Computational complexity for the two-point block method

In this paper, we discussed and compared the computational complexity for two-point block method and one-point method of Adams type. The computational complexity for both methods is determined based on the number of arithmetic operations performed and expressed in O(n). These two methods will be used to solve two-point second order boundary value problem directly and implemented using variable step size strategy adapted with the multiple shooting technique via three-step iterative method. Two numerical examples will be tested. The results show that the computational complexity of these methods is reliable to estimate the cost of these methods in term of the execution time. We conclude that the two-point block method has better computational performance compare to the one-point method as the total number of steps is larger

On the solution of two point boundary value problems with two point direct method.

The two point boundary value problems (BVPs) occur in a wide variety of applications especially in sciences such as chemistry and biology. In this paper, we propose two point direct method of order six for solving nonlinear two point boundary value problems directly. This method is presented in a simple form of Adams Mouton type and determines the approximate solution at two point simultaneously. The method will be implemented using constant step size via shooting technique adapted with three-step iterative method. Numerical results are given to compare the efficiency of the proposed method with the Runge-Kutta and bvp4c method

Solving directly two point non-linear boundary value problems using direct Adams Moulton method.

Problem statement: In this study, a direct method of Adams Moulton type was developed for solving non linear two point Boundary Value Problems (BVPs) directly. Most of the existence researches involving BVPs will reduced the problem to a system of first order Ordinary Differential Equations (ODEs). This approach is very well established but it obviously will enlarge the systems of first order equations. However, the direct method in this research will solved the second order BVPs directly without reducing it to first order ODEs. Approach: Lagrange interpolation polynomial was applied in the derivation of the proposed method. The method was implemented using constant step size via shooting technique in order to determine the approximated solutions. The shooting technique will employ the Newton’s method for checking the convergent of the guessing values for the next iteration. Results: Numerical results confirmed that the direct method gave better accuracy and converged faster compared to the existing method. Conclusion: The proposed direct method is suitable for solving two point non linear boundary value problems

Study of predictor corrector block method via multiple shooting to Blasius and Sakiadis flow

In this paper, a predictor corrector two-point block method is proposed to solve the well-known Blasius and Sakiadis flow numerically. The Blasius and Sakiadis flow will be modeled by a third order boundary value problem. The main motivation of this study is to provide a new method that can solve the higher order BVP directly without reducing it to a system of first order equation. Two approximate solutions will be obtained simultaneously in a single step by using predictor corrector two-point block method able to solve the third order boundary value problem directly. The proposed direct predictor corrector two-point block method will be adapted with multiple shooting techniques via a three-step iterative method. The advantage of the proposed code is that the multiple shooting will converge faster than the shooting method that has been implemented in other software. The developed code will automatically choose the guessing values in order to solve the given problems. Some numerical results are presented and a comparison to the existing methods has been included to show the performance of the proposed method for solving Blasius and Sakiadis flow

Solving boundary layer problem using fifth order block method

A fifth order block method of Adam’s type is presented to obtain the numerical solution of the boundary layer problem. The boundary layer problem we handle in this research is nanofluid over a moving surface in a flowing fluid. It is modelled as a system of combination of third order and second order differential equations subject to the two point boundary conditions. The block method will implemented with variable step size strategy and multiple shooting technique to solve this boundary layer problem. Two approximate solutions will obtain simultaneously using the same back value. The boundary layer problem will be solved directly without reducing to the system of first order equations. The numerical results are presented and compared to the existing method

Solving nonlinear two point boundary value problem using two step direct method

In this paper, we present two step direct method of Adams Moulton type (2PDAM4 and 2PDAM5) for solving nonlinear two point boundary value problems (BVPs) directly. The two step direct method will be utilised to obtain a series solution of the initial value problems at two steps simultaneously. These methods will solve the nonlinear second order BVPs by shooting technique using constant step size. Three step iterative method is considered as a procedure for solving the nonlinear equations and the convergence of the shooting technique. Numerical results are given to illustrate the efficiency and performance of the direct method by the shooting technique with root finding via three-step iterative method for solving boundary value problems. The results clearly show that the two step direct method is able to produce good results compared to the existing method

Solving nonlinear system of third-order boundary value problems using block method

In this paper, we propose an algorithm of two-point block method to solve the nonlinear system of third-order boundary value problems directly. The proposed method is presented in a simple form of Adams type and two approximate solutions will be obtained simultaneously with the block method using variable step size strategy. The method will be implemented with the multiple shooting technique via the three-step iterative method to generate the missing initial value. Most of the existence method will reduce the third-order boundary value problems to a system of first order equations where the systems of six equations need to be solved. The method we proposed in this paper will solve the third-order boundary value problems directly. Two numerical examples are given to illustrate the efficiency of the proposed method

New algorithm of two-point block method for solving boundary value problem with dirichlet and Neumann boundary conditions

Two-point block method with variable step-size strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the three-step iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy

Study of a quadrupole ion trap with damping force by the two-point one block method

RATIONALE: The capabilities and performances of a quadrupole ion trap under damping force based on collisional cooling is of particular importance in high-resolution mass spectrometry and should be analyzed by Mathieu's differential solutions. These solutions describe the stability and instability of the ion's trajectories confined in quadrupole devices. One of the methods for solving Mathieu's differential equation is a two-point one block method. In this case, Mathieu's stability diagram, trapping parameters az and qz and the secular frequency of the ion motion wz, can be derived in a precise manner. The two-point one block method (TPOBM) of Adams Moulton type is presented to study these parameters with and without the effect of damping force and compared to the 5th-order Runge–Kutta method (RKM5). The simulated results show that the TPOBM is more accurate and 10 times faster than the RKM5. The physical properties of the confined ions in the r and z axes are illustrated and the fractional mass resolutions m/Δm of the confined ions in the first stability region were analyzed by the RKM5 and the TPOBM. METHODS: The Lagrange interpolation polynomial was applied in the derivation of the proposed method. The proposed method will be utilized to obtain a series solution directly without reducing it to first order equations. RESULTS: The problem was tested with the ion trajectories in real time with and without the effect of damping force using constant step size. Numerical results from the two-point one block method have been compared with the fifth order Runge–Kutta method. CONCLUSIONS: The proposed two-point one block method has a potential application to solve complicated linear and nonlinear equations of the charged particle confinement in a quadrupole field especially in fine tuning accelerators, and, generally speaking, in physics of high energy

Solving second order delay differential equations by direct two and three point one-step block method

In this paper we present a two point and three point one-step block method for solving second order delay differential equations (DDEs). The one-step block method will solve directly the second order DDEs without reducing to first order equations. The two point and three point one-step block method will compute the solutions for the DDEs at two and three points simultaneously along the interval. These methods will solve the retarded type of DDE of single delay using constant step size. The P- stability and Q-stability are also discussed. The numerical results are presented to illustrate the performance of those block method for solving delay differential equations