Initial Condition(redirected from Initial value problem)
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initial condition[i′nish·əl kən′dish·ən]
in the mathematical analysis of a process the state of the process at a given moment of time, taken as the initial moment. If the process is described by a differential equation, then the problem of finding solutions for a given initial condition is called the Cauchy problem. An initial condition for the equation
consists in specifying y, dy/dt, . . . , (dn-1y)/dtn-1 for a value t = t0. If n = 2 and y = y(t) is the law of motion of a point mass, then the initial condition specifies the position and velocity of the point at the moment t = t0 An initial condition for a partial differential equation is similarly defined. Thus, for the equation
of a free vibrating string, where u(t, x) is the deviation of the point x of the string at the moment t from the position of equilibrium on the x-axis, the initial condition specifies the initial shape uǀt = t0 = f(x) of the string and the initial velocities δu/ δ/1,= t0 of the points of the string. Any other argument may play the role of time. An initial condition is then specified for some value of that argument.