Runge-kutta type methods for solving special third and fourth order ordinary differential equations

This thesis is focused on developing Runge-Kutta type methods for solving two types of ordinary differential equations (ODEs). The first type is the special third-order ODEs which do not depend on the first derivative y′(x) and the second derivative y′′(x) explicitly. The second is the special fourth-order ODEs which are not dependent on the first derivative y′(x), the second derivative y′′(x) and third derivative y′′′(x) explicitly. These types of ODE often used to describe the mathematical models for problems arises in several fields of applied sciences and engineering. Traditionally, these ODEs are solved by reducing them to an equivalent system of first-order ordinary differential equations. However, it is more efficient in terms of accuracy, the number of function evaluations as well as computational time, if they can be solved directly by using numerical methods. The first part of the thesis described the construction of the Improved Runge-Kutta type method for directly solving the special third-order ODEs where the method is denoted as IRKD method. Taylor series expansion is used to derive the order conditions of the IRKD method. Based on these order conditions, three-stage fourth-order and four-stage fifth-order IRKD methods are derived. Codes based on these methods are developed and then used to solve the special third-order ODEs. The IRKD methods are also used to solve physical problem in thin film flow. The second part of the thesis is focused on the derivation of the direct Runge-Kuttatype method denoted as RKFD method for solving the special fourth-order ODEs.The order conditions of the RKFD methods are derived by using two approaches; the first approach is using the Taylor series expansion and the second approach is using the B-series and the associated relevant-colored trees. Based on the order conditions, three-stage fourth-order, three-stage fifth-order and four-stage sixth-order RKFD methods are derived. Codes based on the RKFD methods are developed and used for solving the special fourth-order ODEs. The RKFD methods are also applied to solve engineering problem which is the ill-posed problem in a beam on elastic foundation. Then two embedded RFKD pairs of order four in five and order five in six are derived. Based on the embedded RKFD methods, the variable step-size codes are developed and used to solve the special fourth-order ODEs. In conclusion, the new IRKD and RKFD methods developed in this thesis are suit-able for directly solving special third-order and fourth-order ODEs respectively. The methods are also more efficient than the existing Runge-Kutta type methods in terms of accuracy, computational time and number of function evaluations.

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