Abstract
This paper is concerned with the numerical stability of Runge-Kutta methods for a class of nonlinear functional differential and functional equations. The sufficient conditions for the stability and asymptotic stability of(k,l)-algebraically stable Runge-Kutta methods are derived. A numerical test is given to confirm the theoretical results.
Highlights
This paper is concerned with the numerical solution of the following nonlinear functional differential and functional equations (FDFEs): y (t) = f (t, y (t), y (t − τ), z (t − τ)), z (t) = g (t, y (t), y (t − τ), z (t − τ)), (1)
In terms of Corollary 7, the corresponding methods are globally stable for solving the nonlinear FDFEs of the class D(α, β1, β2, γ1, γ2, δ) which satisfy the condition α + β1 + β2 + β2(γ1 + γ2)/(1 − δ) ≤ 0
In terms of Corollary 10, the corresponding methods are asymptotically stable for solving the nonlinear FDFEs of the class D(α, β1, β2, γ1, γ2, δ) which satisfy the condition α + β1 + β2 + β2(γ1 + γ2)/(1 − δ) < 0
Summary
This paper is concerned with the numerical solution of the following nonlinear functional differential and functional equations (FDFEs): y (t) = f (t, y (t) , y (t − τ) , z (t − τ)) , z (t) = g (t, y (t) , y (t − τ) , z (t − τ)) ,. Systems of the form (1) are sometimes called hybrid systems [1] or coupled delay differential and difference equations [2, 3] They arise widely in the fields of science and technology, such as control systems, physics, and biology (see [1,2,3,4,5,6] and the references therein).
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