I have little bit knowledge about mapple and tried to solve nonlinear system of partial differential equation by numerical method. The same process I have done for solving system of ordinary partial differential equation.

For the system of ordinary differential equations, I have used the following process

ode:=(ⅆ)/(ⅆt) x[1](t)=-x[1](t)+95* x[2](t), (ⅆ)/(ⅆt) x[2](t)=-x[1](t)-97*x[2](t): ics:=x[1](0)=1,x[2](0)=1:

dsol := dsolve({ics, ode}, numeric, method = rosenbrock, range = 0 .. 20, stiff = true)

p1:=odeplot`(dsol,[t,x[1](t)-x1E(t)],color=blue):

p2:=odeplot`(dsol,[t,x[2](t)-x2E(t)],color=green) :

display({p1,p2});

points := 8000;

aa := 0; bb := 10;

#`SAMPLING OF NUMERICAL SOLUTION`

ti:=[seq(rhs(dsol((i*bb)/(points))[1]),i=0..points)];

x1i:=[seq(rhs(dsol((i*bb)/(points))[2]),i=0..points)];

x2i:=[seq(rhs(dsol((i*bb)/(points))[3]),i=0..points)];

T:=Vector(ti,datatype=float):

X1:=Vector(x1i,datatype=float):

X2:=Vector(x2i,datatype=float):

nfop:=optimalitytolerance=1e-20,iterationlimit=10000,method=modifiednewton: newU:=unapply(convert(NonlinearFit(eval(U(t),p=1), T,X1,t, nfop),rational),t): newV:=unapply(convert(NonlinearFit(eval(V(t),p=1), T,X2,t, nfop),rational) ,t)

Finally i have solution for U(t), V(t)

newU(t);

newV(t);

But in case of PDE I have no idea how the similar process will be excute?

**The PDE I would like to solve is**

PDESYS := [diff(U(x, t), t)-(diff(U(x, t), x, x))-2*U(x, t)*(diff(U(x, t), x))+diff(U(x, t)*V(x, t), x), diff(V(x, t), t)-(diff(V(x, t), x, x))-2*V(x, t)*(diff(V(x, t), x))+diff(U(x, t)*V(x, t), x)]

ICs := [u(x, 0) = sin(x), v(x, 0) = sin(x)]

M:=5;N:=4:

U:=t->((∑)u[j](t)*p^(j))/(1+(∑)alpha[j]*t^(j)p^(j)): V:=t->((∑)v[j](t)*p^(j))/(1+(∑)beta[j]*t^(j)p^(j)):

U(x, t) = (sin(x)+sin(x)*a[1]*x*t-sin(x)*t+(cos(x)*sin(x)*b[1]*t^2*x+(1/2)*sin(x)*t^2-sin(x)*((1/2)*cos(x)*a[1]*x*t^2+(1/2)*sin(x)*a[1]*t^2-(1/2)*cos(x)*t^2)-cos(x)*((1/2)*sin(x)*b[1]*x*t^2-(1/2)*sin(x)*t^2)+cos(x)*((1/2)*sin(x)*a[1]*x*t^2-(1/2)*sin(x)*t^2)-sin(x)*((1/2)*cos(x)*b[1]*x*t^2+(1/2)*sin(x)*b[1]*t^2-(1/2)*cos(x)*t^2)+a[1]*x*((1/2)*sin(x)*a[1]*x*t^2-(1/2)*sin(x)*t^2)-cos(x)*sin(x)*t^2+sin(x)*a[2]*t^2*x^2+(1/2*(sin(x)*a[1]*x-sin(x)))*a[1]*t^2*x-sin(x)*a[1]^2*x^2*t^2+(1/2)*sin(x)^2*a[1]*t^2+(1/2)*sin(x)^2*b[1]*t^2))/(t^2*x^2*a[2]+t*x*a[1]+1)

V(x, t) = sin(x)+sin(x)*b[1]*x*t-sin(x)*t+(cos(x)*sin(x)*a[1]*t^2*x+sin(x)*b[2]*t^2*x^2+(1/2*(sin(x)*b[1]*x-sin(x)))*b[1]*t^2*x-sin(x)*b[1]^2*x^2*t^2+(1/2)*sin(x)*t^2-sin(x)*((1/2)*cos(x)*a[1]*x*t^2+(1/2)*sin(x)*a[1]*t^2-(1/2)*cos(x)*t^2)+cos(x)*((1/2)*sin(x)*b[1]*x*t^2-(1/2)*sin(x)*t^2)-cos(x)*((1/2)*sin(x)*a[1]*x*t^2-(1/2)*sin(x)*t^2)-sin(x)*((1/2)*cos(x)*b[1]*x*t^2+(1/2)*sin(x)*b[1]*t^2-(1/2)*cos(x)*t^2)-cos(x)*sin(x)*t^2+(1/2)*sin(x)^2*a[1]*t^2+(1/2)*sin(x)^2*b[1]*t^2+b[1]*x*((1/2)*sin(x)*b[1]*x*t^2-(1/2)*sin(x)*t^2)

For numerical process

PDE := [diff(w(x, t), t)-(diff(w(x, t), x, x))-2*w(x, t)*(diff(w(x, t), x))+diff(w(x, t)*r(x, t), x), diff(r(x, t), t)-(diff(r(x, t), x, x))-2*r(x, t)*(diff(r(x, t), x))+diff(w(x, t)*r(x, t), x)]

IBC := [u(x, 0) = sin(x), v(x, 0) = sin(x)]

**Which numerical method is suitable for this type of equation and how that method is execute in mapple?**

**I am attaching a code in case of ordinary differential equ .**

Odecode.mw