Numerical solutions of some partial differential equations with B-spline differential quadrature method

This thesis consists of five chapters. In the first chapter, after giving some general information about the differential quadrature method which will be used in the thesis, fundamental concepts about spline functions, and B-spline functions, Thomas algorithms, fourth order Runge-Kutta algorithm, stability, and rate of convergence are presented. B-spline differential quadrature methods are presented in the second chapter. The weighting coefficients, necessary to approximate the derivatives, are determined by using B-spline functions. In the third chapter, numerical solutions of the mKdV equation are obtained by quintic B-spline differential quadrature method. This method is applied to four model problems. The obtained numerical results are compared with existing results in the literature, the error norms and the invariants are given in the form of tables. The figures of the numerical solutions and eigenvalues of the solutions are given and the stability analysis of the approximation obtained by applying quintic B-spline differential quadrature method is also investigated. In the fourth chapter, besides numerical solutions of the KdVB equation, numerical solutions of the KdV and Burgers' equations are also obtained by quintic B-spline differential quadrature method. The method is applied to four model problems. The obtained numerical results are compared with existing results in the literature, the error norms and the invariants are given in the form of tables. The figures of the numerical solutions and eigenvalues of the solutions are given and the stability analysis of the approximation obtained by applying quintic B-spline differential quadrature method is also investigated. In the fifth chapter, numerical solutions of the mBurgers' equation are obtained by quintic and quartic B-spline differential quadrature methods. Both methods are applied to one model problem. The obtained numerical results are compared with existing results in the literature, the error norms are given in the form of tables. The figures of the numerical solutions and eigenvalues of the solutions are given and the stability analysis of the approximation obtained by applying quintic and quartic B-spline differential quadrature methods is also investigated.