Validated solutions of initial value problems for ordinary differential equations
The purpose of the paper is to review the most significant developments in the area of validated solutions of initial value problems (IVPs) for ordinary differential equations (ODEs), especially interval methods based on Taylor series, and to identify areas for future research. After a short introduction, Section 2 introduces interval-arithmetic operations, interval-valued functions, interval integration and discuss a method for efficient generation of Taylor series coefficients. Section 3 states in more detail the IVP that is the subject of the paper. Section 4 presents an overview of Taylor series methods for validated solutions of IVPs for ODEs. Section 5 examines three techniques for computing an a priori enclosure of the solution and selecting a stepsize. Section 6 explains the wrapping effect. Section 7 describes five algorithms for computing a tight enclosure and reducing the wrapping effect. Section 8 consider some particular stepsize and order control strategies. Section 9 describes a method based on controlling the defect of the computed solution.
scientific article; zbMATH DE number 1355447
scientific article