Ordinary Differential Equations
Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations
1.0 Introduction
In mathematics, if y is a function of x, then an equation that involves x, y and one or more derivatives of y with respect to x is called an ordinary differential equation (ODE). The ODEs which do not have additive solutions are non-linear, and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary function in close form. In addition, the ODEs is use to solve many problems in real life such as cooling or warming law, radio-active decay, carbon dating and in social issue like predator-prey models and exponential growth model.
In this proposal, we are concerned with the numerical solution of initial value problem (IVP) with two fixed points for ODE. The general form is y '=(y-v1)(y-v2)gy, (1) given initial values yxn=yn, where v1<v2 ϵ R and g(y)≠0 is a bounded real-valued function with continuous derivatives. Assume the fixed points are y1x=0 and y2(x)=1.
Diagonally Implicit Two-point Block Backward Differentiation Formulas (DI2BBDF) is a new method from the continuation of the previous methods. One of the methods is block method which is used to compute k blocks and to calculate the current block where each block contain r points. The general form for r point k block is j=0kAjyn+j=hj=0kBjfn+j, (2) where Aj and Bj are r by r matrices.
The researchers then continue with backward differentiation formula, multiblock methods, implicit block backward differentiation formula (BBDF), fixed coefficients BBDF and finally stimulate diagonally implicit block method. The method is called diagonally implicit because the coefficients of the upper triangular matrix entries are zero.
The aim of this proposal, we are interested to compare the accuracy of the DI2BBDF with the one-step
1.0 Introduction
In mathematics, if y is a function of x, then an equation that involves x, y and one or more derivatives of y with respect to x is called an ordinary differential equation (ODE). The ODEs which do not have additive solutions are non-linear, and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary function in close form. In addition, the ODEs is use to solve many problems in real life such as cooling or warming law, radio-active decay, carbon dating and in social issue like predator-prey models and exponential growth model.
In this proposal, we are concerned with the numerical solution of initial value problem (IVP) with two fixed points for ODE. The general form is y '=(y-v1)(y-v2)gy, (1) given initial values yxn=yn, where v1<v2 ϵ R and g(y)≠0 is a bounded real-valued function with continuous derivatives. Assume the fixed points are y1x=0 and y2(x)=1.
Diagonally Implicit Two-point Block Backward Differentiation Formulas (DI2BBDF) is a new method from the continuation of the previous methods. One of the methods is block method which is used to compute k blocks and to calculate the current block where each block contain r points. The general form for r point k block is j=0kAjyn+j=hj=0kBjfn+j, (2) where Aj and Bj are r by r matrices.
The researchers then continue with backward differentiation formula, multiblock methods, implicit block backward differentiation formula (BBDF), fixed coefficients BBDF and finally stimulate diagonally implicit block method. The method is called diagonally implicit because the coefficients of the upper triangular matrix entries are zero.
The aim of this proposal, we are interested to compare the accuracy of the DI2BBDF with the one-step
References: Aguiar, J. V. and Ramos, H. 2011. “A numerical ODE solver that preserves the fixed points and their stability,” Journal of Computational and Applied Mathematics, vol. 235, no. 7, pp. 1856–1867. Cash, J. R. 1983. “The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae,” Computers & Mathematics with Applications, vol. 9, no. 5, pp. 645–657. Chu, M. T. and Hamilton, H. 1987. “Parallel solution of ODEs by multiblock methods,” SIAM Journal on Scientific and Statistical Computing, vol. 8, no. 3, pp. 342–353. Ibrahim, Z. B., Othman, K. I. and Suleiman, M. 2007. “Implicit r-point block backward differentiation formula for solving first-order stiff ODEs,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 558–565. Ibrahim, Z. B., Suleiman, M. B. and Othman, K. I. 2008. “Fixed coefficient block backward differentiation formula for the numerical solution of stiff odes,” Applied Mathematics and Computation, vol. 21, no. 3, pp. 508–520. Shampine, L. F. and Watts, H. A. 1969. “Block implicit one-step methods,” Mathematics of Computation, vol. 23, pp. 731–740.