Abstract
In this research, a six-order, fully implicit Block Backward Differentiation Formula with two off-step points (BBDFO(6)), for the integration of first-order ordinary differential equations (ODEs) that exhibit stiffness, is proposed. The order, consistency and stability properties of the method are discussed, and the method is found to be zero stable and consistent. Hence, the method is convergent. The numerical comparisons with the existing methods of a similar type are given to demonstrate the accuracy of the derived method.
Highlights
We consider the numerical solution of the first order initial value problem (IVP) of ordinary differential equations (ODEs) in the form of y0 (x) = f (x, y), y(a) = μ, x ∈ [a, b]
We considered the three points yn−2, yn−1, yn of equal ste y h = xn+1 − xn as the starting values and the two off-step points as yn+1⁄2 and yn+3
The stability analysis of the BBDFO indicates that the method is consistent and zero stable; the BBDFO(6) is convergent
Summary
We consider the numerical solution of the first order initial value problem (IVP) of ordinary differential equations (ODEs) in the form of y0 (x) = f (x, y), y(a) = μ, x ∈ [a, b] (1). Equation (1) is said to be linear if f (x, y) = A(x) y + Φ(x), where A(x) is a constant d × d matrix and Φ(x) is a d-dimensional vector assumed to be continuously differentiable, and if it satisfies the Lipschitz conditions as given in [1], which guarantees the existence and uniqueness of the solution of Equation (1). The problems in the form of Equation (1) can be classified into two types. The first type is non-stiff ODEs, for which all of the components evolve simultaneously and on comparable time-scales. The second type is stiff ODEs. The first appearance of the term “stiff” is in the paper by [2] on the problems in chemical kinetics. There are various definitions of stiffness given in the literature since there is no universally accepted definition
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