A promising exponentially-fitted two-derivative Runge–Kutta–Nystrom method for solving y′′=f(x,y): Application to Verhulst logistic growth model

Explicit exponentially-fitted two-derivative Runge–Kutta–Nystrom method with single f-function and multiple third derivatives is proposed for solving special type of second-order ordinary differential equations with exponential solutions. B-series and rooted tree theory for the proposed method are developed for the derivation of order conditions. Then, we build frequency-dependent coefficients for the proposed method by integrating the second-order initial value problem exactly with solution in the linear composition of set functions eλt and e−λt with λ∈R. An exponentially-fitted two-derivative Runge–Kutta–Nystrom method with three stages fifth order is derived. Linear stability and stability region of the proposed method are analyzed. The numerical tests show that the proposed method is more effective than other existing methods with similar algebraic order in the integration of special type of second-order ordinary differential equations with exponential solutions. Also, the proposed method is used to solve a famous application problem, Verhulst logistic growth model and the result shows the proposed method still works effectively for solving this model.

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