Thanks @Preben Alsholm for your valuable suggestions. I know both equations have highest order is four. Is there any numerical method available to find the solution of above equations associated with 6 BCs.?

Dear @Kitonum I also attach the file from where I took the system of equation please see fix my problem please. 

HELP.mw

Dear @Kitonum  thanks for your valueable comment. How can I find the solution?

I used this command

fsolve([seq(`$`(F1[l1, l2], l1 = 2 .. 2^K*M-1), l2 = 2 .. 2^K*M), seq(`$`(F2[l1], l1 = 2 .. 2^K*M)), seq(`$`(F3[l1], l1 = 2 .. 2^K*M)), seq(`$`(F4[l1], l1 = 1 .. 2^K*M))]);

@Carl Love @acer

Sir I need you help to fix my above querry @acer @Carl Love @Preben Alsholm

Thanks for your answer @tomleslie . I want to know how we can solve the PDEs have x,y,t by numeric? Waiting your positive and valueable comments.

@tomleslie also the points obtained from BCs and ICs are not in the matrices which you generate.

@tomleslie dear thanks for your positive response. for 1/(h=5) and 1/(k=5) the program give good results but for higher values (arbitrary values) of h and k results seems not good please see your sent program.

@tomleslie thanks for your answer. Now I want to minimize the size of 3D plot. I know how to do it for 2D plot, but don't know in 3D. Please help me to do this. I am waiting your response.

@Carl Love yes I also need legend please guide me how to do. Give nine curve arbitrary name.

Dear @Carl Love thanks for your deep answer about alpha. I got it and do the changing as per your suggestion. But you said you solve the system for N=100 in on mint. Can you share me that maple? I want to know who computational work reduces. 

Dear @Markiyan Hirnyk first of all I am very thankful your positive response. I did the changing according your instruction but could'nt find results becuase maple sheet take alot of time to evalute. Please see the attachment and see why they takes to much time. I want to produce reuslts for M=N=100. Again thanks in advance.

ImprovedEquations.mw

Dear @Carl Love first of all I am very thankful your positive response. I did the changing according your instruction but could'nt find results becuase maple sheet take alot of time to evalute. Please see the attachment and see why they takes to much time. I want to produce reuslts for M=N=100. Again thanks in advance.

Equations.mw

Dear @Kitonum first of all thanks alot for your positive answer. You almost fix my problem but I need answer in form that the denominator seprate with each term then simple.

dear @taro first of all thanks for your answer! I want to covert diff(u,y) into diff(U,Y) with u=v/d^2*U(Y), and y=d*Y. Manually I do in the attachment, which can help to fix the problem. 

manual.pdf