Runge-kutta type methods for solving high-order ordinary differential equations - Ghawadri, Nizam Ghannam Fayez (2018)
This study is focused on developing Runge-Kutta type methods to solve two types of ordinary differential equations (ODEs). The first type is the special third-order ODEs in the forms of y000 = f (x;y) and y000 = f (x;y;y0): The second type is the special fourth-order in the forms of y(4) = f (x;y;y0) and y(4) = f (x;y;y0;y00) and the general fourth-order ODEs. These types of ODEs often used to describe the mathematical models for problems arises in several fields of applied sciences and engineering. The first part of this thesis is focused on construction an exponentially-fitted explicit modified Runge-Kutta type ( MRKT) method for solving special third-order ordinary differential equations (ODEs) in the form of y000 = f (x;y): The new three-stage fourth-order explicit MRKT method is called EFMRKT4 for solving initial value problems is derived. Meanwhile, exponentially- and trigonometrically-fitted explicit modified Runge-Kutta type methods denoted as EFMRKT and TFMRKT respectively for solving special third-order ODEs in the form of y000 = f (x;y;y0) are derived. The new four-stage fifth-order explicit MRKT methods are called EFMRKT5 and TFMRKT5 respectively for solving initial value problems whose solutions involving exponential or trigonometric form. The second part of this thesis is focused on derivation of new explicit Runge-Kutta type RKDF, RKTF and RKTGF methods for directly solving y(4) = f (x;y;y0); y(4) = f (x;y;y0;y00) and y(4) = f (x;y;y0;y00;y000) respectively . The order conditions of the RKDF, RKTF and RKTGF approaches are constructed by using two methods; the first method is using the Taylor series expansion and the second method is B-series and quad-colored trees. Based on algebraic order conditions, fourth- and fifth-order explicit RKDF and RKTF methods using constant step length and an embedded explicit RKDF and RKTF methods of 5(4) pair for variable step size have been derived respectively. The new three-stage fourth- and fifth-order explicit RKDF methods are called RKDF4 and RKDF5 for solving y(4) = f (x;y;y0) is constructed respectively and three-stage fourth-order and four-stage fifth-order explicit RKTF methods are called RKTF4 and RKTF5 for y(4) = f (x;y;y0;y00) is developed respectively. The new fourth-order four-stage explicit RKTGF method that denoted as RKTGF4 using constant step size have been constructed. The third part of this thesis is focused on derivation of diagonally implicit Runge-Kutta type (DIRKT) approach to solve special fourth-order ODEs. To see the accurocy and effectiveness of the method, the constant step size code is developed and numerical results are compared with current methods given in literature. In conclusion, the proposed methods constructed in this study are suitable to solve special third-order, special fourth-order and general fourth-order ODEs. The proposed methods are also more efficient than the existing RK type methods in the terms of accuracy, maximum global error and number of function evaluations.