# solve three coupled linear partial differential equations with space and time variables using NDSolve

Clear[t]
(*define some constants*)
Pr = 0.71;
R = 0.5;
Sc = 0.6;
nn = 0.2;
pde = {[D[u[x, t], {t, 1}] ==
D[u[x, t], {x, 2}] + v[x, t] + nn*w[x, t],
3*R*Pr*D[v[x, t], {t, 1}] == (3*R + 4)*D[v[x, t], {x, 2}],
Sc*D[w[x, t], {t, 1}] == D[w[x, t], {x, 2}]]};
ICs = {u[0, t] == 0, v[0, t] == 0, w[0, t] == 0};
BCs = {u[0, t] == 0, u[1, t] == 0, v[0, t] == 0, v[1, t] == 0,
w[0, t] == 0, w[1, t] == 0};
sol = NDSolve[{pde, ICs, BCs}, u[x, t], {t = 0.2}, {x, 0, 1}];
Plot[Evaluate [u[x, t] /. sol], {x, 0, 1}]


I am unable to plot and getting this error, kindly help me any one,

Syntax::sntxf: "{" cannot be followed by "[D[u[x,t],{t,1}]==D[u[x,t],{x,2}]+v[x,t]+nn*w[x,t],3*RPrD[v[x,t],{t,1}]==(3*R+4)D[v[x,t],{x,2}],ScD[w[x,t],{t,1}]==D[w[x,t],{x,2}]]}"

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• – user9660 Jul 28 '16 at 8:22
• Have a look How to | Use Brackets and Braces Correctly and check pde and ff there are some more misinterpretations. – user9660 Jul 28 '16 at 8:26

Something like this:

Clear[t]
(*define some constants*)
Pr = 0.71;
R = 0.5;
Sc = 0.6;
nn = 0.2;
pde = {D[u[x, t], {t, 1}] ==
D[u[x, t], {x, 2}] + v[x, t] + nn*w[x, t],
3*R*Pr*D[v[x, t], {t, 1}] == (3*R + 4)*D[v[x, t], {x, 2}],
Sc*D[w[x, t], {t, 1}] == D[w[x, t], {x, 2}]};
ICs = {u[x, 0] == x, v[x, 0] == 1 - x, w[x, 0] == 0};
BCs = {u[0, t] == 0, u[1, t] == 1, v[0, t] == 1, v[1, t] == 0,
w[0, t] == 0, w[1, t] == 0};
sol = NDSolveValue[{pde, ICs, BCs}, u, {t, 0, 0.2}, {x, 0, 1}];
Plot[sol[x, 0.2], {x, 0, 1}]


I changed the initial and boundary conditions a bit to make it more interesting.

• (D[u[x, t], {t, 1}]) this is written in the answer above what is this command meaning ? – Andreas Hadjigeorgiou May 25 '17 at 12:49