Md. Amirul Islam - Academia.edu (original) (raw)
Assistant professor,
Dept of mathematics
Uttara university ,Dhaka
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In this paper, We have used Euler method and Runge-kutta method for finding approximate solutions... more In this paper, We have used Euler method and Runge-kutta method for finding approximate solutions of ordinary differential equations(ODE) in initial value problems(IVP). Numerical examples are considered to illustrate the efficiency and convergence ofthe two methods. Numerical results show that the proposed two method are very effective and efficient.We have investigated and computed the error of the proposed two methods. The approximated solutions with different step-size of the methods and analytical solutions are computed in Mathematica software. Approximation accuracy comparison between Euler method and Runge-Kutta methods for ordinary differential equations are done by finding the absolute error. The absolute errors produced by the proposed methods can then be analyzed to determine which one provides more accurate results.
This paper mainly presents Euler method and fourth‐order Runge Kutta Method (RK4) for solving ini... more This paper mainly presents Euler method and fourth‐order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac‐ curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computa‐ tional effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed meth‐ ods for different step sizes to examine superiority. Several numerical examples are given to dem‐ onstrate the reliability and efficiency.
In this paper, we solve numerically initial value problems for ordinary differential equations by... more In this paper, we solve numerically initial value problems for ordinary differential equations by Euler method .The proposed method is quite efficient and is practically well suited for solving these problems. Several examples are presented to demonstrate the accuracy and easy implementation of the proposed method. Our approximate solutions are compared with the exact solutions. The approximate solutions converge to the exact solutions monotonically. Finally we investigate and compute the error of proposed method for different step size.
In this paper, We have used Euler method and Runge-kutta method for finding approximate solutions... more In this paper, We have used Euler method and Runge-kutta method for finding approximate solutions of ordinary differential equations(ODE) in initial value problems(IVP). Numerical examples are considered to illustrate the efficiency and convergence ofthe two methods. Numerical results show that the proposed two method are very effective and efficient.We have investigated and computed the error of the proposed two methods. The approximated solutions with different step-size of the methods and analytical solutions are computed in Mathematica software. Approximation accuracy comparison between Euler method and Runge-Kutta methods for ordinary differential equations are done by finding the absolute error. The absolute errors produced by the proposed methods can then be analyzed to determine which one provides more accurate results.
This paper mainly presents Euler method and fourth‐order Runge Kutta Method (RK4) for solving ini... more This paper mainly presents Euler method and fourth‐order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac‐ curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computa‐ tional effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed meth‐ ods for different step sizes to examine superiority. Several numerical examples are given to dem‐ onstrate the reliability and efficiency.
In this paper, we solve numerically initial value problems for ordinary differential equations by... more In this paper, we solve numerically initial value problems for ordinary differential equations by Euler method .The proposed method is quite efficient and is practically well suited for solving these problems. Several examples are presented to demonstrate the accuracy and easy implementation of the proposed method. Our approximate solutions are compared with the exact solutions. The approximate solutions converge to the exact solutions monotonically. Finally we investigate and compute the error of proposed method for different step size.