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

This Ph.D. thesis contains six sections. In the first section, some basic concepts used in the thesis are introduced and general information about linear formation equations is given. Spline interpolation and B-spline interpolation functions are defined together with finite difference and finite element methods. Finally, wave equations, whose numerical solutions are to be calculated, are introduced together with the test problems. In the second section; Gilson-Pickering (GP) equation given with boundary conditions from model problems is solved numerically by using septic B-spline collocation method. The test problem in which single solitary wave motion is examined is solved by comparing the exact solution and numerical results. Obtained results are tabulated and stability analysis is performed for the equation. In the third section; To calculate the approximate solution of the generalized Oskolkov equation, the collocation finite element method based on quintic B-spline functions is applied. The proposed method is solved using single solitary wave motion and Gaussian and Undular Bore initial conditions. Obtained results are tabulated and stability analysis is performed for the equation. In the fourth section, the finite element model of the Kudryashov-Sinelschkov equation is constructed using septic B-spline functions. Shock wave motion of Kudryashov-Sinelschkov equation, interaction of two solitary waves, Gaussian condition and Undular bore initial condition are discussed. Obtained results are tabulated and stability analysis is performed for the equation. In the fifth section, numerical solutions of the fifth order Korteweg de Vries (fKdV) equations introduced in the first section are investigated. Numerical solutions of Sawada-Kotera (SK), Caudrey-Dodd-Gibbon (CDG), Lax, Kaup-Kuperschmit (KK) and Ito equations are obtained by using septic B-spline functions. For each equation, single solitary wave motion is examined and the results are tabulated. Again, stability analysis is performed for each equation. In the sixth section, the results and suggestions obtained by the collocation finite element method used in the thesis for each equation are given.