About one characteristic initial value problem with prehistory and its investigation

Characteristic initial value problem and its application in quasi-local energy expression

A characteristic initial value problem for a strictly hyperbolic system

Consider the system Autt +Cuxx = f(x,t), (x,t) ∈ T for u(x,t) in R2, where A and C are real constant 2× 2 matrices, and f is a continuous function in R2. We assume that detC ≠ 0 and that the system is strictly hyperbolic in the sense that there are four distinct characteristic curves Γi, i= 1, . . . ,4, in xt-plane whose gradients (ξ1i,ξ2i) satisfy det[Aξ2 1i+ Cξ2 2i]= 0. We allow the character...

On a characteristic initial value problem in plasma physics

The relativistic Vlasov-Maxwell system of plasma physics is considered with initial data on a past light cone. This characteristic initial value problem arises in a natural way as a mathematical framework to study the existence of solutions isolated from incoming radiation. Conservation laws and a priori estimates of smooth solutions are first derived in a formal way. In the spherically symmetr...

Everything you need to know about Initial Value Problem Solvers

2.1 Definition: Lipschitz Function A real valued function f : D ⊆ R → R is called Lipschitz continuous or is said to satisfy a Lipschitz condition if there exists a constant K ≥ 0 such that for all x1, x2 in D |f(x1)− f(x2)| ≤ K|x1 − x2| (3) The inequality is (trivially) satisfied if x1 = x2. Otherwise, for x1 6= x2, one can equivalently define a function to be Lipschitz if and only if |f(x1)−f...

Solving initial value problem by different numerical methods: Practical investigation

Our aim was to study what kind of bases can be provided to understand the basic terms of differential equation through teaching mathematical material in secondary school and to what extent this basis has to be expanded so that we can help the demonstration of differential equation. So if we give up the usual expansion of mathematical device we have to find another method which is easy to algori...