![]() 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. Chapter-6 contains the proposal for the modification of two numerical methods. 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. Derivations of cubic B-splines have represented. Moreover, the B-Spline method for solving two-point boundary value problems of order Four is introduced in this chapter at length. Also, the applications of Green’s function to solve boundary value problems are discussed in detail with the application. It provides a brief discussion of the finite-difference approximation method and shooting method with their applications. ![]() Chapter-5 deals with the solution of the boundary value problems in both ordinary differential equations and partial differential equations. 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. The solution of vibrations of the rectangular membrane by the Rayleigh-Ritz method has given here. The solution of vibrations of a stretched string is mentioned as a method of solution of hyperbolic equations. Schmidt's method and the Crank-Nicholson method are discussed to solve parabolic equations. To solve the method of the elliptic equation of iterations and relaxation are discussed. Three types of partial differential equations such as elliptic equations, parabolic equations and hyperbolic equations with methods of their solutions are discussed at length. Chapter-4 gives a review of the solution of partial differential equations. Also, the advantages and disadvantages of these two methods discussed in it. Comparison between the Predictor-Corrector method and the Runge-Kutta method discussed in detail. 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. Also, the general form of the Runge-Kutta method has given here. ![]() Some improved extensions of the Runge-Kutta method are explained. Solutions of ordinary differential equations by the Runge-Kutta method with error estimation are studied in this chapter. Derivation of Milne’s predictor-corrector formula and Adams-Moulton Predictor-Corrector formula with their local truncation errors and applications are discussed here. Chapter-3 provides a complete idea of the Predictor-Corrector method. Moreover the advantages and disadvantages of these three methods narrated in it. Also in it, the comparison between Taylor’s series method and Picard’s method of successive approximation has given. 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. The definition of Euler’s method is mentioned, the simple pendulum problem is solved to demonstrate Euler’s method. The solution of ordinary differential equations by Picard’s method of successive approximations and its application is discussed in detail. Derivation of Taylor’s series method with truncation error and application are discussed here. 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. 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. Also, Chapter-5 highlights the boundary value problems. 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. 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.
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