Overview of Initial (IVPs) and Boundary Value Problems (BVPs)

DSolve can be used for finding the general solution to a differential equation or system of differential equations. The general solution gives information about the structure of the complete solution space for the problem. However, in practice, one is often interested only in particular solutions that satisfy some conditions related to the area of application. These conditions are usually of two types.

• The solution and/or its derivatives are required to have specific values at a single point, for example, and . Such problems are traditionally called initial value problems (IVPs) because the system is assumed to start evolving from the fixed initial point (in this case, 0).
• The solution is required to have specific values at a pair of points, for example, and . These problems are known as boundary value problems (BVPs) because the points 0 and 1 are regarded as boundary points (or edges) of the domain of interest in the application.

The symbolic solution of both IVPs and BVPs requires knowledge of the general solution for the problem. The final step, in which the particular solution is obtained using the initial or boundary values, involves mostly algebraic operations, and is similar for IVPs and for BVPs.

IVPs and BVPs for linear differential equations are solved rather easily since the final algebraic step involves the solution of linear equations. However, if the underlying equations are nonlinear, the solution could have several branches, or the arbitrary constants from the general solution could occur in different arguments of transcendental functions. As a result, it is not always possible to complete the final algebraic step for nonlinear problems. Finally, if the underlying equations have piecewise (that is, discontinuous) coefficients, an IVP naturally breaks up into simpler IVPs over the regions in which the coefficients are continuous.

Linear IVPs and BVPs

To begin, consider an initial value problem for a linear first-order ODE.

It should be noted that, in contrast to initial value problems, there are no general existence or uniqueness theorems when boundary values are prescribed, and there may be no solution in some cases.

The previous discussion of linear equations generalizes to the case of higher-order linear ODEs and linear systems of ODEs.

Nonlinear IVPs and BVPs

Many real-world applications require the solution of IVPs and BVPs for nonlinear ODEs. For example, consider the logistic equation, which occurs in population dynamics.

It may not always be possible to obtain a symbolic solution to an IVP or BVP for a nonlinear equation. Numerical methods may be necessary in such cases.

IVPs with Piecewise Coefficients

The differential equations that arise in modern applications often have discontinuous coefficients. DSolve can handle a wide variety of such ODEs with piecewise coefficients. Some of the functions used in these equations are UnitStep, Max, Min, Sign, and Abs. These functions and combinations of them can be converted into Piecewise objects.

A piecewise ODE can be thought of as a collection of ODEs over disjoint intervals such that the expressions for the coefficients and the boundary conditions change from one interval to another. Thus, different intervals have different solutions, and the final solution for the ODE is obtained by patching together the solutions over the different intervals.

If there are a large number of discontinuities in a problem, it is convenient to use Piecewise directly in the formulation of the problem.