Abstract:

In mathematics, in the field of partial differential equations, an initial value problem (also called the Cauchy problem) is a partial differential equation together with a specified value called the initial condition of the unknown function at a given point in the domain of the solution. In physics or other sciences, modelling a system frequently amounts to solving an initial value problem. Boundary value problems arise in several branches of physics, as any physical differential equation will have them. Problems involving the wave equation such as the determination of the normal nodes are often stated as boundary value problems. Boundary value problems are similar to initial value problems. A boundary value problem has conditions specified at the extremes (boundaries) of the independent variable in the equation, whereas an initial value problem has all the conditions specified at the same value of the independent variable (the value is at the lower boundary of the domain thus, the term initial value). To be useful in applications, an initial value problem as well as a boundary value problem should be well posed. This means that given the input to the problem, there exists a unique solution which depends continuously on the input.

Description:

Mathematical Formulation, Initial-Boundary, Value Problems, First Order PDES, Unbounded Domains, differential equations