@PatrickT You see that I made one of the ODEs myself and so I had to use implicitdiff. Now, how can I rewrite my system in the above form. I have an ordinary second order differential equation (ode2) and an equation where I exchanged it to another differential equation.

Any other suggestion? Thanks

Thank you very much. By order I don't mean what you think. I'm doing a best-fitting analysis, so I have to run the program many times for diffeent range of numbers. I don't change the initial conditions, at all. I just want to ensure me that every time I run the program the first result is using for best-fit analysis. Not, one time the first result and another time the second one.

I mean one time "diff(phi(z), z)=first result" is used as an ODE and another time "diff(phi(z), z)=second result".

thanks Mr. Israel!

can someone in maple team help me?

 I have two equations:

diff(diff(diff(a(t,y),t),y),y) = (-2*a(t,y)*diff(a(t,y),t)*diff(diff(a(t,y),y),y)^2+2*diff(a(t,y),t)*diff(diff(a(t,y),y),y)
*diff(a(t,y),y)^2-2*diff(a(t,y),t)*diff(a(t,y),y)^4*a(t,y)+2*diff(a(t,y),t)*diff(a(t,y),y)^4-6*diff(a(t,y),t)
*diff(diff(a(t,y),y),y)*diff(a(t,y),y)^2*a(t,y)-a(t,y)*diff(a(t,y),t)*diff(a(t,y),y)*diff(diff(diff(a(t,y),y),y),y))
/(diff(a(t,y),y)^2*a(t,y)+a(t,y)*diff(diff(a(t,y),y),y))

and

diff(diff(diff(a(t,y),t),y),y) = (-3*a(t,y)*diff(a(t,y),t)*diff(diff(a(t,y),y),y)^2-a(t,y)*diff(a(t,y),t)*diff(a(t,y),y)
*diff(diff(diff(a(t,y),y),y),y)-6*diff(a(t,y),t)*diff(diff(a(t,y),y),y)*diff(a(t,y),y)^2)/(a(t,y)^2*diff(diff(a(t,y),y),y)
+diff(a(t,y),y)^2*a(t,y))

and in both cases the boundary and initial conditions are

{a(0,y) = y, a(t,0) = 0, a(t,1) = 1+2*t, D[2](a)(t,0) = 1+t}

but maple solves one of them and for another one gives an error:"error, (in pdsolve/numeric/par_hyp) incorrect number of boundary conditions, expected 2, got 3"

but both two equations have the same order in "y". what is the problem?

I'm sorry, I forgot that. I need both "A" and "N", not one of them. I tried to solve the right equations by casesplit as your work but I have some errors:

"error, (in pdsolve/numeric/par_hyp) incorrect number of boundary conditions, expected 2, got 3"  

"error, 'sol2' doesn't evaluate to a module"

The new equation for "Neq" is first order in "t" and second order in "y", and the new one for "Aeq" is first order in "t" and third order in "y" as same as your work. I also used your initial conditions:

"ibc2:= {a(0,y)=y,a(t,0)=0,D[2](a)(t,0)=1+t,a(t,1)=1+2*t};
  Sol2:= pdsolve(Aeq,ibc2,numeric);
  Sol2:-plot3d(A,t=0..1,y=0..1,axes=box);"

but I have the above errors. please help me. thanks.

I'm sorry Mr. Israel!

I think you didn't use the right equations. you missed one "a" at the first equation and also you added one more "a" at the second one. the right equations are:

sys:= {diff(A,t)^2 - N^2*A*diff(A,y$2) - N^2*diff(A,y)^2 = 0,
   N^3*diff(A,y)^2 + N^2*A*diff(A,y)*diff(N,y) -N*A*diff(A,t$2)- N*diff(A,t)^2+A*diff(A,t)*diff(N,t)=0};

can you help me again, please?

thanks Mr. Israel.

I added your idea in my worksheet and it worked. I have one more question now. How can I solve the next equation too and get "Neq"?

Dear Israel

To be honest with you I try to duplicate your maple commands on my comupter but it does not work.  Does it need any particulat library to run it or I am missing something.

for example: I run

> with (PDEtools);  with (DEtools);

>sys:= {diff(A,t)^2 - N^2*diff(A,y$2) - N^2*diff(A,y)^2 = 0,
   N^3*diff(A,y)^2 + N^2*A*diff(A,y)*diff(N,y) -N*A*diff(A,t$2)- N*A*diff(A,t)^2+A*diff(A,t)*diff(N,t)=0};
  pdsolve(sys);
 

 butit does not give me the solution you obtanied. Instead, it gives me

>sys:={0=0}

Error, (in pdsolve/info) first argument is not a differential equation

Would you help me with this one please

thank you. I use your idea but it doesn't work. I don't get a result as you. If I write me equations and then I use casesplit, I have the same equation for "n", but a different one for "a". also, if I use your equations, I have an error. Do I have to use a special package? I've "used pdetools" and "detools".  

thanks Mr. Israel. I'm working on physics of extra dimensions and so I don't have the initial and boundary conditions explicitly. anyway I removed "a(10,y) = 0" and "D[2](a)(t,0) = 1", but a new error:

"Error, (in pdsolve/numeric/match_PDEs_BCs) cannot handle systems with multiple PDE describing the time dependence of the same dependent variable, or having no time dependence" 

I think my initial/boundary conditions were good.

but thanks for you Alec. my PDEs are as: (remember "a" and "n" are functions of both "t" and "y". like this: a(t,y) and n(t,y))

1. (da/dt)^2 - n^2*a*(d^2a/dy^2) - n^2*(da/dy)^2 = 0

2. n^3*(da/dy)^2 + n^2*a*da/dy*dn/dy - n*a*(d^2a/dt^2) - n*(da/dt)^2 + a*da/dt*dn/dt = 0

1.if I erase my end condition  a(10,y) = 0, then what can I replace instead of it? you said me I have to have a more initial condition for "t", then what is it?

2.what do you mean " unless you want to program your own numerical method " ? can you explain more or introduce some examples?

thanks Mr. Israel. I corrected my initial conditions. I added: 

 D[2](a)(t,0) = 1, a(10,y) = 0 and n(0,y) = 1

but now I have this error:

"Error, (in pdsolve/numeric) unable to handle elliptic PDEs"

what can I do now?

I have two functions a(t,y) and n(t,y) in my PDEs. one of my PDEs is first order in time for "a" and second order in "y" for "a", and another is second order in time and first order in "y" for "a" and also first order in both time and "y" for "n" . I have considered these initial and boundary conditions: 

a(t,0)=1, a(t,1)=2, a(0,y)=0, n(t,0)=1, D[1](a)(0,y)=1

but  I can't solve it. I have an error as:

"Error, (in pdsolve/numeric/par_hyp) Incorrect number of initial conditions, expected 3, got 2"

 o.k. Mr. Israel. thank you. I was wrong. I got a same error. but if I change my initial conditions as you say I get the same error:

Error, (in pdsolve/numeric/process_IBCs) initial/boundary condition is not valid: {a(0, y) = 0, a(t, 0) = 1, (diff(a(t, y), t))(t, 0) = 1}

any initial conditions lead to similar errors.

thanks Mr. Israel for your reply. but I have a new problem. after inserting your corrections I receive a new error as: "Error, (in pdsolve/indexlex) invalid arguments to sort:" what can I do now?