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In multivariable calculus, an initial value problem [a] (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem.
For instance, the differential equation dy / dt = y 2 with initial condition y(0) = 1 has the solution y(t) = 1/(1-t), which is not defined at t = 1. Nevertheless, if f is a differentiable function defined over a compact subset of R n , then the initial value problem has a unique solution defined over the entire R . [ 6 ]
In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps:
In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides the interval over which a solution is sought into several smaller intervals, solves an initial value problem in each of the smaller intervals ...
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
Initial value problem is an ordinary differential equation together with a specified value, called the initial condition; Intravenous pyelogram, a radiological procedure used to visualize abnormalities of the urinary system, including the kidneys, ureters, and bladder
Direct calculation shows that this is a solution of the differential equation at every point, including = and =. Uniqueness fails for these solutions on the interval c 1 ≤ x ≤ c 2 {\displaystyle c_{1}\leq x\leq c_{2}} , and the solutions are singular, in the sense that the second derivative fails to exist, at x = c 1 {\displaystyle x=c_{1 ...
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 of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
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