Runge-Kutta type direct methods for solving second and third order boundary value problems

In this thesis, methods for solving higher-order two-point boundary value problems (BVPs) directly are developed. These methods are known as one- and two-step explicit Runge-Kutta type methods. Conventionally, higher-order BVPs are solved by converting them to a system of first-order BVPs. However, it is more efficient in terms of accuracy, the number of function evaluations as well as computational time, if these problems can be solved directly by using the proposed methods with constant step length via shooting technique. In the first part of the thesis, one-step Runge-Kutta type methods are constructed to solve second- and third-order BVPs. An exponentially-fitted technique is implemented in four stages fourth-order Runge-Kutta-Nystr¨om (EFMRKN4) for solving special second-order BVPs which possesses an exponential solution. Meanwhile, four-stage fourth-order general Runge-Kutta-Nystr¨om (RKNG4) method is constructed for solving general second-order BVPs. Thereafter, two-stage third-order and three-stage fourth-order explicit Runge-Kutta type (RKT2s3) and (RKT3s4) methods are constructed respectively for solving special third-order BVPs. Besides, two-stage third-order exponentially-fitted modified Runge-Kutta type (EFMRKT2s3) method is derived in order to improve the efficiency of RKT2s3 method. The Local Truncation Error (LTE) of the fitted methods is computed, the absolute stability of the EFMRKN method is discussed. The numerical results obtained show that the developed methods are more efficient in terms of accuracy and number of function evaluations in comparison with the existing methods in the literature for the same order. In the second part of the thesis, two-step Runge-Kutta-Nystr¨om (TSRKN) method is derived for the direct solution of special second-order BVPs. The two-step method has an advantage that it can estimate the solution with fewer function evaluations compared to the one-step method. The order conditions are provided and three stages fourth-order two-step Runge-Kutta-Nystr¨om (TSRKN3s4) method is derived. The stability of TSRKN method is analyzed and the numerical results show a clear advantage of the TSRKN method as compared with the existing methods in terms of number of function evaluations per step and time. In conclusion, the developed methods are able to solve the second- and special thirdorder BVPs directly.

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