# bvpcol: Solves multipoint boundary value problems of ordinary... In bvpSolve: Solvers for Boundary Value Problems of Differential Equations

## Description

Solves Boundary Value Problems For Ordinary Differential Equations (ODE) or semi-explicit Differential-Algebraic Equations (DAE) with index at most 2.

It is possible to solve stiff ODE systems, by using an automatic continuation strategy

This is an implementation of the fortran codes colsys.f, colnew.f and coldae.f written by respectively U. Ascher, J. christiansen and R.D. Russell (colsys), U. Ascher and G. Bader (colnew) and U. Ascher and C. Spiteri.

The continuation strategy is an implementation of the fortran code colmod written by J.R. Cash, M.H. Wright and F. Mazzia.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```bvpcol (yini = NULL, x, func, yend = NULL, parms = NULL, order = NULL, ynames = NULL, xguess = NULL, yguess = NULL, jacfunc = NULL, bound = NULL, jacbound = NULL, leftbc = NULL, posbound = NULL, islin = FALSE, nmax = 1000, ncomp = NULL, atol = 1e-8, colp = NULL, bspline = FALSE, fullOut = TRUE, dllname = NULL, initfunc = dllname, rpar = NULL, ipar = NULL, nout = 0, outnames = NULL, forcings = NULL, initforc = NULL, fcontrol = NULL, verbose = FALSE, epsini = NULL, eps = epsini, dae = NULL, ...) ```

## Arguments

 `yini ` either a vector with the initial (state) variable values for the ODE system, or `NULL`. If `yini` is a vector, use `NA` for an initial value which is not specified. If `yini` has a `names` attribute, the names will be available within `func` and used to label the output matrix. If `yini = NULL`, then the boundary conditions must be specified via function `bound`; if not `NULL` then `yend` should also be not `NULL`. `x ` sequence of the independent variable for which output is wanted; the first value of `x` must be the initial value (at which `yini` is defined), the final value the end condition (at which `yend` is defined). `func ` either an R-function that computes the values of the derivatives in the ODE system (the model definition) at point `x`, or a character string giving the name of a compiled function in a dynamically loaded shared library. If `func` is an R-function, it must be defined as: `func = function(x, y, parms,...)`. `x` is the current point of the independent variable in the integration, `y` is the current estimate of the (state) variables in the ODE system. If the initial values `yini` has a names attribute, the names will be available inside `func`. `parms` is a vector or list of parameters; ... (optional) are any other arguments passed to the function. The return value of `func` should be a list, whose first element is a vector containing the values of the equations. In case where the equations are first-order, this will be the derivatives of `y` with respect to `x`. After this can come global values that are required at each point in `x`. If the problem is a DAE, then the algebraic equations should be the last. If `func` is a string, then `dllname` must give the name of the shared library (without extension) which must be loaded before `bvptwp` is called. See package vignette "bvpSolve" for more details. `yend ` either a vector with the final (state) variable values for the ODE system, or `NULL`; if `yend` is a vector, use `NA` for a final value which is not specified. If `yend` has a `names` attribute, and `yini` does not, the names will be available within the functions and used to label the output matrix. If `yend = NULL`, then the boundary conditions must be specified via function `bound`; if not `NULL` then `yini` should also be not `NULL`. `parms ` vector or a list with parameters passed to `func`, `jacfunc`, `bound` and `jacbound` (if present). If `eps` is given a value then it should be the **first** element in `parms`. `epsini ` the initial value of the continuation parameter. If `NULL` and `eps` is given a value, then `epsini` takes the default starting value of 0.5. For many singular perturbation type problems, the choice of 0.1 < `eps` < 1 represents a (fairly) easy problem. The user should attempt to specify an initial problem that is not ‘too’ challenging. `epsini` must be initialised strictly less than 1 and greater than 0. `eps ` the desired value of precision for which the user would like to solve the problem. `eps` must be less than or equal to `epsini`. If this is given a value, it must be the first value in `parms`. `ynames ` The names of the variables; used to label the output, and avaliable within the functions. If `ynames` is `NULL`, names can also be passed via `yini`, `yend` or `yguess`. `xguess ` Initial grid `x`, a vector. If `xguess` is given, so should `yguess` be. Supplying `xguess` and `yguess`, based on results from a previous (simpler) BVP-ODE can be used for model continuation, see example 2 of `bvptwp`. `yguess ` First guess values of `y`, corresponding to initial grid `xguess`; a matrix with number of rows equal to the number of variables, and whose number of columns equals the length of `xguess`. if the rows of `yguess` have a names attribute, the names will be available within the functions and used to label the output matrix. It is also allowed to pass the output of a previous run for continuation. This will use the information that is stored in the attributes `istate` and `rstate`. It will only work when for the previous run, `fullOut` was set equal to `TRUE` (the default). In this case, `xguess` need not be provided. See example 3b. `jacfunc ` jacobian (optional) - either an R-function that evaluates the jacobian of `func` at point `x`, or a string with the name of a function or subroutine in `dllname` that computes the Jacobian (see vignette `"bvpSolve"` for more about this option). If `jacfunc` is an R-function, it must be defined as: `jacfunc = function(x, y, parms,...)`. It should return the partial derivatives of `func` with respect to `y`, i.e. df(i,j) = dfi/dyj. See last example. If `jacfunc` is `NULL`, then a numerical approximation using differences is used. This is the default. `bound ` boundary function (optional) - only if `yini` and `yend` are not available. Either an R function that evaluates the i-th boundary element at point `x`, or a string with the name of a function or subroutine in `dllname` that computes the boundaries (see vignette `"bvpSolve"` for more about this option). If `bound` is an R-function, it should be defined as: `bound = function(i, y, parms, ...)`. It should return the i-th boundary condition. See last example. `jacbound ` jacobian of the boundary function (optional) - only if `bound` is defined. Either an R function that evaluates the gradient of the i-th boundary element with respect to the state variables, at point `x`, or a string with the name of a function or subroutine in `dllname` that computes the boundary jacobian (see vignette `"bvpSolve"` for more about this option). If `jacbound` is an R-function, it should be defined as: `jacbound = function(i, y, parms, ...)`. It should return the gradient of the i-th boundary condition. See examples. If `jacbound` is `NULL`, then a numerical approximation using differences is used. This is the default. `leftbc ` only if `yini` and `yend` are not available and `posbound` is not specified: the number of left boundary conditions. `posbound ` only used if `bound` is given: a vector with the position (in the mesh) of the boundary conditions - its values should be sorted - and it should be within the range of `x`; (posbound corresponds to fortran input "Zeta" in the colnew/colsys FORTRAN codes. ) See last example. Note that two-point boundary value problems can also be specified via `leftbc` (which is simpler). `islin ` set to `TRUE` if the problem is linear - this will speed up the simulation. `nmax ` maximal number of subintervals during the calculation. `order ` the order of each derivative in `func`. The default is that all derivatives are 1-st order, in which case `order` can be set = `NULL`. For higher-order derivatives, specifying the order can improve computational efficiency, but this interface is more complex. If `order` is not `NULL`, the number of equations in `func` must equal the length of `order`; the summed values of `order` must equal the number of variables (ncomp). The jacobian as specified in `jacfunc` must have number of rows = number of equations and number of columns = number of variables. `bound` and `jacbound` remain defined in the number of variables. See example 3 and 3b. `ncomp ` used if the model is specified by compiled code, the number of components (or equations). See package vignette `"bvpSolve"`. Also to be used if the boundary conditions are specified by `bound`, and there is no `yguess` `atol ` error tolerance, a scalar. `colp ` number of collocation points per subinterval. `bspline ` if `FALSE`, then code colnew is used the default, if `TRUE`, then fortran code colsys is used. Code colnew incorporates a new basis representation, while colsys uses b-splines. `fullOut ` if set to `TRUE`, then the collocation output required e.g. for continuation will be returned in attributes `rwork` and `iwork`. Use `attributes(out)\\$rwork, attributes(out)\\$rwork` to see their contents `dllname ` a string giving the name of the shared library (without extension) that contains all the compiled function or subroutine definitions referred to in `func`, `jacfunc`, `bound` and `jacbound`. Note that ALL these subroutines must be defined in the shared library; it is not allowed to merge R-functions with compiled functions. See package vignette `"bvpSolve"` or deSolve's package vignette `"compiledCode"`. `initfunc ` if not `NULL`, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See package vignette `"bvpSolve"`. `rpar ` only when ‘dllname’ is specified: a vector with double precision values passed to the dll-functions whose names are specified by `func` and `jacfunc`. `ipar ` only when ‘dllname’ is specified: a vector with integer values passed to the dll-functions whose names are specified by `func` and `jacfunc`. `nout ` only used if `dllname` is specified and the model is defined in compiled code: the number of output variables calculated in the compiled function `func`, present in the shared library. Note: it is not automatically checked whether this is indeed the number of output variables calculated in the dll - you have to perform this check in the code. See deSolve's package vignette `"compiledCode"`. `outnames ` only used if function is specified in compiled code and `nout` > 0: the names of output variables calculated in the compiled function. These names will be used to label the output matrix. The length of `outnames` should be = `nout`. `forcings ` only used if ‘dllname’ is specified: a list with the forcing function data sets, each present as a two-columned matrix, with (time,value); interpolation outside the interval [min(`times`), max(`times`)] is done by taking the value at the closest data extreme. This feature is included for consistency with the initial value problem solvers from package `deSolve`. See package vignette `"compiledCode"` from package `deSolve`. `initforc ` if not `NULL`, the name of the forcing function initialisation function, as provided in ‘dllname’. It MUST be present if `forcings` has been given a value. See package vignette `"compiledCode"` from package `deSolve`. `fcontrol ` A list of control parameters for the forcing functions. See package vignette `"compiledCode"` from package `deSolve`. `verbose ` when `TRUE`, writes messages (warnings) to the screen. `dae ` if the problem is a DAE, should be a list containing the `index` of the problem and the number of algebraic equations `nalg`. See example 5 `... ` additional arguments passed to the model functions.

## Details

If `eps` does not have a value and `dae` = NULL, then the method is based on an implementation of the Collocation methods called "colnew" and "colsys" to solve multi-point boundary value problems of ordinary differential equations.

The ODEs and boundary conditions are made available through the user-provided routines, `func` and vectors `yini` and `yend` or (optionally) `bound`. `bvpcol` can also solve multipoint boundary value problems (see one but last example).

The corresponding partial derivatives are optionally available through the user-provided routines, `jacfunc` and `jacbound`. Default is that they are automatically generated by R, using numerical differences.

The user-requested tolerance is provided through `atol`.

If the function terminates because the maximum number of subintervals was exceeded, then it is recommended that 'the program be run again with a larger value for this maximum.'

If `eps` does have a value, then the method is based on an implementation of the Collocation methods called "colmod". The type of problems which this is designed to solve typically involve a small positive parameter 0 < eps << 1. As eps becomes progressively smaller, the problem normally becomes increasingly difficult to approximate numerically (for example, due to the appearance of narrow transition layers in the profile of the analytic solution).

The idea of continuation is to solve a chain of problems in which the parameter eps decreases monotonically towards some desired value. That is, a sequence of problems is attempted to be solved:

epsini > eps1 > eps2 > eps3 > ..... > eps > 0

where `epsini` is a user provided starting value and `eps` is a user desired final value for the parameter.

If `dae` is not NULL, then it is assumed that a DAE has to be solved. In that case, `dae` should contain give the `index` of the DAE and the number of algebraic equations (`nalg`).

(this part comes from the comments in the code coldae). With respect to the dae, it should be noted that the code does not explicitly check the index of the problem, so if the index is > 2 then the code will not work well. The number of boundary conditions required is independent of the index. it is the user's responsibility to ensure that these conditions are consistent with the constraints. The conditions at the left end point must include a subset equivalent to specifying the index-2 constraints there. For an index-2 problem in hessenberg form, the projected collocation method of Ascher and Petzold  is used.

## Value

A matrix of class `bvpSolve`, with up to as many rows as elements in `x` and as many columns as elements in `yini` plus the number of "global" values returned in the second element of the return from `func`, plus an additional column (the first) for the `x`-value.

There will be one row for each element in `x` unless the solver returns with an unrecoverable error.

If `ynames` is given, or `yini`, `yend` has a names attribute, or `yguess` has named rows, the names will be used to label the columns of the output value.

The output will also have `attributes` `istate` and `rstate` which contain the collocation output required e.g. for continuation of a problem, unless `fullOutput` is `FALSE`

## Note

colnew.f (Bader and Ascher, 1987), is a modification of the code colsys.f (Ascher, Christiansen and Russell, 1981), which incorporates a new basis representation replacing b-splines, and improvements for the linear and nonlinear algebraic equation solvers. To toggle on/off colsys, set `bspline` = `TRUE`/`FALSE`

colmod is a revised version of the package colnew by Bader and Ascher (1987), which in turn is a modification of the package colsys by Ascher, Christiansen and Russell (1981). Colmod has been adapted to allow an automatic continuation strategy to be used (Cash et al., 1995).

The mesh selection algorithm used in colmod differs from that used in colnew

## Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

## References

U. Ascher, J. Christiansen and R. D. Russell, (1981) collocation software for boundary-value odes, acm trans. math software 7, 209-222.

G. Bader and U. Ascher, (1987) a new basis implementation for a mixed order boundary value ode solver, siam j. scient. stat. comput. 8, 487-483.

U. Ascher, J. Christiansen and R.D. Russell, (1979) a collocation solver for mixed order systems of boundary value problems, math. comp. 33, 659-679.

U. Ascher, J. Christiansen and R.D. Russell, (1979) colsys - a collocation code for boundary value problems, lecture notes comp.sc. 76, springer verlag, B. Childs et. al. (eds.), 164-185.

J. R. Cash, G. Moore and R. W. Wright, (1995) an automatic continuation strategy for the solution of singularly perturbed linear two-point boundary value problems, j. comp. phys. 122, 266-279.

U. Ascher and R. Spiteri, 1994. collocation software for boundary value differential-algebraic equations, siam j. scient. stat. comput. 15, 938-952.

U. Ascher and L. Petzold, 1991. projected implicit runge-kutta methods for differential- algebraic equations, siam j. num. anal. 28 (1991), 1097-1120.

`bvpshoot` for the shooting method

bvptwp for a MIRK formula

`diagnostics.bvpSolve`, for a description of diagnostic messages

`approx.bvpSolve`, for approximating solution in new values

`plot.bvpSolve`, for a description of plotting the output of the BVP solvers.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316``` ```## ============================================================================= ## Example 1: simple standard problem ## solve the BVP ODE: ## d2y/dt^2=-3py/(p+t^2)^2 ## y(t= -0.1)=-0.1/sqrt(p+0.01) ## y(t= 0.1)= 0.1/sqrt(p+0.01) ## where p = 1e-5 ## ## analytical solution y(t) = t/sqrt(p + t^2). ## ## The problem is rewritten as a system of 2 ODEs: ## dy=y2 ## dy2=-3p*y/(p+t^2)^2 ## ============================================================================= #-------------------------------- # Derivative function #-------------------------------- fun <- function(t, y, pars) { dy1 <- y dy2 <- - 3 * p * y / (p+t*t)^2 return(list(c(dy1, dy2))) } # parameter value p <- 1e-5 # initial and final condition; second conditions unknown init <- c(-0.1 / sqrt(p+0.01), NA) end <- c( 0.1 / sqrt(p+0.01), NA) # Solve bvp sol <- bvpcol(yini = init, yend = end, x = seq(-0.1, 0.1, by = 0.001), func = fun) plot(sol, which = 1) # add analytical solution curve(x/sqrt(p+x*x), add = TRUE, type = "p") diagnostics(sol) zoom <- approx(sol, xout = seq(-0.005, 0.005, by = 0.0001)) plot(zoom, which = 1, main = "zoom in on [-0.0005,0.0005]") ## ============================================================================= ## Example 1b: ## Same problem, now solved as a second-order equation ## and with different value of "p". ## ============================================================================= fun2 <- function(t, y, pars) { dy <- - 3 * p * y / (p+t*t)^2 list(dy) } p <- 1e-4 sol2 <- bvpcol(yini = init, yend = end, order = 2, x = seq(-0.1, 0.1, by = 0.001), func = fun2) # plot both runs at once: plot(sol, sol2, which = 1) ## ============================================================================= ## Example 1c: simple ## solve d2y/dx2 + 1/x*dy/dx + (1-1/(4x^2)y = sqrt(x)*cos(x), ## on the interval [1,6] and with boundary conditions: ## y(1)=1, y(6)=-0.5 ## ## Write as set of 2 odes ## dy/dx = y2 ## dy2/dx = - 1/x*dy/dx - (1-1/(4x^2)y + sqrt(x)*cos(x) ## ============================================================================= f2 <- function(x, y, parms) { dy <- y dy2 <- -1/x * y- (1-1/(4*x^2))*y + sqrt(x)*cos(x) list(c(dy, dy2)) } x <- seq(1, 6, 0.1) sol <- bvpcol(yini = c(1, NA), yend = c(-0.5, NA), bspline = TRUE, x = x, func = f2) plot(sol, which = 1) # add the analytic solution curve(0.0588713*cos(x)/sqrt(x) + 1/4*sqrt(x)*cos(x)+0.740071*sin(x)/sqrt(x)+ 1/4*x^(3/2)*sin(x), add = TRUE, type = "l") ## ============================================================================= ## Example 2. Uses continuation ## Test problem 24 ## ============================================================================= Prob24<- function(t, y, ks) { #eps is called ks here A <- 1+t*t AA <- 2*t ga <- 1.4 list(c(y,(((1+ga)/2 -ks*AA)*y*y-y/y- (AA/A)*(1-(ga-1)*y^2/2))/(ks*A*y))) } ini <- c(0.9129, NA) end <- c(0.375, NA) xguess <- c(0, 1) yguess <- matrix(nrow = 2, ncol = 2, 0.9 ) # bvpcol works with eps NOT too small, and good initial condition ... sol <- bvpcol(yini = ini, yend = end, x = seq(0, 1, by = 0.01), xguess = xguess, yguess = yguess, parms = 0.1, func = Prob24, verbose = FALSE) # when continuation is used: does not need a good initial condition sol2 <- bvpcol(yini = ini, yend = end, x = seq(0, 1, by = 0.01), parms = 0.05, func = Prob24, eps = 0.05) #zoom <- approx(sol2, xout = seq(0.01, 0.02, by = 0.0001)) #plot(zoom, which = 1, main = "zoom in on [0.01, 0.02]") sol3 <- bvpcol(yini = ini, yend = end, x = seq(0, 1, by = 0.01), parms = 0.01, func = Prob24 , eps = 0.01) sol4 <- bvpcol(yini = ini, yend = end, x = seq(0, 1, by = 0.01), parms = 0.001, func = Prob24, eps = 0.001) # This takes a long time ## Not run: print(system.time( sol5 <- bvpcol(yini = ini, yend = end, x = seq(0, 1, by = 0.01), parms = 1e-4, func = Prob24, eps = 1e-4) )) ## End(Not run) plot(sol, sol2, sol3, sol4, which = 1, main = "test problem 24", lwd = 2) legend("topright", col = 1:4, lty = 1:4, lwd = 2, legend = c("0.1", "0.05", "0.01", "0.001"), title = "eps") ## ============================================================================= ## Example 3 - solved with specification of boundary, and jacobians ## d4y/dx4 =R(dy/dx*d2y/dx2 -y*dy3/dx3) ## y(0)=y'(0)=0 ## y(1)=1, y'(1)=0 ## ## dy/dx = y2 ## dy2/dx = y3 (=d2y/dx2) ## dy3/dx = y4 (=d3y/dx3) ## dy4/dx = R*(y2*y3 -y*y4) ## ============================================================================= # derivative function: 4 first-order derivatives f1st<- function(x, y, S) { list(c(y, y, y, 1/S*(y*y - y*y) )) } # jacobian of derivative function df1st <- function(x, y, S) { matrix(nrow = 4, ncol = 4, byrow = TRUE, data = c( 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1*y/S, y/S, y/S, -y/S)) } # boundary g2 <- function(i, y, S) { if (i == 1) return (y) if (i == 2) return (y) if (i == 3) return (y - 1) if (i == 4) return (y) } # jacobian of boundary dg2 <- function(i, y, S) { if (i == 1) return(c(1, 0, 0, 0)) if (i == 2) return(c(0, 1, 0, 0)) if (i == 3) return(c(1, 0, 0, 0)) if (i == 4) return(c(0, 1, 0, 0)) } # we use posbound to specify the position of boundary conditions # we can also use leftbc = 2 rather than posbound = c(0,0,1,1) S <- 1/100 sol <- bvpcol(x = seq(0, 1, by = 0.01), ynames = c("y", "dy", "d2y", "d3y"), posbound = c(0, 0, 1, 1), func = f1st, parms = S, eps = S, bound = g2, jacfunc = df1st, jacbound = dg2) plot(sol) ## ============================================================================= ## Example 3b - solved with specification of boundary, and jacobians ## and as a higher-order derivative ## d4y/dx4 =R(dy/dx*d2y/dx2 -y*dy3/dx3) ## y(0)=y'(0)=0 ## y(1)=1, y'(1)=0 ## ============================================================================= # derivative function: one fourth-order derivative f4th <- function(x, y, S) { list(1/S * (y*y - y*y)) } # jacobian of derivative function df4th <- function(x, y, S) { matrix(nrow = 1, ncol = 4, byrow = TRUE, data = c( -1*y/S, y/S, y/S, -y/S)) } # boundary function - same as previous example # jacobian of boundary - same as previous # order = 4 specifies the equation to be 4th order # solve with bspline false S <- 1/100 sol <- bvpcol (x = seq(0, 1, by = 0.01), ynames = c("y", "dy", "d2y", "d3y"), posbound = c(0, 0, 1, 1), func = f4th, order = 4, parms = S, eps = S, bound = g2, jacfunc = df4th, jacbound = dg2 ) plot(sol) # Use (manual) continuation to find solution of a more difficult example # Previous solution collocation from sol passed ("guess = sol") sol2 <- bvpcol(x = seq(0, 1, by = 0.01), ynames = c("y", "dy", "d2y", "d3y"), posbound = c(0, 0, 1, 1), func = f4th, parms = 1e-6, order = 4, eps = 1e-6, bound = g2, jacfunc = df4th, jacbound = dg2 ) # plot both at same time plot(sol, sol2, lwd = 2) legend("bottomright", leg = c(100, 10000), title = "R = ", col = 1:2, lty = 1:2, lwd = 2) ## ============================================================================= ## Example 4 - a multipoint bvp ## dy1 = (y2 - 1)/2 ## dy2 = (y1*y2 - x)/mu ## over interval [0,1] ## y1(1) = 0; y2(0.5) = 1 ## ============================================================================= multip <- function (x, y, p) { list(c((y - 1)/2, (y*y - x)/mu)) } bound <- function (i, y, p) { if (i == 1) y -1 # at x=0.5: y2=1 else y # at x= 1: y1=0 } mu <- 0.1 sol <- bvpcol(func = multip, bound = bound, x = seq(0, 1, 0.01), posbound = c(0.5, 1)) plot(sol) # check boundary value sol[sol[,1] == 0.5,] ## ============================================================================= ## Example 5 - a bvp DAE ## ============================================================================= bvpdae <- function(t, x, ks, ...) { p1 <- p2 <- sin(t) dp1 <- dp2 <- cos(t) dx1 <- (ks + x - p2)*x + dp1 dx2 <- dp2 dx3 <- x res <- (x - p1)*(x - exp(t)) list(c(dx1, dx2, dx3, res), res = res) } boundfun <- function(i, x, par, ...) { if (i == 1) return(x - sin(0)) if (i == 2) return(x - 1) if (i == 3) return(x - sin(1)) if (i == 4) return((x - sin(1))*(x - exp(1))) # Not used here.. } x <- seq(0, 1, by = 0.01) mass <- diag(nrow = 4) ; mass[4, 4] <- 0 # solved using boundfun out <- bvpcol (func = bvpdae, bound = boundfun, x = x, parms = 1e-4, ncomp = 4, leftbc = 2, dae = list(index = 2, nalg = 1)) # solved using yini, yend out1 <- bvpcol (func = bvpdae, x = x, parms = 1e-4, yini = c(sin(0), NA, 1, NA), yend = c(NA, sin(1), NA, NA), dae = list(index = 2, nalg = 1)) # the analytic solution ana <- cbind(x, "1" = sin(x), "2" = sin(x), "3" = 1, "4" = 0, res = 0) plot(out, out1, obs = ana) ```

bvpSolve documentation built on June 30, 2020, 3:02 a.m.