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      General description of the method of solving underdetermined systems of equations. As a particular application of the idea proposed a universal method of calculation for all kinds of linkage (lever) mechanisms. With the description and examples.
      The method can be used for powerful CAD linkages.

Description: Calculation_method_of_linkages.pdf

Attachment:
figure_1.mw
figure_2.mw

Or all in one
Calculation_method_of_linkages_(with_attach.).pdf


        Some examples of a much larger number calculated by the proposed method. Examples gathered here not to look for them on the forum and opportunity to demonstrate the method.  Among the examples, I think, there are very complicated.

https://vk.com/doc242471809_408704758
https://vk.com/doc242471809_408704572
https://vk.com/doc242471809_376439263
https://vk.com/doc242471809_402619761
https://vk.com/doc242471809_402610228
https://vk.com/doc242471809_401188803
https://vk.com/doc242471809_400465891
https://vk.com/doc242471809_400711315
https://vk.com/doc242471809_387358164
https://vk.com/doc242471809_380837279
https://vk.com/doc242471809_379935473
https://vk.com/doc242471809_380217387
https://vk.com/doc242471809_363266817
https://vk.com/doc242471809_353980472
https://vk.com/doc242471809_375452868
https://vk.com/doc242471809_353988163 
https://vk.com/doc242471809_353986884 
https://vk.com/doc242471809_353987119
https://vk.com/doc242471809_324249241
https://vk.com/doc242471809_324102889
https://vk.com/doc242471809_322219275
https://vk.com/doc242471809_437298137
https://vk.com/doc242471809_437308238
https://vk.com/doc242471809_437308241
https://vk.com/doc242471809_437308243
https://vk.com/doc242471809_437308245
https://vk.com/doc242471809_437308246
https://vk.com/doc242471809_437401651
https://vk.com/doc242471809_437664558

 

 

HI.please help me for solve differenrtial equation with finite difference method not dsolve numeric solver in maple

thanks

FDM2.mw

L := 1/50000000; -1; eta := 1; -1; PDE[111] := 7.65692307692309*10^(-8)*(diff(f1(x), x, x, x, x))-1.56784615384616*10^12*(diff(f1(x), x, x))+220.592307692308*(diff(f2(x), x, x, x))-3.52947692307693*10^21*(diff(f2(x), x))+43.7538461538462*(diff(f3(x), x, x, x))+4.81292307692309*10^20*(diff(f3(x), x))+6.50473846153848*10^30*f1(x)-7.90000000000000*10^(-8)*eta*f1(x)

0.7656923077e-7*(diff(diff(diff(diff(f1(x), x), x), x), x))-0.1567846154e13*(diff(diff(f1(x), x), x))+220.592307692308*(diff(diff(diff(f2(x), x), x), x))-0.3529476923e22*(diff(f2(x), x))+43.7538461538462*(diff(diff(diff(f3(x), x), x), x))+0.4812923077e21*(diff(f3(x), x))+0.6504738462e31*f1(x)

(1)

PDE[222] := 2.14211538461539*10^(-8)*(diff(f2(x), x, x, x, x))-1.64988461538462*10^12*(diff(f2(x), x, x))+7.90486153846156*10^30*f2(x)-220.592307692308*(diff(f1(x), x, x, x))+3.52947692307693*10^21*(diff(f1(x), x))-5.94323076923080*10^11*(diff(f3(x), x, x))+5.13378461538463*10^30*f3(x)-7.90000000000000*10^(-8)*eta*f2(x)

0.2142115385e-7*(diff(diff(diff(diff(f2(x), x), x), x), x))-0.1649884615e13*(diff(diff(f2(x), x), x))+0.7904861538e31*f2(x)-220.592307692308*(diff(diff(diff(f1(x), x), x), x))+0.3529476923e22*(diff(f1(x), x))-0.5943230769e12*(diff(diff(f3(x), x), x))+0.5133784615e31*f3(x)

(2)

PDE[333] := -6.38076923076924*10^(-31)*(diff(f3(x), x, x, x, x, x, x))+9.66537046153848*10^(-8)*(diff(f3(x), x, x, x, x))-3.10154753538461*10^12*(diff(f3(x), x, x))-43.7538461538462*(diff(f1(x), x, x, x))-4.81292307692309*10^20*(diff(f1(x), x))-5.94323076923080*10^11*(diff(f2(x), x, x))+5.13378461538463*10^30*f2(x)+2.29989058707693*10^31*f3(x)-7.90105333333333*10^(-8)*omega^2*f3(x)+6.58333333333333*10^(-31)*eta*(diff(f3(x), x, x))

-0.6380769231e-30*(diff(diff(diff(diff(diff(diff(f3(x), x), x), x), x), x), x))+0.9665370462e-7*(diff(diff(diff(diff(f3(x), x), x), x), x))-0.3101547535e13*(diff(diff(f3(x), x), x))-43.7538461538462*(diff(diff(diff(f1(x), x), x), x))-0.4812923077e21*(diff(f1(x), x))-0.5943230769e12*(diff(diff(f2(x), x), x))+0.5133784615e31*f2(x)+0.2299890587e32*f3(x)-0.7901053333e-7*omega^2*f3(x)

(3)

bcs := {f1(0) = 0, f1(L) = 0, f2(0) = 0, f2(L) = 0, f3(0) = 0, f3(L) = 0, ((D@@1)(f1))(0) = 0, ((D@@1)(f1))(L) = 0, ((D@@1)(f2))(0) = 0, ((D@@1)(f2))(L) = 0, ((D@@1)(f3))(0) = 0, ((D@@1)(f3))(L) = 0, ((D@@2)(f3))(0) = 0, ((D@@2)(f3))(L) = 0}

{f1(0) = 0, f1(1/50000000) = 0, f2(0) = 0, f2(1/50000000) = 0, f3(0) = 0, f3(1/50000000) = 0, (D(f1))(0) = 0, (D(f1))(1/50000000) = 0, (D(f2))(0) = 0, (D(f2))(1/50000000) = 0, (D(f3))(0) = 0, (D(f3))(1/50000000) = 0, ((D@@2)(f3))(0) = 0, ((D@@2)(f3))(1/50000000) = 0}

(4)

``


Download FDM2.mw

Hi, i am trying to solve my PDEs with HPM method ,but i get strange errors.

first one is :Error, (in trig/reduce/reduce) Maple was unable to allocate enough memory to complete this computation.  Please see ?alloc,

but when i run my last function again,the error chages,let me show you.


restart;
lambda:=0.5;K[r]:=0.5;Sc:=0.5;Nb:=0.1;Nt:=0.1;Pr:=10;
                              0.5
                              0.5
                              0.5
                              0.1
                              0.1
                               10
> EQUATIONS;


equ1:=diff(f(eta),eta$4)-R*(diff(f(eta),eta)*diff(f(eta),eta$2)-f(eta)*diff(f(eta),eta$2))-2*K[r]*diff(g(eta),eta)=0;

equ2:=diff(g(eta),eta$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta)=0;

equ3:=diff(theta(eta),eta$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2=0;

equ4:=diff(phi(eta),eta$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta$2)*(Nt/Nb)=0;
/  d   /  d   /  d   /  d         \\\\     //  d         \ /  d  
|----- |----- |----- |----- f(eta)|||| - R ||----- f(eta)| |-----
\ deta \ deta \ deta \ deta       ////     \\ deta       / \ deta

   /  d         \\          /  d   /  d         \\\
   |----- f(eta)|| - f(eta) |----- |----- f(eta)|||
   \ deta       //          \ deta \ deta       ///

         /  d         \    
   - 1.0 |----- g(eta)| = 0
         \ deta       /    
     /  d   /  d         \\
     |----- |----- g(eta)||
     \ deta \ deta       //

            //  d         \                 /  d         \\
        - R ||----- f(eta)| g(eta) - f(eta) |----- g(eta)||
            \\ deta       /                 \ deta       //

              /  d         \    
        + 1.0 |----- f(eta)| = 0
              \ deta       /    
  /  d   /  d             \\               /  d             \
  |----- |----- theta(eta)|| + 10 R f(eta) |----- theta(eta)|
  \ deta \ deta           //               \ deta           /

           /  d           \ /  d             \
     + 0.1 |----- phi(eta)| |----- theta(eta)|
           \ deta         / \ deta           /

                             2    
           /  d             \     
     + 0.1 |----- theta(eta)|  = 0
           \ deta           /     
    /  d   /  d           \\                /  d           \
    |----- |----- phi(eta)|| + 0.5 R f(eta) |----- phi(eta)|
    \ deta \ deta         //                \ deta         /

                     /  d   /  d             \\    
       + 1.000000000 |----- |----- theta(eta)|| = 0
                     \ deta \ deta           //    
> BOUNDARY*CONDITIONS;


ics:=
f(0)=0,D(f)(0)=1,g(0)=0,theta(0)=1,phi(0)=1;
f(1)=lambda,D(f)(1)=0,g(1)=0,theta(1)=0,phi(1)=0;
   f(0) = 0, D(f)(0) = 1, g(0) = 0, theta(0) = 1, phi(0) = 1
  f(1) = 0.5, D(f)(1) = 0, g(1) = 0, theta(1) = 0, phi(1) = 0
> HPMs;


hpm1:=(1-p)*(diff(f(eta),eta$4)-2*K[r]*diff(g(eta),eta))+p*(diff(f(eta),eta$4)-R*(diff(f(eta),eta)*diff(f(eta),eta$2)-f(eta)*diff(f(eta),eta$2))-2*K[r]*diff(g(eta),eta))=0;

hpm2:=(1-p)*(diff(g(eta),eta$2)+2*K[r]*diff(f(eta),eta))+p*(diff(g(eta),eta$2)-R*(diff(f(eta),eta)*g(eta)-f(eta)*diff(g(eta),eta))+2*K[r]*diff(f(eta),eta))=0;

hpm3:=(1-p)*(diff(theta(eta),eta$2))+p*(diff(theta(eta),eta$2)+Pr*R*f(eta)*diff(theta(eta),eta)+Nb*diff(phi(eta),eta)*diff(theta(eta),eta)+Nt*diff(theta(eta),eta)^2)=0;

hpm4:=(1-p)*(diff(phi(eta),eta$2)+diff(theta(eta),eta$2)*(Nt/Nb))+p*(diff(phi(eta),eta$2)+R*Sc*f(eta)*diff(phi(eta),eta)+diff(theta(eta),eta$2)*(Nt/Nb))=0;

        //  d   /  d   /  d   /  d         \\\\
(1 - p) ||----- |----- |----- |----- f(eta)||||
        \\ deta \ deta \ deta \ deta       ////

         /  d         \\     //  d   /  d   /  d   /  d         \
   - 1.0 |----- g(eta)|| + p ||----- |----- |----- |----- f(eta)|
         \ deta       //     \\ deta \ deta \ deta \ deta       /

  \\\     //  d         \ /  d   /  d         \\
  ||| - R ||----- f(eta)| |----- |----- f(eta)||
  ///     \\ deta       / \ deta \ deta       //

            /  d   /  d         \\\       /  d         \\    
   - f(eta) |----- |----- f(eta)||| - 1.0 |----- g(eta)|| = 0
            \ deta \ deta       ///       \ deta       //    
        //  d   /  d         \\       /  d         \\     //  d  
(1 - p) ||----- |----- g(eta)|| + 1.0 |----- f(eta)|| + p ||-----
        \\ deta \ deta       //       \ deta       //     \\ deta

   /  d         \\
   |----- g(eta)||
   \ deta       //

       //  d         \                 /  d         \\
   - R ||----- f(eta)| g(eta) - f(eta) |----- g(eta)||
       \\ deta       /                 \ deta       //

         /  d         \\    
   + 1.0 |----- f(eta)|| = 0
         \ deta       //    
                                       /                         
        /  d   /  d             \\     |/  d   /  d             \
(1 - p) |----- |----- theta(eta)|| + p ||----- |----- theta(eta)|
        \ deta \ deta           //     \\ deta \ deta           /

  \               /  d             \
  | + 10 R f(eta) |----- theta(eta)|
  /               \ deta           /

         /  d           \ /  d             \
   + 0.1 |----- phi(eta)| |----- theta(eta)|
         \ deta         / \ deta           /

                           2\    
         /  d             \ |    
   + 0.1 |----- theta(eta)| | = 0
         \ deta           / /    
        //  d   /  d           \\
(1 - p) ||----- |----- phi(eta)||
        \\ deta \ deta         //

                 /  d   /  d             \\\     //  d   /  d   
   + 1.000000000 |----- |----- theta(eta)||| + p ||----- |-----
                 \ deta \ deta           ///     \\ deta \ deta

          \\                /  d           \
  phi(eta)|| + 0.5 R f(eta) |----- phi(eta)|
          //                \ deta         /

                 /  d   /  d             \\\    
   + 1.000000000 |----- |----- theta(eta)||| = 0
                 \ deta \ deta           ///    
f(eta)=sum(f[i](eta)*p^i,i=0..1);
                f(eta) = f[0](eta) + f[1](eta) p
g(eta)=sum(g[i](eta)*p^i,i=0..1);
                g(eta) = g[0](eta) + g[1](eta) p
theta(eta)=sum(theta[i](eta)*p^i,i=0..1);
          theta(eta) = theta[0](eta) + theta[1](eta) p
phi(eta)=sum(phi[i](eta)*p^i,i=0..1);
             phi(eta) = phi[0](eta) + phi[1](eta) p
> FORequ1;


A:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm1)),p);
/      /  d            \ /  d   /  d            \\
|-1. R |----- f[1](eta)| |----- |----- f[1](eta)||
\      \ deta          / \ deta \ deta          //

                 /  d   /  d            \\\  3   /
   + R f[1](eta) |----- |----- f[1](eta)||| p  + |
                 \ deta \ deta          ///      \
      /  d            \ /  d   /  d            \\
-1. R |----- f[0](eta)| |----- |----- f[1](eta)||
      \ deta          / \ deta \ deta          //

          /  d            \ /  d   /  d            \\
   - 1. R |----- f[1](eta)| |----- |----- f[0](eta)||
          \ deta          / \ deta \ deta          //

                 /  d   /  d            \\
   + R f[0](eta) |----- |----- f[1](eta)||
                 \ deta \ deta          //

                 /  d   /  d            \\\  2   //  d   /  d   /
   + R f[1](eta) |----- |----- f[0](eta)||| p  + ||----- |----- |
                 \ deta \ deta          ///      \\ deta \ deta \

    d   /  d            \\\\       /  d            \
  ----- |----- f[1](eta)|||| - 1.0 |----- g[1](eta)|
   deta \ deta          ////       \ deta          /

          /  d            \ /  d   /  d            \\
   - 1. R |----- f[0](eta)| |----- |----- f[0](eta)||
          \ deta          / \ deta \ deta          //

                 /  d   /  d            \\\  
   + R f[0](eta) |----- |----- f[0](eta)||| p
                 \ deta \ deta          ///  

     /  d   /  d   /  d   /  d            \\\\
   + |----- |----- |----- |----- f[0](eta)||||
     \ deta \ deta \ deta \ deta          ////

         /  d            \    
   - 1.0 |----- g[0](eta)| = 0
         \ deta          /    
A1:=diff(f[0](eta),eta$4)-2*K[r]*(diff(g[0](eta),eta))=0;
A2:=diff(f[1](eta),eta$4)-2*K[r]*(diff(g[1](eta),eta))-R*(diff(f[0](eta),eta))*(diff(f[0](eta),eta$2))+R*f[0](eta)*(diff(f[0](eta),eta$2))=0;
/  d   /  d   /  d   /  d            \\\\       /  d            \   
|----- |----- |----- |----- f[0](eta)|||| - 1.0 |----- g[0](eta)| =
\ deta \ deta \ deta \ deta          ////       \ deta          /   

  0
/  d   /  d   /  d   /  d            \\\\       /  d            \
|----- |----- |----- |----- f[1](eta)|||| - 1.0 |----- g[1](eta)|
\ deta \ deta \ deta \ deta          ////       \ deta          /

       /  d            \ /  d   /  d            \\
   - R |----- f[0](eta)| |----- |----- f[0](eta)||
       \ deta          / \ deta \ deta          //

                 /  d   /  d            \\    
   + R f[0](eta) |----- |----- f[0](eta)|| = 0
                 \ deta \ deta          //    
icsA1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0;
icsA2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;
   f[0](0) = 0, D(f[0])(0) = 1, g[0](0) = 0, f[0](1) = 0.5,

     D(f[0])(1) = 0, g[0](1) = 0
    f[1](0) = 0, D(f[1])(0) = 0, g[1](0) = 0, f[1](1) = 0,

      D(f[1])(1) = 0, g[1](1) = 0
>
FORequ2;


B:=collect(expand(subs(f(eta)=f[0](eta)+f[1](eta)*p,g(eta)=g[0](eta)+g[1](eta)*p,hpm2)),p);
/      /  d            \          
|-1. R |----- f[1](eta)| g[1](eta)
\      \ deta          /          

                 /  d            \\  3   /
   + R f[1](eta) |----- g[1](eta)|| p  + |
                 \ deta          //      \
      /  d            \          
-1. R |----- f[0](eta)| g[1](eta)
      \ deta          /          

          /  d            \          
   - 1. R |----- f[1](eta)| g[0](eta)
          \ deta          /          

                 /  d            \
   + R f[0](eta) |----- g[1](eta)|
                 \ deta          /

                 /  d            \\  2   //  d   /  d            
   + R f[1](eta) |----- g[0](eta)|| p  + ||----- |----- g[1](eta)
                 \ deta          //      \\ deta \ deta          

  \\       /  d            \        /  d            \          
  || + 1.0 |----- f[1](eta)| - 1. R |----- f[0](eta)| g[0](eta)
  //       \ deta          /        \ deta          /          

                 /  d            \\     /  d   /  d            \\
   + R f[0](eta) |----- g[0](eta)|| p + |----- |----- g[0](eta)||
                 \ deta          //     \ deta \ deta          //

         /  d            \    
   + 1.0 |----- f[0](eta)| = 0
         \ deta          /    
B1:=diff(g[0](eta),eta$2)+2*K[r]*(diff(f[0](eta),eta))=0;
B2:=diff(g[1](eta),eta$2)+2*K[r]*(diff(f[1](eta),eta))-R*(diff(f[0](eta),eta))*g[0](eta)+R*f[0](eta)*(diff(g[0](eta),eta))=0;
     /  d   /  d            \\       /  d            \    
     |----- |----- g[0](eta)|| + 1.0 |----- f[0](eta)| = 0
     \ deta \ deta          //       \ deta          /    
       /  d   /  d            \\       /  d            \
       |----- |----- g[1](eta)|| + 1.0 |----- f[1](eta)|
       \ deta \ deta          //       \ deta          /

              /  d            \          
          - R |----- f[0](eta)| g[0](eta)
              \ deta          /          

                        /  d            \    
          + R f[0](eta) |----- g[0](eta)| = 0
                        \ deta          /    
icsB1:=f[0](0)=0,D(f[0])(0)=1,g[0](0)=0,f[0](1)=lambda,D(f[0])(1)=0,g[0](1)=0;
icsB2:=f[1](0)=0,D(f[1])(0)=0,g[1](0)=0,f[1](1)=0,D(f[1])(1)=0,g[1](1)=0;
   f[0](0) = 0, D(f[0])(0) = 1, g[0](0) = 0, f[0](1) = 0.5,

     D(f[0])(1) = 0, g[0](1) = 0
    f[1](0) = 0, D(f[1])(0) = 0, g[1](0) = 0, f[1](1) = 0,

      D(f[1])(1) = 0, g[1](1) = 0
> FORequ3;


C:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm3)),p);
 /                                     
 |                /  d                \
 |10. R f[1](eta) |----- theta[1](eta)|
 \                \ deta              /

          /  d              \ /  d                \
    + 0.1 |----- phi[1](eta)| |----- theta[1](eta)|
          \ deta            / \ deta              /

                               2\                              
          /  d                \ |  3   /                /  d   
    + 0.1 |----- theta[1](eta)| | p  + |10. R f[0](eta) |-----
          \ deta              / /      \                \ deta

                \                   /  d                \
   theta[1](eta)| + 10. R f[1](eta) |----- theta[0](eta)|
                /                   \ deta              /

          /  d              \ /  d                \
    + 0.1 |----- phi[0](eta)| |----- theta[1](eta)|
          \ deta            / \ deta              /

          /  d              \ /  d                \
    + 0.1 |----- phi[1](eta)| |----- theta[0](eta)|
          \ deta            / \ deta              /

                                                            /
          /  d                \ /  d                \\  2   |/
    + 0.2 |----- theta[0](eta)| |----- theta[1](eta)|| p  + ||
          \ deta              / \ deta              //      \\

     d   /  d                \\
   ----- |----- theta[1](eta)||
    deta \ deta              //

                      /  d                \
    + 10. R f[0](eta) |----- theta[0](eta)|
                      \ deta              /

          /  d              \ /  d                \
    + 0.1 |----- phi[0](eta)| |----- theta[0](eta)|
          \ deta            / \ deta              /

                               2\  
          /  d                \ |  
    + 0.1 |----- theta[0](eta)| | p
          \ deta              / /  

      /  d   /  d                \\    
    + |----- |----- theta[0](eta)|| = 0
      \ deta \ deta              //    
C1:=diff(theta[0](eta),eta$2)=0;
C2:=diff(theta[1](eta), eta, eta)+Pr*R*f[0](eta)*(diff(theta[0](eta), eta))+Nb*(diff(phi[0](eta), eta))*(diff(theta[0](eta), eta))+Nt*(diff(theta[0](eta), eta))^2=0;
                  d   /  d                \    
                ----- |----- theta[0](eta)| = 0
                 deta \ deta              /    
       /  d   /  d                \\
       |----- |----- theta[1](eta)||
       \ deta \ deta              //

                           /  d                \
          + 10 R f[0](eta) |----- theta[0](eta)|
                           \ deta              /

                /  d              \ /  d                \
          + 0.1 |----- phi[0](eta)| |----- theta[0](eta)|
                \ deta            / \ deta              /

                                     2    
                /  d                \     
          + 0.1 |----- theta[0](eta)|  = 0
                \ deta              /     
icsC1:=theta[0](0)=1,theta[0](1)=0;
icsC2:=theta[1](0)=0,theta[1](1)=0,phi[0](0)=0,phi[0](1)=0;
                theta[0](0) = 1, theta[0](1) = 0
 theta[1](0) = 0, theta[1](1) = 0, phi[0](0) = 0, phi[0](1) = 0
> FORequ4;


E:=collect(expand(subs(theta(eta)=theta[0](eta)+theta[1](eta)*p,phi(eta)=phi[0](eta)+phi[1](eta)*p,f(eta)=f[0](eta)+f[1](eta)*p,hpm4)),p);
                 3 /  d              \   /                /  d   
0.5 R f[1](eta) p  |----- phi[1](eta)| + |0.5 R f[0](eta) |-----
                   \ deta            /   \                \ deta

             \                   /  d              \\  2   //
  phi[1](eta)| + 0.5 R f[1](eta) |----- phi[0](eta)|| p  + ||
             /                   \ deta            //      \\

    d   /  d              \\
  ----- |----- phi[1](eta)||
   deta \ deta            //

                 /  d   /  d                \\
   + 1.000000000 |----- |----- theta[1](eta)||
                 \ deta \ deta              //

                     /  d              \\  
   + 0.5 R f[0](eta) |----- phi[0](eta)|| p
                     \ deta            //  

     /  d   /  d              \\
   + |----- |----- phi[0](eta)||
     \ deta \ deta            //

                 /  d   /  d                \\    
   + 1.000000000 |----- |----- theta[0](eta)|| = 0
                 \ deta \ deta              //    
E1:=diff(phi[0](eta),eta$2)+Nt*(diff(theta[0](eta),eta$2))/Nb=0;
E2:=diff(phi[1](eta),eta$2)+Nt*(diff(theta[1](eta),eta$2))/Nb+R*Sc*f[0](eta)*(diff(phi[0](eta),eta))=0;
       /  d   /  d              \\
       |----- |----- phi[0](eta)||
       \ deta \ deta            //

                        /  d   /  d                \\    
          + 1.000000000 |----- |----- theta[0](eta)|| = 0
                        \ deta \ deta              //    
         /  d   /  d              \\
         |----- |----- phi[1](eta)||
         \ deta \ deta            //

                          /  d   /  d                \\
            + 1.000000000 |----- |----- theta[1](eta)||
                          \ deta \ deta              //

                              /  d              \    
            + 0.5 R f[0](eta) |----- phi[0](eta)| = 0
                              \ deta            /    
icsE1:=theta[0](0)=1,theta[0](1)=0,phi[0](0)=1,phi[0](1)=0;
icsE2:=theta[1](0)=0,theta[1](1)=0,phi[1](0)=0,phi[1](1)=0;
 theta[0](0) = 1, theta[0](1) = 0, phi[0](0) = 1, phi[0](1) = 0
 theta[1](0) = 0, theta[1](1) = 0, phi[1](0) = 0, phi[1](1) = 0
       
theta[0](eta) = -(152675527/100000000)*eta+1;
                                152675527        
              theta[0](eta) = - --------- eta + 1
                                100000000        
U:=f[1](eta)=0;
                         f[1](eta) = 0
Dsolve(A1,B1,icsA1,icsB1);
                  Dsolve(A1, B1, icsA1, icsB1)


sys:={ diff(g[0](eta), eta, eta)+1.0*(diff(f[0](eta), eta)) = 0, diff(f[0](eta), eta, eta, eta, eta)-1.0*(diff(g[0](eta), eta)) = 0};
    //  d   /  d   /  d   /  d            \\\\
   { |----- |----- |----- |----- f[0](eta)||||
    \\ deta \ deta \ deta \ deta          ////

            /  d            \      
      - 1.0 |----- g[0](eta)| = 0,
            \ deta          /      

     /  d   /  d            \\       /  d            \    \
     |----- |----- g[0](eta)|| + 1.0 |----- f[0](eta)| = 0 }
     \ deta \ deta          //       \ deta          /    /
IC_1:={ f[0](0) = 0, (D(f[0]))(0) = 1, g[0](0) = 0, f[0](1) = .5, (D(f[0]))(1) = 0, g[0](1) = 0,f[0](0) = 0, (D(f[0]))(0) = 1, g[0](0) = 0, f[0](1) = .5, (D(f[0]))(1) = 0, g[0](1) = 0};
    {f[0](0) = 0, f[0](1) = 0.5, g[0](0) = 0, g[0](1) = 0,

      D(f[0])(0) = 1, D(f[0])(1) = 0}
ans1 := combine(dsolve(sys union IC_1,{f[0](eta),g[0](eta)}),trig);
Error, (in dsolve) expecting an ODE or a set or list of ODEs. Received `union`(IC_1, sys)
>

     Example of the equidistant surface at a distance of 0.25 to the surface
x3
-0.1 * (sin (4 * x1) + sin (3 * x2 + x3) + sin (2 * x2)) = 0
Constructed on the basis of universal parameterization of surfaces.

equidistant_surface.mw 


Hi. Is there a maple function which can interpolate values from a simple table:

given (x,y) compute z value

eg z(0.5,0.5)=1.5

x:=[0,1,2];y:=[0,1,2]

array:=[0,2,4;1,3,5;2,4,6] #probably the wrong input, but you get the idea

ArrayInterpolation?

there is a method in vba:

http://www.tushar-mehta.com/excel/newsgroups/interpolation/#Two_dimensional_interpolation_

I have to use the optimization package. 

- The objective function is non linear,

- I have constrains and bounds,

- The constrains are not linear.

 

I have reading the help page on maplesoft.com

 

My question are:
Can you confirm me that the only algorithm I can use is : NLPSolve with method ''sqp''?


And if I would like to use the gradient method how can I do?

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations. 
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables for example the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw 
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf






Another_way_to_calculate_the_definite_integral.mw

ptin the file I upload is a complicated expression containing θ and r. Will the accuracy of the result be affected if I use the second method in the file to calculate the definite integral?

 

and plot  function I? This I is the area which I wrote at the paper.

Could you give me the code which can be used to solve the ODE by numerical method and plot I with respect to t?

I think I have write down everything clearly but if you feel confused please ask me.

I am eager to know the code. Thanks very much!

Dear all,

I need you help to finish some steps of this idea to approximate the roots of a given equation (polynom). Thanks in advance for your help. 

I have a sturm sequence, I would like to use Bisection method to approximation the roots using Sturm decomposition of my polynom. For example, my polynom is  P=x^6-4*x^3+x-2

s := sturmseq(x^6-4*x^3+x-2,x);

sturm(s,x,-2,2); # The number of roots in the interval (-2,2)

Here, i would like to find the roots in (-M,M) :

Bounding all roots in [-M,M] where M = max{1, sum^(n-1) |ai|/an}.

f0 = f, f1 = f', then use -remainder,

I know that  sturm(s,x,-M,M); gives the number of roots in (-M,M)  but is it possible to use the variation of sign like :

      gives a Sturm sequence for f.

      variation of sign, varsign(a0,a1,...,ar).

      Thm: (Sturm) varsign(f0(alpha),...,fr(alpha)) - varsign(f0(beta),..., fr(beta))

      is the number of distinct roots of f in [alpha,beta].

then i would like Isolating roots of rational polynomials

 

Method: reduce, remove rational roots, divide and conquer in [-M,M],

      then use bisection  in disjoint closed intervals ctg one root each

 Bisection method :

Bisection
      Setup: f(a) < 0, f(b) > 0 (or conversely).
      Repeated subdivision of [a,b] guaranteed to get close to a root.

      Error analysis: for error eps, solve (b-a)/ 2^(n+1)  < tol for n. where tol is the tolerance

Thanks

Dear all;

I need you help for solving this problem, and thanks in advantage for your help.

I have a polynom like  P =x^6-4*x^3+x-2;  and i would like to find an approximate value of the roots in some interval [a,b] =[-2,2] using sturm sequence.

The method is based on:

1) first construct the sturm sequence:

For given polynom P =x^6-4*x^3+x-2;

Let S0=P;

S1=diff(p,x);

let   s:=quo(S0,S1,x);
       S2:=-rem(S0,S1,x);

.... S[k+1-rem(S[k-1],S[k]);

 

S[k] is the sturm sequence.

2) let f(a)= number of change of sign in the sturm sequence and f(b) the same . so f(b)-f(a) give the number of roots in the interval [a,b].

3) If f(b)-f(a) =0 so there are no roots

and if f(a)-f(b)=1 one can find the root

4) if f(a) -f(b) >2  :

given toterance tol=0.001; for example

if the abs(a-b)<2*epsilon we display a message that there are k roots at (b+a)/2

with our error tolerance

5) otherwise if c=(b+a)/2 is not a root of P_k(x)  for any k, ( where p_k is an element of the sturm sequence ) 

we divide the interval into equal halves [a,c] and [x,b] and we run step 2 on each interval

else if c is a root of one of these p_k(x) add any time account to c so that c lies close the middle of [a,b] and not a root

6) Give all the roots ( approximate the rrots with small error epsilon).

 

I kindly  appreciate your help

 

 

 

Completely new to maple. Not even sure where to start with this. Can't get the Simpson's or Trapezoidal Rule to work for me. Any help would be much appreciated

Hi everyone. Can enuone help with small parameter method?

restart;

eq := diff(x(t), `$`(t, 2))-epsilon(alpha*x(t)^4-beta)*(diff(x(t), t))-x(t)+x(t)^3;

subs(x(t) = 1+epsilon*x[1](t)+epsilon^2*x[2](t), eq);

pls help with this...

Hello. I have an inequality and I need to prove or negate if it is true or false. This inequality has 8 variables. I simplify it and try to see if it is ture or false. I tried "test relation" in maple and it seems I can't say it is always true or false. For some values of the variables it is true and for some others its false. Is there a method I can show if this inequlity is hold under some assumptions? I mean I want to keep some variables as constant and prove it up to a point. My inequlity is below. Thank you for the help in advance.


(P[A]*(p-w)/(1-P[A])-c)*H[A]+(w-P[A]*(p-w)/(1-P[A]))*P[A]*H[A]+w[u]*P[B]*(1-P[A])*H[B] < (P[B]*(p-w)/(1-P[B])-c)*H[B]+(w-P[B]*(p-w)/(1-P[B]))*P[B]*H[B]+w[u]*P[A]*(1-P[B])*H[B]

And this is how it looks on maple:

Hi, I am completely new to Maple, and I need to use it to optimize my equations in order to make my PLC codes more compressed. I am calculating forward kinematics with the Denavit-Hartenberg method and as such I get long expressions. After a lot of google'ing and frustration, I thought I'd ask here in the hope that one of you might be able to assist me.

I have the following equations;

X := L10*cos(q5) - L16*(sin(q10)*(sin(q5)*sin(q8) - cos(q8)*(cos(q5)*cos(q6)*cos(q7) - cos(q5)*sin(q6)*sin(q7))) - cos(q10)*(sin(q9)*(cos(q8)*sin(q5) + sin(q8)*(cos(q5)*cos(q6)*cos(q7) - cos(q5)*sin(q6)*sin(q7))) + cos(q9)*(cos(q5)*cos(q6)*sin(q7) + cos(q5)*cos(q7)*sin(q6)))) - d2*(cos(q10)*(sin(q5)*sin(q8) - cos(q8)*(cos(q5)*cos(q6)*cos(q7) - cos(q5)*sin(q6)*sin(q7))) + sin(q10)*(sin(q9)*(cos(q8)*sin(q5) + sin(q8)*(cos(q5)*cos(q6)*cos(q7) - cos(q5)*sin(q6)*sin(q7))) + cos(q9)*(cos(q5)*cos(q6)*sin(q7) + cos(q5)*cos(q7)*sin(q6)))) + L15*(sin(q9)*(cos(q8)*sin(q5) + sin(q8)*(cos(q5)*cos(q6)*cos(q7) - cos(q5)*sin(q6)*sin(q7))) + cos(q9)*(cos(q5)*cos(q6)*sin(q7) + cos(q5)*cos(q7)*sin(q6))) - L11*cos(q5)*sin(q6) + d1*cos(q5)*cos(q6) - L13*sin(q5)*sin(q8) + L14*cos(q9)*(cos(q8)*sin(q5) + sin(q8)*(cos(q5)*cos(q6)*cos(q7) - cos(q5)*sin(q6)*sin(q7))) + L13*cos(q8)*(cos(q5)*cos(q6)*cos(q7) - cos(q5)*sin(q6)*sin(q7)) - L14*sin(q9)*(cos(q5)*cos(q6)*sin(q7) + cos(q5)*cos(q7)*sin(q6)) + L12*cos(q5)*cos(q6)*cos(q7) - L12*cos(q5)*sin(q6)*sin(q7);

Y := L10*sin(q5) - L9 + L16*(sin(q10)*(cos(q5)*sin(q8) - cos(q8)*(sin(q5)*sin(q6)*sin(q7) - cos(q6)*cos(q7)*sin(q5))) - cos(q10)*(sin(q9)*(cos(q5)*cos(q8) + sin(q8)*(sin(q5)*sin(q6)*sin(q7) - cos(q6)*cos(q7)*sin(q5))) - cos(q9)*(cos(q6)*sin(q5)*sin(q7) + cos(q7)*sin(q5)*sin(q6)))) + d2*(cos(q10)*(cos(q5)*sin(q8) - cos(q8)*(sin(q5)*sin(q6)*sin(q7) - cos(q6)*cos(q7)*sin(q5))) + sin(q10)*(sin(q9)*(cos(q5)*cos(q8) + sin(q8)*(sin(q5)*sin(q6)*sin(q7) - cos(q6)*cos(q7)*sin(q5))) - cos(q9)*(cos(q6)*sin(q5)*sin(q7) + cos(q7)*sin(q5)*sin(q6)))) - L15*(sin(q9)*(cos(q5)*cos(q8) + sin(q8)*(sin(q5)*sin(q6)*sin(q7) - cos(q6)*cos(q7)*sin(q5))) - cos(q9)*(cos(q6)*sin(q5)*sin(q7) + cos(q7)*sin(q5)*sin(q6))) + L13*cos(q5)*sin(q8) - L11*sin(q5)*sin(q6) + d1*cos(q6)*sin(q5) - L14*cos(q9)*(cos(q5)*cos(q8) + sin(q8)*(sin(q5)*sin(q6)*sin(q7) - cos(q6)*cos(q7)*sin(q5))) - L13*cos(q8)*(sin(q5)*sin(q6)*sin(q7) - cos(q6)*cos(q7)*sin(q5)) - L14*sin(q9)*(cos(q6)*sin(q5)*sin(q7) + cos(q7)*sin(q5)*sin(q6)) + L12*cos(q6)*cos(q7)*sin(q5) - L12*sin(q5)*sin(q6)*sin(q7);

Z := L15*(cos(q9)*(cos(q6)*cos(q7) - sin(q6)*sin(q7)) - sin(q8)*sin(q9)*(cos(q6)*sin(q7) + cos(q7)*sin(q6))) - L11*cos(q6) - L8 - d1*sin(q6) + L16*(cos(q10)*(cos(q9)*(cos(q6)*cos(q7) - sin(q6)*sin(q7)) - sin(q8)*sin(q9)*(cos(q6)*sin(q7) + cos(q7)*sin(q6))) - cos(q8)*sin(q10)*(cos(q6)*sin(q7) + cos(q7)*sin(q6))) - d2*(sin(q10)*(cos(q9)*(cos(q6)*cos(q7) - sin(q6)*sin(q7)) - sin(q8)*sin(q9)*(cos(q6)*sin(q7) + cos(q7)*sin(q6))) + cos(q8)*cos(q10)*(cos(q6)*sin(q7) + cos(q7)*sin(q6))) - L13*cos(q8)*(cos(q6)*sin(q7) + cos(q7)*sin(q6)) - L14*sin(q9)*(cos(q6)*cos(q7) - sin(q6)*sin(q7)) - L12*cos(q6)*sin(q7) - L12*cos(q7)*sin(q6) - L14*cos(q9)*sin(q8)*(cos(q6)*sin(q7) + cos(q7)*sin(q6));

 

I need to optimize these equations, but still keep them separate. I would like to use mutual expressions for the calculations within, but still as I said keep the outputs of X, Y and Z separate.

This is MATLAB code.

 

Thanks in advance for any help.

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