A new embedded Two Derivative Runge-Kutta method (TDRK) based on First Same As Last (FSAL) technique for the numerical solution of first order Initial Value Problems (IVPs) is derived. We present an embedded 4(3) pair explicit fourth order TDRK method with a 'small' principal local truncation error coefficient. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of our method in comparison with other existing embedded Runge-Kutta methods (RK) of the same order.
A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.
The stability properties of fourth and fifth-order Diagonally Implicit Two Derivative Runge-Kutta method (DITDRK) combined with Lagrange interpolation when applied to the linear Delay Differential Equations (DDEs) are investigated. This type of stability is known as P-stability and Q-stability. Their stability regions for (λ,μ∈R) and (μ∈C,λ=0) are determined. The superiority of the DITDRK methods over other same order existing Diagonally Implicit Runge-Kutta (DIRK) methods when solving DDEs problems are clearly demonstrated by plotting the efficiency curves of the log of both maximum errors versus function evaluations and the CPU time taken to do the integration.
In this paper, a trigonometrically-fitted Two Derivative Runge-Kutta method (TFTDRK) of high algebraic order for the numerical integration of first order Initial Value Problems (IVPs) which possesses oscillatory solutions is constructed. Using the trigonometrically-fitted property, a sixth order four stage Two Derivative Runge-Kutta (TDRK) method is designed. The numerical experiments are carried out with the comparison with other existing Runge-Kutta methods (RK) to show the accuracy and efficiency of the derived methods.
This study concentrates on crack that occurs at PRT beam which the beam then retrofitted with carbon fibre reinforced plastic (CFRP) sheets. The beam was tested to determine crack modes and ultimate capacity before and after retrofitted. The beam was tested under static test with displacement control of 0.009mm/sec. Once the first crack appeared, the beam was unloaded and took out for retrofitting process. The beam was loaded again to failure to evaluate the effect of CFRP on the behaviour of the long span beam. Therefore, the finding of this study will guide engineers and construction practitioners to further understand the effects of CFRP sheets used in long span beam in term of load capacity and crack behaviour after retrofitting work.
A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.
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