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Boundary Value Problem

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A one-dimensional boundary value problem (BVP) is an ordinary differential equation, plus some boundary conditions (constraints) equal to the order of the differential equation (the order is the number of the highest derivative).

These problems are similar to initial value problems, which have conditions specified for the lowest end of the domain (the “initial” values). Boundary values are minimum or maximum values for some physical boundary.
boundary value problem

The conditions might involve solution values at two or more points, its derivatives, or both. For boundary value problems with some kind of physical relevance, conditions are usually imposed at two separate points. When solving boundary value problems, we are only interested in a solution between the two points.

Simple Example of a Boundary Value Problem

Example question: Find a function that satisfies the equation f′(x) = 2x for any x-values between 0 and 1. The function has a boundary value of 3 when x = 1.

Solution: Many functions can satisfy the equation f′(x) = 2x for any x-values between 0 and 1, but not all will meet the requirement for the stated boundary values (3, when x = 1).


For example, the function f(x) = x2 satisfies the differential equation, but it fails to satisfy the specified boundary values (as stated in the question, the function has a boundary value of 3 when x = 1). Plugging in x = 1, we get: f(1) = 12 = 1.

The function f(x) = x2 + 2 satisfies the differential equation and the given boundary values. The first derivative of f(x) = x2 + 2 = x2, and (plugging in the boundary values):
12 + 2 = 3.

References

Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C. Computational Differential Equations. Lund: Studentlitteratur, 1996.
Powers, D. Boundary Value Problems: And Partial Differential Equations. Academic Press. (2006).
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. “Two Point Boundary Value Problems” and “Boundary Values Problems.” Ch. 17 and part of §19.0 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 753-787 and 829-833, 1992.

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Stephanie Glen. "Boundary Value Problem" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/boundary-value-problem/
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