# solving coupled pde with conditions

I have two coupled PDE

eqns = {w D[c[x, t], t] + u D[c[x, t], x] -
v D[c[x, t], {x, 2}] == -k c[x, t] s[x, t],
D[s[x, t], t] == -p k c[x, t] s[x, t]};

sol = DSolve[eqns, {c, s}, {x, t}]


I couldn't add the boundaries conditions to find the solution, the conditions are

c(0, t) = cd(t), for  t in [0,T],
c(L, t) = 0, for t in [0,T],
c(x,0) = 0, for x in [0,L],
s(x,0) = s0, for x in [0,L].


c and s are two functions defined in [0, T]x[0, L]

w,u,v,k, and p are constants

I suspect that DSolve will be able to give you a particular solution. Here is a numerical solution,

cd[t_] = Sin[t]; (*I choose this randomly*)

s0 = 1; u = 1; w = 1; v = 1; k = 1; p = 1; L = 1; (*I choose these randomly*)

pde1 = w*D[c[x, t], t] + u*D[c[x, t], x] - v*D[c[x, t], {x, 2}] == -k*
c[x, t]*s[x, t];

pde2 = D[s[x, t], t] == -p*k*c[x, t]*s[x, t];

DSolve[{pde1, pde2, c[0, t] == cd[t], c[L, t] == 0, c[x, 0] == 0, s[x, 0] == s0}, {c, s}, {x, 0, L},
{t, 0, 1}]; (*No output by DSolve*)

sol = NDSolve[{pde1, pde2, c[0, t] == cd[t], c[L, t] == 0, c[x, 0] == 0, s[x, 0] == s0}, {c, s},
{x, 0, L}, {t, 0, 10}]

Plot3D[{c[x, t], s[x, t]} /. sol, {t, 0, 10}, {x, 0, L}]
`

• so i can't have a symbolic solution for my system ? Commented Dec 20, 2018 at 6:23
• @ElyesAbidi I am unable to find one.
– zhk
Commented Dec 20, 2018 at 6:26