Boundary Value Problems

Defines an BVP problem. Documentation Page: https://docs.sciml.ai/DiffEqDocs/stable/types/bvp_types/

Mathematical Specification of a BVP Problem

To define a BVP Problem, you simply need to give the function $f$ and the initial condition $u_0$ which define an ODE:

\[\frac{du}{dt} = f(u,p,t)\]

along with an implicit function bc which defines the residual equation, where

\[bc(u,p,t) = 0\]

is the manifold on which the solution must live. A common form for this is the two-point BVProblem where the manifold defines the solution at two points:

\[u(t_0) = a u(t_f) = b\]

Problem Type

Constructors

TwoPointBVProblem{isinplace}(f, bc, u0, tspan, p=NullParameters(); kwargs...)
BVProblem{isinplace}(f, bc, u0, tspan, p=NullParameters(); kwargs...)

or if we have an initial guess function initialGuess(p, t) for the given BVP, we can pass the initial guess to the problem constructors:

TwoPointBVProblem{isinplace}(f, bc, initialGuess, tspan, p=NullParameters(); kwargs...)
BVProblem{isinplace}(f, bc, initialGuess, tspan, p=NullParameters(); kwargs...)

For any BVP problem type, bc must be inplace if f is inplace. Otherwise it must be out-of-place.

If the bvp is a StandardBVProblem (also known as a Multi-Point BV Problem) it must define either of the following functions

bc!(residual, u, p, t)
residual = bc(u, p, t)

where residual computed from the current u. u is an array of solution values where u[i] is at time t[i], while p are the parameters. For a TwoPointBVProblem, t = tspan. For the more general BVProblem, u can be all of the internal time points, and for shooting type methods u=sol the ODE solution. Note that all features of the ODESolution are present in this form. In both cases, the size of the residual matches the size of the initial condition.

If the bvp is a TwoPointBVProblem then bc must be a Tuple (bca, bcb) and each of them must define either of the following functions:

begin
    bca!(resid_a, u_a, p)
    bcb!(resid_b, u_b, p)
end
begin
    resid_a = bca(u_a, p)
    resid_b = bcb(u_b, p)
end

where resid_a and resid_b are the residuals at the two endpoints, u_a and u_b are the solution values at the two endpoints, and p are the parameters.

Parameters are optional, and if not given, then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

Fields

• f: The function for the ODE.
• bc: The boundary condition function.
• u0: The initial condition. Either the initial condition for the ODE as an initial value problem, or a Vector of values for $u(t_i)$ for collocation methods.
• tspan: The timespan for the problem.
• p: The parameters for the problem. Defaults to NullParameters
• kwargs: The keyword arguments passed onto the solves.

Special Keyword Arguments

• nlls: Specify that the BVP is a nonlinear least squares problem. Use Val(true) or Val(false) for type stability. By default this is automatically inferred based on the size of the input and outputs, however this is type unstable for any array type that doesn't store array size as part of type information. If we can't reliably infer this, we set it to Nothing. Downstreams solvers must be setup to deal with this case.

struct TwoPointBVProblem{iip}

Defines a second order BVP problem. Documentation Page: https://docs.sciml.ai/DiffEqDocs/stable/types/bvp_types/

Mathematical Specification of a second order BVP Problem

To define a second order BVP Problem, you simply need to give the function $f$ and the initial condition $u_0$ which define an ODE:

\[\frac{ddu}{dt} = f(du,u,p,t)\]

along with an implicit function bc which defines the residual equation, where

\[bc(du,u,p,t) = 0\]

is the manifold on which the solution must live. A common form for this is the two-point SecondOrderBVProblem where the manifold defines the solution at two points:

\[g(u(t_0),u'(t_0)) = 0 g(u(t_f),u'(t_f)) = 0\]

Problem Type

Constructors

TwoPointSecondOrderBVProblem{isinplace}(f, bc, u0, tspan, p=NullParameters(); kwargs...)
SecondOrderBVProblem{isinplace}(f, bc, u0, tspan, p=NullParameters(); kwargs...)

or if we have an initial guess function initialGuess(p, t) for the given BVP, we can pass the initial guess to the problem constructors:

TwoPointSecondOrderBVProblem{isinplace}(f, bc, initialGuess, tspan, p=NullParameters(); kwargs...)
SecondOrderBVProblem{isinplace}(f, bc, initialGuess, tspan, p=NullParameters(); kwargs...)

For any BVP problem type, bc must be inplace if f is inplace. Otherwise it must be out-of-place.

If the bvp is a StandardSecondOrderBVProblem (also known as a Multi-Point BV Problem) it must define either of the following functions

bc!(residual, du, u, p, t)
residual = bc(du, u, p, t)

where residual computed from the current u. u is an array of solution values where u[i] is at time t[i], while p are the parameters. For a TwoPointBVProblem, t = tspan. For the more general BVProblem, u can be all of the internal time points, and for shooting type methods u=sol the ODE solution. Note that all features of the ODESolution are present in this form. In both cases, the size of the residual matches the size of the initial condition.

If the bvp is a TwoPointSecondOrderBVProblem then bc must be a Tuple (bca, bcb) and each of them must define either of the following functions:

begin
    bca!(resid_a, du_a, u_a, p)
    bcb!(resid_b, du_b, u_b, p)
end
begin
    resid_a = bca(du_a, u_a, p)
    resid_b = bcb(du_b, u_b, p)
end

where resid_a and resid_b are the residuals at the two endpoints, u_a and u_b are the solution values at the two endpoints, du_a and du_b are the derivative of solution values at the two endpoints, and p are the parameters.

Parameters are optional, and if not given, then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

Fields

• f: The function for the ODE.
• bc: The boundary condition function.
• u0: The initial condition. Either the initial condition for the ODE as an initial value problem, or a Vector of values for $u(t_i)$ for collocation methods.
• tspan: The timespan for the problem.
• p: The parameters for the problem. Defaults to NullParameters
• kwargs: The keyword arguments passed onto the solves.

struct TwoPointSecondOrderBVProblem{iip}