Comparing solutions to initial value problems

Computing Exact Solutions of Initial Value Problems

Finding the exact solution of initial value problems is the main topic of this thesis. Because differential equations in practice can be hard to solve, we require algorithms to solve them on a computer. Since the solution of the equation needs to be stored on a computer, we first discuss the representation of real functions. We show that approximating a function by polynomials gives a suitable ...

Solutions of initial and boundary value problems via F-contraction mappings in metric-like space

We present sufficient conditions for the existence of solutions of second-order two-point boundary value and fractional order functional differential equation problems in a space where self distance is not necessarily zero. For this, first we introduce a Ciric type generalized F-contraction and F- Suzuki contraction in a metric-like space and give relevance to fixed point results. To illustrate...

Approximating Initial-Value Problems with Two-Point Boundary-Value Problems: BBM-Equation

Validated solutions of initial value problems for ordinary differential equations

Compared to standard numerical methods for initial value problems (IVPs) for ordinary diierential equations (ODEs), validated methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. The authors survey Taylor series methods for validated so...

Nonlinear Initial Value Problems with Positive Global Solutions

We give conditions on m(t), p(t), and f(t, y, z) so that the nonlinear initial value problem (1/m(t)(p(t)y0)0 + f(t, y, p(t)y0) = 0, for t > 0, y(0) = 0, limt→0+ p(t)y0(t) = B > 0, has at least one positive solution which exists for all t > 0, if B is sufficiently small. We allow a singularity at t = 0 so the solution y0(t) may be unbounded near t = 0.