a total discussion of Rayleigh-Ritz with methods of iteration and relaxation reviewed in this chapter.

Chapter-5 deals with the solution of the boundary value problems in both ordinary differential equations and partial differential equations. It provides a brief discussion of the finite-difference approximation method and shooting method with their applications. Also, the applications of Green’s function to solve boundary value problems are discussed in detail with the application.

Moreover, the B-Spline method for solving two-point boundary value problems of order Four is introduced in this chapter at length. Derivations of cubic B-splines have represented. Cubic B-spline solutions of the special linear fourth-order boundary value problems, the general case of the boundary value problem, treatment of non-linear problems and singular problems have discussed here.

Chapter-6 contains the proposal for the modification of two numerical methods. One of which proposed a modification of Milne’s predictor-corrector formula for solving ordinary differential equations of the first order and first degree, namely Milne’s (modified) Predictor-Corrector formula. One more step-length and one more term in Newton’s interpolation formula being calculated for deriving the predictor and corrector formulae of Milne’s (modified) Predictor-Corrector formula. Also, a modified formula for solving the elliptic equation by finite-difference approximation is proposed, namely surrounding 9-point formula. This formula is obtained by combining standard 5-point formula and diagonal 5-point formula, which gives a more contributive to find mesh points of a given domain in a certain region.

Moreover, the advantages of proposed methods over previous methods are mentioned at the end of this chapter.

Chapter-7 provides us the conclusions of this thesis paper. In this chapter, we have chosen the better methods in every chapter by comparing them. Also, the advantages and limitations of Milne’s (modified) predictor-corrector formulae and surrounding 9-point formula are given here.

Finally, recommendations for future research and a list of few further works have mentioned.