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Solve boundary value problems for ordinary differential equations
sol = bvp4c(odefun,bcfun,solinit)
sol = bvp4c(odefun,bcfun,solinit,options)
solinit = bvpinit(x, yinit,
params)
A function handle that evaluates the differential equations f(x,y). It can have the form dydx = odefun(x,y) dydx = odefun(x,y,parameters) For a scalar x and a column vector y, odefun(x,y) must return a column vector, dydx, representing f(x,y). parameters is a vector of unknown parameters. | ||
A function handle that computes the residual in the boundary conditions. For two-point boundary value conditions of the form bc(y(a),y(b)), bcfun can have the form res = bcfun(ya,yb) res = bcfun(ya,yb,parameters) where ya and yb are column vectors corresponding to y(a) and y(b). parameters is a vector of unknown parameters. The output res is a column vector. See Multipoint Boundary Value Problems for a description of bcfun for multipoint boundary value problems. | ||
A structure containing the initial guess for a solution. You create solinit using the function bvpinit. solinit has the following fields. | ||
Ordered nodes of the initial mesh. Boundary conditions are imposed at a = solinit.x(1) and b = solinit.x(end). | ||
Initial guess for the solution such that solinit.y(:,i) is a guess for the solution at the node solinit.x(i). | ||
Optional. A vector that provides an initial guess for unknown parameters. | ||
The structure can have any name, but the fields must be named x, y, and parameters. You can form solinit with the helper function bvpinit. See bvpinit for details. | ||
Optional integration argument. A structure you create using the bvpset function. See bvpset for details. |
sol = bvp4c(odefun,bcfun,solinit) integrates a system of ordinary differential equations of the form
y′ = f(x,y)
on the interval [a,b] subject to two-point boundary value conditions bc(y(a),y(b)) = 0.
odefun and bcfun are function handles. See the function_handle reference page for more information.
Parameterizing Functions explains how to provide additional parameters to the function odefun, as well as the boundary condition function bcfun, if necessary.
bvp4c can also solve multipoint boundary value problems. See Multipoint Boundary Value Problems. You can use the function bvpinit to specify the boundary points, which are stored in the input argument solinit. See the reference page for bvpinit for more information.
The bvp4c solver can also find unknown parameters p for problems of the form
y′ = f(x,y, p)
0 = bc(y(a),y(b),p)
where p corresponds to parameters. You provide bvp4c an initial guess for any unknown parameters in solinit.parameters. The bvp4c solver returns the final values of these unknown parameters in sol.parameters.
bvp4c produces a solution that is continuous on [a,b] and has a continuous first derivative there. Use the function deval and the output sol of bvp4c to evaluate the solution at specific points xint in the interval [a,b].
sxint = deval(sol,xint)
The structure sol returned by bvp4c has the following fields:
Mesh selected by bvp4c | |
Approximation to y(x) at the mesh points of sol.x | |
Approximation to y′(x) at the mesh points of sol.x | |
Values returned by bvp4c for the unknown parameters, if any | |
Computational cost statistics (also displayed when the stats option is set with bvpset). |
The structure sol can have any name, and bvp4c creates the fields x, y, yp, parameters, and solver.
sol = bvp4c(odefun,bcfun,solinit,options) solves as above with default integration properties replaced by the values in options, a structure created with the bvpset function. See bvpset for details.
solinit = bvpinit(x, yinit, params) forms the initial guess solinit with the vector params of guesses for the unknown parameters.
bvp4c solves a class of singular boundary value problems, including problems with unknown parameters p, of the form
y′ = S · y/x + F(x,y,p)
0 = bc(y(0),y(b),p)
The interval is required to be [0, b] with b > 0. Often such problems arise when computing a smooth solution of ODEs that result from partial differential equations (PDEs) due to cylindrical or spherical symmetry. For singular problems, you specify the (constant) matrix S as the value of the 'SingularTerm' option of bvpset, and odefun evaluates only f(x,y,p). The boundary conditions must be consistent with the necessary condition S · y(0) = 0 and the initial guess should satisfy this condition.
bvp4c can solve multipoint boundary value problems where a = a_{0} < a_{1} < a_{2} < ...< a_{n} = b are boundary points in the interval [a,b]. The points a_{1},a_{2},...,a_{n−1} represent interfaces that divide [a,b] into regions. bvp4c enumerates the regions from left to right (from a to b), with indices starting from 1. In region k, [a_{k−1},a_{k}], bvp4c evaluates the derivative as
yp = odefun(x,y,k)
In the boundary conditions function
bcfun(yleft,yright)
yleft(:,k) is the solution at the left boundary of [a_{k−1},a_{k}]. Similarly, yright(:,k) is the solution at the right boundary of region k. In particular,
yleft(:,1) = y(a)
and
yright(:,end) = y(b)
When you create an initial guess with
solinit = bvpinit(xinit,yinit),
use double entries in xinit for each interface point. See the reference page for bvpinit for more information.
If yinit is a function, bvpinit calls y = yinit(x,k) to get an initial guess for the solution at x in region k. In the solution structure sol returned by bpv4c, sol.x has double entries for each interface point. The corresponding columns of sol.y contain the left and right solution at the interface, respectively.
To see an example that solves a three-point boundary value problem, type threebvp at the MATLAB^{®} command prompt.
Note: The bvp5c function is used exactly like bvp4c, with the exception of the meaning of error tolerances between the two solvers. If S(x) approximates the solution y(x), bvp4c controls the residual |S′(x) – f(x,S(x))|. This controls indirectly the true error |y(x) – S(x)|. bvp5c controls the true error directly. bvp5c is more efficient than bvp4c for small error tolerances. |
Boundary value problems can have multiple solutions. One purpose of the initial guess is to indicate which solution, among several, that you want.
The second-order differential equation
has exactly two solutions that satisfy the boundary conditions
Before using bvp4c to solve the problem, you need to rewrite the differential equation as a system of two first-order ODEs,
where and . This system has the required form
The function, f, and the boundary conditions, bc, are coded in MATLAB as the functions twoode and twobc.
function dydx = twoode(x,y)
dydx = [ y(2); -abs(y(1)) ];
function res = twobc(ya,yb)
res = [ ya(1); yb(1) + 2 ];
Form a guess structure consisting of an initial mesh of five equally spaced points in [0,4] and a guess of the constant values
solinit = bvpinit(linspace(0,4,5),[1 0]);
Solve the problem using bvp4c.
sol = bvp4c(@twoode,@twobc,solinit);
Evaluate the numerical solution at 100 equally spaced points and plot y(x).
x = linspace(0,4); y = deval(sol,x); plot(x,y(1,:))
To obtain the other solution of this problem, use the initial guess
solinit = bvpinit(linspace(0,4,5),[-1 0]); sol = bvp4c(@twoode,@twobc,solinit); x = linspace(0,4); y = deval(sol,x); plot(x,y(1,:))
This boundary value problem involves an unknown parameter. The task is to compute the fourth (q = 5) eigenvalue λ of Mathieu's equation
y" + (λ – 2q cos2x)y = 0.
Because the unknown parameter λ is present, this second-order differential equation is subject to three boundary conditions:
y′(0) = 0
y′(π) = 0
y(0) = 1
It is convenient to use local functions to place all the functions required by bvp4c in a single file.
function mat4bvp lambda = 15; solinit = bvpinit(linspace(0,pi,10),@mat4init,lambda); sol = bvp4c(@mat4ode,@mat4bc,solinit); fprintf('The fourth eigenvalue is approximately %7.3f.\n',... sol.parameters) xint = linspace(0,pi); Sxint = deval(sol,xint); plot(xint,Sxint(1,:)) axis([0 pi -1 1.1]) title('Eigenfunction of Mathieu''s equation.') xlabel('x') ylabel('solution y') % ------------------------------------------------------------ function dydx = mat4ode(x,y,lambda) q = 5; dydx = [ y(2) -(lambda - 2*q*cos(2*x))*y(1) ]; % ------------------------------------------------------------ function res = mat4bc(ya,yb,lambda) res = [ ya(2) yb(2) ya(1)-1 ]; % ------------------------------------------------------------ function yinit = mat4init(x) yinit = [ cos(4*x) -4*sin(4*x) ];
The differential equation (converted to a first-order system) and the boundary conditions are coded as local functions mat4ode and mat4bc, respectively. Because unknown parameters are present, these functions must accept three input arguments, even though some of the arguments are not used.
The guess structure solinit is formed with bvpinit. An initial guess for the solution is supplied in the form of a function mat4init. We chose y = cos 4x because it satisfies the boundary conditions and has the correct qualitative behavior (the correct number of sign changes). In the call to bvpinit, the third argument (lambda = 15) provides an initial guess for the unknown parameter λ.
After the problem is solved with bvp4c, the field sol.parameters returns the value λ = 17.097, and the plot shows the eigenfunction associated with this eigenvalue.
[1] Shampine, L.F., M.W. Reichelt, and J. Kierzenka, "Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c," available at http://www.mathworks.com/bvp_tutorial