The tool FLOW* performs Taylor model-based flowpipe construction for non-linear (polynomial) hybrid systems. FLOW* combines well-known Taylor model arithmetic techniques for guaranteed approximations of the continuous dynamics in each mode with a combination of approaches for handling mode invariants and discrete transitions. FLOW* supports a wide variety of optimizations including adaptive ste...

In this work, we present our numerical results of the application of Galerkin-based Isogeometric Analysis (IGA) to incompressible Navier–Stokes–Cahn–Hilliard (NSCH) equations in velocity–pressure– phase field–chemical potential formulation. For the approximation of the velocity and pressure fields, LBB compatible non-uniform rational B-spline spaces are used which can be regarded as smooth gene...

A CMOS current-mode companding multiplier/divider composed of three compact logarithm circuits and one compact exponential circuit is proposed. Approaches for constructing the logarithm circuits and the exponential circuit are based on second-order Taylor series approximations. By gathering the three logarithm results and passing through the exponential circuit, we can obtain the result of zp(x...

8. First-Order Equations: Numerical Methods 8.1. Numerical Approximations 2 8.2. Explicit and Implicit Euler Methods 3 8.3. Explicit One-Step Methods Based on Taylor Approximation 4 8.3.1. Explicit Euler Method Revisited 4 8.3.2. Local and Global Errors 4 8.3.3. Higher-Order Taylor-Based Methods (not covered) 5 8.4. Explicit One-Step Methods Based on Quadrature 6 8.4.1. Explicit Euler Method Re...

We consider second-order linear differential equations φ(x)y′′ + f(x)y′ + g(x)y = h(x) in the interval (−1, 1) with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. We consider φ(x), f(x), g(x) and h(x) analytic in a Cassini disk with foci at x = ±1 containing the interval (−1, 1). The two-point Taylor expansion of the solution y(x) at the extreme points ±1 is used to give a c...

This talk will present an effort to formalize Taylor Models in the Coq proof assistant. Machinecheckable correctness proofs are facilitated by an abstract viewpoint: Taylor models can be generalized to balls in the Chebyshev metric. Extensions of elementary functions are then explained as compositions of such balls. This approach also accommodates other polynomial approximation methods than Tay...

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the present paper, we demonstrate that such a view is only valid in very special cases, and in general the homotopy analysis method is far more robust. In particular...