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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).
Multivalued Differential Equations. Walter de Gruyter. ISBN 978-3110132120. Andres, J.; Górniewicz, Lech (2003). Topological Fixed Point Principles for Boundary Value Problems. Springer. ISBN 978-9048163182. Filippov, A.F. (1988). Differential equations with discontinuous right-hand sides. Kluwer Academic Publishers Group. ISBN 90-277-2699-X
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly.
R.P. Agarwal and D. O’Regan, An Introduction to Ordinary Differential Equations, Springer, New York, 2008. R.P. Agarwal and D. O’Regan, Ordinary and Partial Differential Equations with Special Functions, Fourier Series and Boundary Value Problems, Springer, New York, 2009.
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 ...
In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to an initial value problem.It involves finding solutions to the initial value problem for different initial conditions until one finds the solution that also satisfies the boundary conditions of the boundary value problem.
Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves ...
The differential equation is called oscillating if it has an oscillating solution. The number of roots carries also information on the spectrum of associated boundary value problems . Examples