This is the initial value problem : Given any s ( x ) in L ^ p ( \ mathbb R ), find a harmonic function \ phi ( x, y ) defined in the upper half-plane satisfying the following conditions:
42.
I am also thinking along the lines of linearization and then checking for the uniqueness of the linear BVP, but am not able to find the conditions for uniqueness of a system of differential equations when constraints can be on either end ( unlike an initial value problem ).
43.
The power series method will give solutions only to initial value problems ( opposed to boundary value problems ), this is not an issue when dealing with linear equations since the solution may turn up multiple linearly independent solutions which may be combined ( by superposition ) to solve boundary value problems as well.
44.
The above definition of an integral curve " ? " for a vector field " X ", passing through " p " at time " t " 0, is the same as saying that " ? " is a local solution to the ordinary differential equation / initial value problem
45.
By 1973 his research group in UMIST had found solutions to the sine-Gordon and the self-induced transparency ( SIT ) equations for their multi-soliton solutions and gone on to both introduce, and to solve the initial value problem for, the system they called the Reduced Maxwell-Bloch ( RMB ) Equations.
46.
Furthermore, we say that the numerical method has " order p " if for any sufficiently smooth solution of the initial value problem, the local truncation error is O ( h ^ { p + 1 } ) ( meaning that there exist constants C and H such that | \ tau _ n | for all h ).
47.
For example, if the independent variable is time over the domain [ 0, 1 ], a boundary value problem would specify values for y ( t ) at both t = 0 and t = 1, whereas an initial value problem would specify a value of y ( t ) and y'( t ) at time t = 0.
48.
Although for technical reasons the precise equations formulated in the original ADM paper are rarely used in numerical simulations, most or all practical approaches to numerical relativity use a " 3 + 1 decomposition " of spacetime into three-dimensional space and one-dimensional time that is closely related to the ADM formulation, because the ADM procedure reformulates the Einstein field equations into a constrained initial value problem that can be addressed encoded on a computer for solution.
49.
These methods proceed by guessing a value of ?, solving an initial value problem defined by the boundary conditions at one endpoint, say, " a ", of the interval [ " a ", " b " ], comparing the value this solution takes at the other endpoint " b " with the other desired boundary condition, and finally increasing or decreasing ? as necessary to correct the original value.
50.
(The expression in parenthesis is just P ^ Tf ( x, t ) in the notation of the general method above . ) So a solution of the original initial value problem is obtained by starting with a solution to the problem with the same prescribed initial values problem but with " zero " external force, and adding to that ( integrating ) the contributions from the added force in the time intervals from " T " to " T " + " dT ":
