d02tvf and its associated routines (d02tlf, d02txf, d02tyf and d02tzf) solve the two-point boundary value problem for a nonlinear system of ordinary differential equations
over an interval subject to () nonlinear boundary conditions at and () nonlinear boundary conditions at , where . Note that is the th derivative of the th solution component. Hence . The left boundary conditions at are defined as
and the right boundary conditions at as
See Section 9 for information on how boundary value problems of a more general nature can be treated.
d02tvf is used to specify an initial mesh, error requirements and other details. d02tlf is then used to solve the boundary value problem.
The solution routine d02tlf proceeds as follows. A modified Newton method is applied to the equations
and the boundary conditions. To solve these equations numerically the components are approximated by piecewise polynomials using a monomial basis on the th mesh sub-interval. The coefficients of the polynomials form the unknowns to be computed. Collocation is applied at Gaussian points
where is the th collocation point in the th mesh sub-interval. Continuity at the mesh points is imposed, that is
where is the right-hand end of the th mesh sub-interval. The linearized collocation equations and boundary conditions, together with the continuity conditions, form a system of linear algebraic equations, an almost block diagonal system which is solved using special linear solvers. To start the modified Newton process, an approximation to the solution on the initial mesh must be supplied via the procedure argument guess of d02tlf.
The solver attempts to satisfy the conditions
where is the approximate solution for the th solution component and tols is supplied by you. The mesh is refined by trying to equidistribute the estimated error in the computed solution over all mesh sub-intervals, and an extrapolation-like test (doubling the number of mesh sub-intervals) is used to check for (1).
Ascher U M and Bader G (1987) A new basis implementation for a mixed order boundary value ODE solver SIAM J. Sci. Stat. Comput.8 483–500
Ascher U M, Christiansen J and Russell R D (1979) A collocation solver for mixed order systems of boundary value problems Math. Comput.33 659–679
Ascher U M, Mattheij R M M and Russell R D (1988) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Prentice–Hall
Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Keller H B (1992) Numerical Methods for Two-point Boundary-value Problems Dover, New York
Schwartz I B (1983) Estimating regions of existence of unstable periodic orbits using computer-based techniques SIAM J. Sci. Statist. Comput.20(1) 106–120
1: – IntegerInput
On entry: , the number of ordinary differential equations to be solved.
2: – Integer arrayInput
On entry: must contain , the order of the th differential equation, for .
, for .
3: – IntegerInput
On entry: , the number of left boundary conditions defined at the left-hand end, ().
4: – IntegerInput
On entry: , the number of right boundary conditions defined at the right-hand end, ().
5: – IntegerInput
On entry: the number of collocation points to be used in each mesh sub-interval.
, where .
6: – Real (Kind=nag_wp) arrayInput
On entry: must contain the error requirement for the th solution component.
, for .
7: – IntegerInput
On entry: the maximum number of mesh points to be used during the solution process.
8: – IntegerInput
On entry: the number of points to be used in the initial mesh of the solution process.
9: – Real (Kind=nag_wp) arrayInput
On entry: the positions of the initial nmesh mesh points. The remaining elements of mesh need not be set. You should try to place the mesh points in areas where you expect the solution to vary most rapidly. In the absence of any other information the points should be equally distributed on .
must contain the left boundary point, , and must contain the right boundary point, .
, for .
10: – Integer arrayInput
On entry: specifies whether or not the initial mesh point defined in , for , should be a fixed point in all meshes computed during the solution process. The remaining elements of ipmesh need not be set.
Indicates that should be a fixed point in all meshes.
Indicates that is not a fixed point.
and , (i.e., the left and right boundary points, and , must be fixed points, in all meshes);
or , for .
11: – Real (Kind=nag_wp) arrayCommunication Array
On exit: contains information for use by d02tlf. This must be the same array as will be supplied to d02tlf. The contents of this array must remain unchanged between calls.
12: – IntegerInput
On entry: the dimension of the array rcomm as declared in the (sub)program from which d02tvf is called. If , a communication array size query is requested. In this case there is an immediate return with communication array dimensions stored in icomm; contains the required dimension of rcomm, while contains the required dimension of icomm.
, or , where and .
13: – Integer arrayCommunication Array
On exit: contains information for use by d02tlf. This must be the same array as will be supplied to d02tlf. The contents of this array must remain unchanged between calls. If , a communication array size query is requested. In this case, on immediate return, will contain the required dimension for rcomm while will contain the required dimension for icomm.
14: – IntegerInput
On entry: the dimension of the array icomm as declared in the (sub)program from which d02tvf is called. If , a communication array size query is requested. In this case icomm need only be of dimension in order to hold the required communication array dimensions for the given problem and algorithmic parameters.
if , ;
15: – IntegerInput/Output
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, or does not equal .
On entry, or for some .
On entry, . Constraint: .
On entry, . Constraint: or .
On entry, .
Constraint: for all .
On entry, and .
On entry, and .
On entry, .
On entry, , and .
On entry, and .
Constraint: and .
On entry, .
On entry, the elements of mesh are not strictly increasing.
On entry, .
Constraint: for all .
An unexpected error has been triggered by this routine. Please
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
Parallelism and Performance
d02tvf is not threaded in any implementation.
For problems where sharp changes of behaviour are expected over short intervals it may be advisable to:
cluster the initial mesh points where sharp changes in behaviour are expected;
maintain fixed points in the mesh using the argument ipmesh to ensure that the remeshing process does not inadvertently remove mesh points from areas of known interest before they are detected automatically by the algorithm.
Nonseparated Boundary Conditions
A boundary value problem with nonseparated boundary conditions can be treated by transformation to an equivalent problem with separated conditions. As a simple example consider the system
on subject to the boundary conditions
By adjoining the trivial ordinary differential equation
which implies , and letting , say, we have a new system
subject to the separated boundary conditions
There is an obvious overhead in adjoining an extra differential equation: the system to be solved is increased in size.
Multipoint Boundary Value Problems
Multipoint boundary value problems, that is problems where conditions are specified at more than two points, can also be transformed to an equivalent problem with two boundary points. Each sub-interval defined by the multipoint conditions can be transformed onto the interval , say, leading to a larger set of differential equations. The boundary conditions of the transformed system consist of the original boundary conditions and the conditions imposed by the requirement that the solution components be continuous at the interior break-points. For example, consider the equation
subject to the conditions
where . This can be transformed to the system
subject to the boundary conditions
In this instance two of the resulting boundary conditions are nonseparated but they may next be treated as described above.
High Order Systems
Systems of ordinary differential equations containing derivatives of order greater than four can always be reduced to systems of order suitable for treatment by d02tvf and its related routines. For example suppose we have the sixth-order equation
Writing the variables and we obtain the system
which has maximal order four, or writing the variables and we obtain the system
which has maximal order three. The best choice of reduction by choosing new variables will depend on the structure and physical meaning of the system. Note that you will control the error in each of the variables and . Indeed, if you wish to control the error in certain derivatives of the solution of an equation of order greater than one, then you should make those derivatives new variables.
Fixed Points and Singularities
The solver routine d02tlf employs collocation at Gaussian points in each sub-interval of the mesh. Hence the coefficients of the differential equations are not evaluated at the mesh points. Thus, fixed points should be specified in the mesh where either the coefficients are singular, or the solution has less smoothness, or where the differential equations should not be evaluated. Singular coefficients at boundary points often arise when physical symmetry is used to reduce partial differential equations to ordinary differential equations. These do not pose a direct numerical problem for using this code but they can severely impact its convergence.
The solver routine d02tlf requires an external routine fjac to evaluate the partial derivatives of with respect to the elements of (). In cases where the partial derivatives are difficult to evaluate, numerical approximations can be used. However, this approach might have a negative impact on the convergence of the modified Newton method. You could consider the use of symbolic mathematic packages and/or automatic differentiation packages if available to you.
See Section 10 in d02tzf for an example using numerical approximations to the Jacobian. There central differences are used and each is assumed to depend on all the components of . This requires two evaluations of the system of differential equations for each component of . The perturbation used depends on the size of each component of and a minimum quantity dependent on the machine precision. The cost of this approach could be reduced by employing an alternative difference scheme and/or by only perturbing the components of which appear in the definitions of the . A discussion on the choice of perturbation factors for use in finite difference approximations to partial derivatives can be found in Gill et al. (1981).
The following example is used to illustrate the treatment of nonseparated boundary conditions. See also d02tlf, d02txf, d02tyf and d02tzf, for the illustration of other facilities.
The following equations model of the spread of measles. See Schwartz (1983). Under certain assumptions the dynamics of the model can be expressed as
subject to the periodic boundary conditions
Here and are respectively the proportions of susceptibles, infectives and latents to the whole population. ( years) is the latent period, ( years) is the infectious period and () is the population birth rate. is the contact rate where .
The nonseparated boundary conditions are treated as described in Section 9 by adjoining the trivial differential equations
that is and are constants. The boundary conditions of the augmented system can then be posed in the separated form
This is a relatively easy problem and an (arbitrary) initial guess of for each component suffices, even though two components of the solution are much smaller than .