Advanced Search

Question: How do I solve stiff nonlinear ODE coupled with independent variable

Question:How do I solve stiff nonlinear ODE coupled with independent variable

kjokil201 5 Maple 2016
ode dsolve numeric stiff odeplot differential-equations
December 17 2017
1

Hi there

I am trying to solve the ODE below

schechter_guo_v2.mw
 

> 

odeSG := {diff(z(t), t) = (-phi*z(t)*sqrt(F*phi*z(t)/(5*t))/(3*t)+1-H/(1-z(t)))/(phi*(S_oi-S_or-sqrt(F*phi*z(t)/(5*t)))), z(t0) = z0}

{diff(z(t), t) = (-(1/15)*phi*z(t)*5^(1/2)*(F*phi*z(t)/t)^(1/2)/t+1-H/(1-z(t)))/(phi*(S_oi-S_or-(1/5)*5^(1/2)*(F*phi*z(t)/t)^(1/2))), z(t0) = z0}

 (1)
> 

solSG := dsolve(odeSG, numeric, method = lsode, parameters = [phi, F, H, S_oi, S_or, t0, z0])

proc (x_lsode) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_lsode) else _xout := evalf(x_lsode) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _n, _y0, _ctl, _octl, _reinit, _errcd, _fcn, _i, _yini, _pars, _ini, _par; option `Copyright (c) 2002 by the University of Waterloo. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _ctl := array( 1 .. 39, [( 1 ) = (1), ( 2 ) = (t0), ( 3 ) = (t0), ( 4 ) = (1), ( 5 ) = (1), ( 6 ) = (10), ( 7 ) = (0), ( 9 ) = (0.1e-6), ( 8 ) = (z0), ( 11 ) = (0), ( 10 ) = (0.1e-6), ( 13 ) = (0), ( 12 ) = (0), ( 15 ) = (0), ( 14 ) = (0), ( 18 ) = (0), ( 19 ) = (0), ( 16 ) = (0), ( 17 ) = (0), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = (0), ( 21 ) = (0), ( 27 ) = (0), ( 26 ) = (0), ( 25 ) = (0), ( 24 ) = (0), ( 31 ) = (-1), ( 30 ) = (0), ( 29 ) = (0), ( 28 ) = (0), ( 36 ) = (0), ( 37 ) = (0), ( 38 ) = (0), ( 39 ) = (0), ( 32 ) = (7), ( 33 ) = (0), ( 34 ) = (0), ( 35 ) = (0) ] ); _octl := array( 1 .. 39, [( 1 ) = (1), ( 2 ) = (t0), ( 3 ) = (t0), ( 4 ) = (1), ( 5 ) = (1), ( 6 ) = (10), ( 7 ) = (0), ( 9 ) = (0.1e-6), ( 8 ) = (z0), ( 11 ) = (0), ( 10 ) = (0.1e-6), ( 13 ) = (0), ( 12 ) = (0), ( 15 ) = (0), ( 14 ) = (0), ( 18 ) = (0), ( 19 ) = (0), ( 16 ) = (0), ( 17 ) = (0), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = (0), ( 21 ) = (0), ( 27 ) = (0), ( 26 ) = (0), ( 25 ) = (0), ( 24 ) = (0), ( 31 ) = (-1), ( 30 ) = (0), ( 29 ) = (0), ( 28 ) = (0), ( 36 ) = (0), ( 37 ) = (0), ( 38 ) = (0), ( 39 ) = (0), ( 32 ) = (7), ( 33 ) = (0), ( 34 ) = (0), ( 35 ) = (0) ] ); _n := trunc(_ctl[1]); _yini := Array(0..8, {(1) = t0, (2) = z0, (3) = undefined, (4) = undefined, (5) = undefined, (6) = undefined, (7) = undefined, (8) = undefined}); _y0 := Array(0..8, {(1) = t0, (2) = z0, (3) = undefined, (4) = undefined, (5) = undefined, (6) = undefined, (7) = undefined, (8) = undefined}); _fcn := proc (N, X, Y, YP) option `[Y[1] = z(t)]`; if Y[3]*Y[2]*Y[1]/X < 0 then YP[1] := undefined; return 0 end if; YP[1] := (-.149071198499986*Y[2]*Y[1]*evalf((Y[3]*Y[2]*Y[1]/X)^(1/2))/X+1-Y[4]/(1-Y[1]))/(Y[2]*(Y[5]-Y[6]-.447213595499958*evalf((Y[3]*Y[2]*Y[1]/X)^(1/2)))); 0 end proc; _pars := [phi = phi, F = F, H = H, S_oi = S_oi, S_or = S_or, t0 = t0, z0 = z0]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then return _y0[0] elif _xout = "method" then return "lsode" elif _xout = "numfun" then return trunc(_ctl[24+trunc(_ctl[1])]) elif _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _ctl[2]-_y0[0] = 0. then error "no information is available on last computed point" else _xout := _ctl[2] end if elif _xout = "enginedata" then return eval(_octl, 1) elif _xout = "function" then return eval(_fcn, 1) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _yini) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n, _ini, _yini, _pars) end if; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_pars))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; _octl[2] := _y0[0]; _octl[3] := _y0[0]; for _i to _n do _octl[_i+7] := _y0[_i] end do; for _i to nops(_pars) do _octl[2*_n+30+_i] := _y0[_n+_i] end do; for _i to 39 do _ctl[_i] := _octl[_i] end do; if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] else return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] end if else return "procname" end if end if; if _xout-_y0[0] = 0. then return [seq(_y0[_i], _i = 0 .. _n)] end if; _reinit := false; if _xin <> "last" then if 0 < 0 and `dsolve/numeric/checkglobals`(0, table( [ ] ), _pars, _n, _yini) then _reinit := true; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_pars))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; for _i to _n do _octl[_i+7] := _y0[_i] end do; for _i to nops(_pars) do _octl[2*_n+30+_i] := _y0[_n+_i] end do end if; if _pars <> [] and select(type, {seq(_yini[_n+_i], _i = 1 .. nops(_pars))}, 'undefined') <> {} then error "parameters must be initialized before solution can be computed" end if end if; if not _reinit and _xout-_ctl[2] = 0 then [_ctl[2], seq(_ctl[_i], _i = 8 .. 7+_n)] else if sign(_xout-_ctl[2]) <> sign(_ctl[2]-_y0[0]) or abs(_xout-_y0[0]) < abs(_xout-_ctl[2]) or _reinit then for _i to 39 do _ctl[_i] := _octl[_i] end do end if; _ctl[3] := _xout; if Digits <= evalhf(Digits) then try _errcd := evalhf(`dsolve/numeric/lsode`(_fcn, var(_ctl))) catch: userinfo(2, `dsolve/debug`, print(`Exception in lsode:`, [lastexception])); if searchtext('evalhf', lastexception[2]) <> 0 or searchtext('real', lastexception[2]) <> 0 or searchtext('hardware', lastexception[2]) <> 0 then _errcd := `dsolve/numeric/lsode`(_fcn, _ctl) else error end if end try else _errcd := `dsolve/numeric/lsode`(_fcn, _ctl) end if; if _errcd < 0 then userinfo(2, {dsolve, `dsolve/lsode`}, `Last values returned:`); userinfo(2, {dsolve, `dsolve/lsode`}, ` t =`, _ctl[2]); _i := 8; userinfo(2, {dsolve, `dsolve/lsode`}, ` y =`, _ctl[_i]); for _i from _i+1 to 7+_n do userinfo(2, {dsolve, `dsolve/lsode`}, ` `, _ctl[_i]) end do; if _errcd+1. = 0. then if _ctl[14+trunc(_ctl[1])] <> 0 then error "an excessive amount of work was done, maxstep may be too small" else error "an excessive amount of work (greater than mxstep) was done" end if elif _errcd+2. = 0. then error "too much accuracy was requested for the machine being used" elif _errcd+3. = 0. then error "illegal input was detected" elif _errcd+4. = 0. then error "repeated error test failures on the attempted step" elif _errcd+5. = 0. then error "repeated convergence test failures on the attempted step" elif _errcd+6. = 0. then error "pure relative error control requested for a variable that has vanished" elif _errcd+7. = 0. then error "cannot evaluate the solution past %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_ctl[2]) else error "unknown error code returned from lsode %1", trunc(_errcd) end if end if; if _Env_smart_dsolve_numeric = true then if _y0[0] < _xout and procname("right") < _xout then procname("right") := _xout elif _xout < _y0[0] and _xout < procname("left") then procname("left") := _xout end if end if; [_xout, seq(_ctl[_i], _i = 8 .. 7+_n)] end if end proc, (2) = Array(0..0, {}), (3) = [t, z(t)], (4) = [phi = phi, F = F, H = H, S_oi = S_oi, S_or = S_or, t0 = t0, z0 = z0]}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_lsode, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_lsode, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_lsode, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_lsode, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_lsode), 'string') = rhs(x_lsode); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_lsode), 'string') = rhs(x_lsode)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_lsode) else _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_lsode) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

 (2)
> 

solSG(parameters = [.1, 1, .1, 1, .1, 0.1e-3, 0])

[phi = .1, F = 1., H = .1, S_oi = 1., S_or = .1, t0 = 0.1e-3, z0 = 0.]

 (3)
> 

``

Loading plots

> 

odeplot(solSG, t = 0.1e-3 .. 10)

 
> 

plots:-odeplot(solSG, t = 0.1e-3 .. 1)

 
> 

plots:-odeplot(solSG, t = 0.1e-3 .. .1)

 
> 

``


 

Download schechter_guo_v2.mw

My questions are:

1. Why does the solution for the longer time span (t<10) looks different from the shorter time span (t<0.1)? I have read about stiff ODEs and probably this has something to do with it. I am trying to figure out what is going on with the solutions.

2. I tried dsolve with stiff methods (lsode and rosenbrock) and both gave me the same solutions as above. I have not tried the advance options yet. How do I set dsolve so that the solution for the longer span (t>10) looks similar to shorter span?

Many thanks for your answers/suggestions.

 
Please Wait...