Initial-Value Problem or Cauchy Problem

x=0

x=0

DEFINITIONINITIAL-VALUE PROBLEM

t=0

The preview below separates an initial-value problem into its main parts. We observe that the partial differential equation gives the rule, while the initial conditions give the starting information. Notice that ordinary finite boundary conditions are not part of the basic initial-value problem structure.

Dynamic Sandbox

Initializing Workspace

u=u(x,t)

t=0

Dynamic Sandbox

Initializing Workspace

NOTECAUCHY PROBLEM

The term Cauchy problem is often used for an initial-value problem. At this introductory level, we may understand a Cauchy problem as a problem where the initial data are prescribed and the solution is then determined from the partial differential equation . A more detailed mathematical definition may be studied later.

The checker below compares an initial-value problem with an initial boundary-value problem . Notice that adding ordinary finite boundary data changes the structure of the problem. A condition at infinity may describe behavior far away, but it is not an ordinary boundary condition at a finite endpoint.

Initializing Workspace

An initial-value problem is different from an initial boundary-value problem . An initial boundary-value problem contains both initial conditions and boundary conditions . An initial-value problem mainly uses initial data.

REMARKUNBOUNDED DOMAINS

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The preview below compares a finite interval with a larger and larger displayed portion of the real line. We observe that a finite interval has ordinary endpoints, while an unbounded domain is represented by extending farther in both directions. This helps explain why initial-value problems on unbounded domains may not use ordinary boundary conditions at finite endpoints.

Dynamic Sandbox

Initializing Workspace

The physical meaning is simple. If the domain is very large or unbounded, the solution may be treated as not being affected by ordinary boundary points. Then the initial data play the main role. However, this idea must be used carefully. Not every problem is determined by initial data alone. The exact conditions depend on the equation and on the physical situation.

NOTECONDITIONS AT INFINITY

Sometimes the behavior of the solution far away from the origin is important. In such cases, we may need to prescribe a condition at infinity. This may be a boundedness condition or another condition describing how the solution behaves when the spatial variable becomes very large. Such conditions are not ordinary boundary conditions at finite points. They describe the behavior of the solution far away in an unbounded domain.

The Initial-Value Problem or Cauchy Problem is closely related to Initial Conditions and Boundary Conditions . It also helps us compare different types of Mathematical Problems in Partial Differential Equations . When we study Well-Posed Problems, we shall ask whether an initial-value problem has a solution, whether that solution is unique, and whether small changes in the initial data produce only small changes in the solution.

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