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Lee := (-1+Int(exp(LambertW(1/(-1+t))*(-1+t)), t=1..x))/(Int(exp(LambertW(1/(-1+t))*(-1+t)), t=1..x));
sum(unknown, n=1..infinity) = Lee
 
how to find unknown?

Please I am having problem with this code particularly the last subroutine

#subroutine 1

restart;
Digits:=30:

f:=proc(n)
    -25*y[n]+12*cos(x[n]):
end proc:

#subroutine 2

e1:=y[n+4] = -y[n]+2*y[n+2]+((1/15)*h^2+(2/945)*h^2*u^2+(1/56700)*h^2*u^4-(1/415800)*h^2*u^6-(167/833976000)*h^2*u^8-(2633/245188944000)*h^2*u^10-(2671/5557616064000)*h^2*u^12-(257857/13304932857216000)*h^2*u^14-(3073333/4215002729166028800)*h^2*u^16)*f(n)+((16/15)*h^2-(8/945)*h^2*u^2-(1/14175)*h^2*u^4+(1/103950)*h^2*u^6+(167/208494000)*h^2*u^8+(2633/61297236000)*h^2*u^10+(2671/1389404016000)*h^2*u^12+(257857/3326233214304000)*h^2*u^14+(3073333/1053750682291507200)*h^2*u^16)*f(n+1)+((26/15)*h^2+(4/315)*h^2*u^2+(1/9450)*h^2*u^4-(1/69300)*h^2*u^6-(167/138996000)*h^2*u^8-(2633/40864824000)*h^2*u^10-(2671/926269344000)*h^2*u^12-(257857/2217488809536000)*h^2*u^14-(3073333/702500454861004800)*h^2*u^16)*f(n+2)+((16/15)*h^2-(8/945)*h^2*u^2-(1/14175)*h^2*u^4+(1/103950)*h^2*u^6+(167/208494000)*h^2*u^8+(2633/61297236000)*h^2*u^10+(2671/1389404016000)*h^2*u^12+(257857/3326233214304000)*h^2*u^14+(3073333/1053750682291507200)*h^2*u^16)*f(n+3)+((1/15)*h^2+(2/945)*h^2*u^2+(1/56700)*h^2*u^4-(1/415800)*h^2*u^6-(167/833976000)*h^2*u^8-(2633/245188944000)*h^2*u^10-(2671/5557616064000)*h^2*u^12-(257857/13304932857216000)*h^2*u^14-(3073333/4215002729166028800)*h^2*u^16)*f(n+4):

e2:=y[n+3] = -y[n+1]+2*y[n+2]+(-(1/240)*h^2-(31/60480)*h^2*u^2-(67/1814400)*h^2*u^4-(109/53222400)*h^2*u^6-(18127/186810624000)*h^2*u^8-(64931/15692092416000)*h^2*u^10-(9701/59281238016000)*h^2*u^12-(20832397/3406062811447296000)*h^2*u^14-(11349439/51876956666658816000)*h^2*u^16)*f(n)+((1/10)*h^2+(31/15120)*h^2*u^2+(67/453600)*h^2*u^4+(109/13305600)*h^2*u^6+(18127/46702656000)*h^2*u^8+(64931/3923023104000)*h^2*u^10+(9701/14820309504000)*h^2*u^12+(20832397/851515702861824000)*h^2*u^14+(11349439/12969239166664704000)*h^2*u^16)*f(n+1)+((97/120)*h^2-(31/10080)*h^2*u^2-(67/302400)*h^2*u^4-(109/8870400)*h^2*u^6-(18127/31135104000)*h^2*u^8-(64931/2615348736000)*h^2*u^10-(9701/9880206336000)*h^2*u^12-(20832397/567677135241216000)*h^2*u^14-(11349439/8646159444443136000)*h^2*u^16)*f(n+2)+((1/10)*h^2+(31/15120)*h^2*u^2+(67/453600)*h^2*u^4+(109/13305600)*h^2*u^6+(18127/46702656000)*h^2*u^8+(64931/3923023104000)*h^2*u^10+(9701/14820309504000)*h^2*u^12+(20832397/851515702861824000)*h^2*u^14+(11349439/12969239166664704000)*h^2*u^16)*f(n+3)+(-(1/240)*h^2-(31/60480)*h^2*u^2-(67/1814400)*h^2*u^4-(109/53222400)*h^2*u^6-(18127/186810624000)*h^2*u^8-(64931/15692092416000)*h^2*u^10-(9701/59281238016000)*h^2*u^12-(20832397/3406062811447296000)*h^2*u^14-(11349439/51876956666658816000)*h^2*u^16)*f(n+4):

e3:=h*delta[n] = (-149/42-(16/245)*u^2-(1324/169785)*u^4-(559246/695269575)*u^6-(14310311/175207932900)*u^8-(170868550903/20641246574949000)*u^10)*y[n]+(128/21+(32/245)*u^2+(2648/169785)*u^4+(1118492/695269575)*u^6+(14310311/87603966450)*u^8+(170868550903/10320623287474500)*u^10)*y[n+1]+(-107/42-(16/245)*u^2-(1324/169785)*u^4-(559246/695269575)*u^6-(14310311/175207932900)*u^8-(170868550903/20641246574949000)*u^10)*y[n+2]+(-(67/1260)*h^2+(1241/198450)*h^2*u^2+(277961/366735600)*h^2*u^4+(26460409/333729396000)*h^2*u^6+(1363374533/168199615584000)*h^2*u^8+(16323847966961/19815596711951040000)*h^2*u^10)*f(n)+((188/105)*h^2+(5078/99225)*h^2*u^2+(556159/91683900)*h^2*u^4+(51834031/83432349000)*h^2*u^6+(67782373/1078202664000)*h^2*u^8+(1854079193287/291405833999280000)*h^2*u^10)*f(n+1)+((31/90)*h^2+(341/33075)*h^2*u^2+(79361/61122600)*h^2*u^4+(23456627/166864698000)*h^2*u^6+(1228061399/84099807792000)*h^2*u^8+(14797833720283/9907798355975520000)*h^2*u^10)*f(n+2)+(-(4/105)*h^2-(46/14175)*h^2*u^2-(809/1871100)*h^2*u^4-(27827/567567000)*h^2*u^6-(637171/122594472000)*h^2*u^8-(33500737/62523180720000)*h^2*u^10)*f(n+3)+((1/252)*h^2+(23/28350)*h^2*u^2+(809/7484400)*h^2*u^4+(27827/2270268000)*h^2*u^6+(637171/490377888000)*h^2*u^8+(33500737/250092722880000)*h^2*u^10)*f(n+4):

e4:=y[3] = -y[1]+2*y[2]+(-(1/240)*h^2-(31/60480)*h^2*u^2-(67/1814400)*h^2*u^4-(109/53222400)*h^2*u^6-(18127/186810624000)*h^2*u^8-(64931/15692092416000)*h^2*u^10-(9701/59281238016000)*h^2*u^12-(20832397/3406062811447296000)*h^2*u^14-(11349439/51876956666658816000)*h^2*u^16)*f(0)+((1/10)*h^2+(31/15120)*h^2*u^2+(67/453600)*h^2*u^4+(109/13305600)*h^2*u^6+(18127/46702656000)*h^2*u^8+(64931/3923023104000)*h^2*u^10+(9701/14820309504000)*h^2*u^12+(20832397/851515702861824000)*h^2*u^14+(11349439/12969239166664704000)*h^2*u^16)*f(1)+((97/120)*h^2-(31/10080)*h^2*u^2-(67/302400)*h^2*u^4-(109/8870400)*h^2*u^6-(18127/31135104000)*h^2*u^8-(64931/2615348736000)*h^2*u^10-(9701/9880206336000)*h^2*u^12-(20832397/567677135241216000)*h^2*u^14-(11349439/8646159444443136000)*h^2*u^16)*f(2)+((1/10)*h^2+(31/15120)*h^2*u^2+(67/453600)*h^2*u^4+(109/13305600)*h^2*u^6+(18127/46702656000)*h^2*u^8+(64931/3923023104000)*h^2*u^10+(9701/14820309504000)*h^2*u^12+(20832397/851515702861824000)*h^2*u^14+(11349439/12969239166664704000)*h^2*u^16)*f(3)+(-(1/240)*h^2-(31/60480)*h^2*u^2-(67/1814400)*h^2*u^4-(109/53222400)*h^2*u^6-(18127/186810624000)*h^2*u^8-(64931/15692092416000)*h^2*u^10-(9701/59281238016000)*h^2*u^12-(20832397/3406062811447296000)*h^2*u^14-(11349439/51876956666658816000)*h^2*u^16)*f(4):

e5:=h*delta[0] = (-149/42-(16/245)*u^2-(1324/169785)*u^4-(559246/695269575)*u^6-(14310311/175207932900)*u^8-(170868550903/20641246574949000)*u^10)*y[0]+(128/21+(32/245)*u^2+(2648/169785)*u^4+(1118492/695269575)*u^6+(14310311/87603966450)*u^8+(170868550903/10320623287474500)*u^10)*y[1]+(-107/42-(16/245)*u^2-(1324/169785)*u^4-(559246/695269575)*u^6-(14310311/175207932900)*u^8-(170868550903/20641246574949000)*u^10)*y[2]+(-(67/1260)*h^2+(1241/198450)*h^2*u^2+(277961/366735600)*h^2*u^4+(26460409/333729396000)*h^2*u^6+(1363374533/168199615584000)*h^2*u^8+(16323847966961/19815596711951040000)*h^2*u^10)*f(0)+((188/105)*h^2+(5078/99225)*h^2*u^2+(556159/91683900)*h^2*u^4+(51834031/83432349000)*h^2*u^6+(67782373/1078202664000)*h^2*u^8+(1854079193287/291405833999280000)*h^2*u^10)*f(1)+((31/90)*h^2+(341/33075)*h^2*u^2+(79361/61122600)*h^2*u^4+(23456627/166864698000)*h^2*u^6+(1228061399/84099807792000)*h^2*u^8+(14797833720283/9907798355975520000)*h^2*u^10)*f(2)+(-(4/105)*h^2-(46/14175)*h^2*u^2-(809/1871100)*h^2*u^4-(27827/567567000)*h^2*u^6-(637171/122594472000)*h^2*u^8-(33500737/62523180720000)*h^2*u^10)*f(3)+((1/252)*h^2+(23/28350)*h^2*u^2+(809/7484400)*h^2*u^4+(27827/2270268000)*h^2*u^6+(637171/490377888000)*h^2*u^8+(33500737/250092722880000)*h^2*u^10)*f(4):

 

#subroutine 3

 


inx:=0:
ind:=0:
iny:=1:
h:=Pi/4.0:
n:=0:
omega:=5:
u:=omega*h:
N:=solve(h*p = 500*Pi/2, p):

c:=1:
for j from 0 to 5 do
    t[j]:=inx+j*h:
end do:
#e||(1..6);
vars:=y[n+1],y[n+2],y[n+3],delta[n],y[n+4]:

printf("%6s%15s%15s%15s\n",
    "h","Num.y","Ex.y","Error y");
for k from 1 to N do

    par1:=x[0]=t[0],x[1]=t[1],x[2]=t[2],x[3]=t[3],x[4]=t[4],x[5]=t[5]:
    par2:=y[n]=iny,delta[n]=ind:
   

res:=eval(<vars>, fsolve(eval({e||(1..5)},[par1,par2]), {vars}));

    for i from 1 to 5 do
        exy:=eval(0.5*cos(5*c*h)+0.5*cos(c*h)):
        printf("%6.5f%17.9f%15.9f%13.5g\n",
        h*c,res[i],exy,abs(res[i]-exy)):
        
        c:=c+1:
    end do:
    iny:=res[5]:
    inx:=t[5]:
    for j from 0 to 5 do
        t[j]:=inx + j*h:
    end do:
end do:

 

 

 

 

A system of algebraic equation

in terms of x, y, z

how draw 3 different circles to show the range of possible values for x, y and z respectively?

it may not be a circle 

It may be 3 bounded area graph to show the range of x , y , z respectively

 

updated

like the graph in many examples in

algebraic and geometric ideas in the theory of discrete optimization

bound area have color

Hi, There is a problem in solving ODE using dsolve/numeric code. You people would like to help me in resolving the error, here is in attached file,

Latif_paper.mw

The following is the PDE I need to solve.

(x*y+1)*(diff(h(x, y), y, y, y))+(x+h(x, y))*(diff(h(x, y), y, y))-(diff(h(x, y), y))^2+k(x, y) = 0, (10.*(x*y+1))*(diff(k(x, y), y,y))+(10.*x+h(x,y))*(diff(k(x, y), y))-(diff(h(x, y), y))*k(x, y) = 0

 

This is the original boundary condition:

h(0, y) = f(y), h(x, 0) = 0, k(0, y) = g(y), k(x, 0) = 1, k(x, 25) = 0, (D[2](h))(x, 0) = 0, (D[2](h))(x, 25) = 0

 

After using pdsolve it come out the error:

pdsolve(eval(pde2, P = .1), pdebc4, numeric, [h(x, y), k(x, y)], spacestep = .1)

Error, (in pdsolve/numeric/par_hyp) Incorrect number of initial conditions, expected 0, got 2

 

If I remove one of the boundary condition when x=0, maybe h(0,y)=f(y), then the error will be this:

Error, (in pdsolve/numeric/par_hyp) Incorrect number of initial conditions, expected 0, got 1

 

However if I remove both when x=0, it come out this error:

Error, (in pdsolve/numeric) initial/boundary conditions must be defined at one or two points for each independent variable

 

May I know what is the problem of this equations?

P/S: I know its only differentiate with respect to y and is consider to be an ODE( I need more explantion on this please) and I'm still new to maple. Thanks!!

 

I am trying to evaluate the following double integral where hypergeom([x,1/2],[3/2],C) is gauss hypergeometric function 2f1. maple gives back it unevaluated. I doubt it may be due to slow convergence of hypergeometric function. 
 

restart; x := (1/6)*Pi; evalf(int(evalf(int(cos(x)*hypergeom([x, 1/2], [3/2], sin(x)/(r*cos(x)+k-2*r*sin(x))^2)/(r*sin(x)^2+r*cos(x)+k)^4, k = 0 .. 10)), r = 1 .. 2))

Int(Int(.8660254040*hypergeom([.5000000000, .5235987758], [1.500000000], .5000000000/(-.1339745960*r+k)^2)/(1.116025404*r+k)^4, k = 0. .. 10.), r = 1. .. 2.)

(1)

``


 

Download DOUBLE_INT_2.mw

Hi every body,

I have a function "p(v,T)" which I evaluated its critical point. after calculating when I want to plot diagram of "p-v" for some values of "T" around critical value of "T" I expect the shape of diagram for "T" bigger and smaller than critical value of "T" be different. but it not happened. Are here anyone can help me? The function "p(v,T)" is in the file. if you want calculate critical point and check I am right or wrong. Thanks criticalpoint.mw

is it possible to change ODE to PDE?

the ODE has diff(a(t),t) and diff(b(t),t)

how to convert to diff(t, a), diff(t, b) ?

I am trying to evaluate the following triple integral but it takes much time so i kill the job.


 

restart; R := 5; KK := proc (theta) options operator, arrow; evalf(int(int(int(1/(R*sin(theta)^2+(R*cos(theta)+Z)^2+(2*R*k.sin(theta))*cos(p))^2, p = 0 .. 2*Pi), Z = 0 .. 60), k = 1 .. 10, numeric)) end proc; evalf(KK((1/6)*Pi))

Warning,  computation interrupted

 

``


 

Download int_maple_prime2.mw

with(DEtools, buildsym, equinv, symtest):
ans := dsolve([eq2,eq3,eq4], Lie);
Error, (in dsolve) too many arguments; some or all of the following are wrong: [{a(t), b(t), c(t)}, Lie]
 
ans := dsolve([eq2+eq3+eq4 = exp(t)], Lie);
Error, (in PDEtools/sdsolve) too many arguments; some or all of the following are wrong: [{a(t), b(t), c(t)}, Lie]
 
ans := dsolve([eq2,eq3,eq4]);
sym2 := buildsym(ans);
Error, (in buildsym) invalid input: `ODEtools/buildsym` expects its 1st argument, sol, to be of type {algebraic, algebraic = algebraic}, but received [{c(t) = ...}, {b(t) = ...}, {a(t) = ...)}]
 
 
PDEtools[declare](a(t), b(t), c(t), prime = t):
symgen(eq2+eq3+eq4=0);
                       a(t) will now be displayed as a
                       b(t) will now be displayed as b
                       c(t) will now be displayed as c
   derivatives with respect to t of functions of one variable will now be
      displayed with 'symgen(....)'
 
 
update
if it can not do for 3 function a(t),b(t),c(t) system of differential equations
then
 
i change to use
eq2 := subs(b(t)=a(t),subs(c(t)=a(t),eq2));
eq3 := subs(b(t)=a(t),subs(c(t)=a(t),eq3));
eq4 := subs(b(t)=a(t),subs(c(t)=a(t),eq4));
 
with(DEtools, buildsym, equinv, symtest):
ans := dsolve(eq2 = 0, Lie);
buildsym(ans[1], a(t));
buildsym(ans[2], a(t));
buildsym(ans[3], a(t));
 
there are 3 answers, can i use one of it to recover the equation eq2 or  eq3 or eq4?
 
ans := dsolve(eq3=0, Lie);
buildsym(ans[1], a(t));
sym2 := buildsym(ans[2], a(t));
buildsym(ans[3], a(t));

sym := [_xi=rhs(sym2[2]),_eta=rhs(sym2[1])];
ODE := equinv(sym, a(t));
eq3 - ODE;
sym := [_xi=rhs(sym2[1]),_eta=rhs(sym2[2])];
ODE := equinv(sym, a(t));
eq3 - ODE;
but ODE is not equal to original eq3
ans := dsolve(eq4=0, Lie);
buildsym(ans[1], a(t));
buildsym(ans[2], a(t));
 
ans := dsolve(eq2+eq3+eq4=0, Lie);
sym := buildsym(ans[1], a(t));
ODE := equinv(sym, a(t));
eq2+eq3+eq4 - ODE;
sym := buildsym(ans[2], a(t));
ODE := equinv(sym, a(t));
eq2+eq3+eq4 - ODE;
sym := buildsym(ans[3], a(t));
ODE := equinv(sym, a(t));
simplify(eq2+eq3+eq4 - - ODE);
 
can not recover the original result

UPDATE

Thanks for checking. I verify I get the error and made screen shots below

When I add the convert() command, the error goes away. Here is screen shot

 

I am using Maple 2017, student version, on windows 7, 64 bit, home edition.

 

Original post

This is using Maple 2017 on windows.

With the following input, Maple pdsolve gives an error

pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2);
bc:=D[1](u)(0,t)=0,D[1](u)(L,t)=0:
assume(L>0):
ic:=u(x,0)=piecewise(0<x and x<=L/2,0,L/2<x and x<L,1):
sol:=pdsolve([pde,bc,ic],u(x,t)):

However, if I add one line to convert the piecewise function above to piecewise, then pdsolve no longer gives an error. So the following input works

restart;
pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2);
bc:=D[1](u)(0,t)=0,D[1](u)(L,t)=0:
assume(L>0):
ic:=u(x,0)=piecewise(0<x and x<=L/2,0,L/2<x and x<L,1):
ic:=convert(ic,piecewise,x):
sol:=pdsolve([pde,bc,ic],u(x,t)):

 

Notice the extra line. Why does one have to convert piecewise to piecewise to make pdsolve accept the input?

sorry did not write down the error message and I am writing this from school library PC. But if you try the first case, you'll see the error.

 

I have final project to make a media for learning mathematic using maple. But I'm so confused to make a net of cuboid, anybody can help me? please :D

Hi,

 

I have a problem of having a problem to solve the following equation with the this error. I found out that it might be due to that I have a differential that is squared and maple could not calculate it as after square root it will have positive and negative. May I know how to overcome this other than changing my equations?

ODE_summarize.mw
 

restart; with(DEtools)

````

sol1 := alpha*(diff(f(y), y, y, y)) = -(1/4)*f(y)*(diff(f(y), y, y))*n+(1/2)*(diff(f(y), y))^2*n-(3/4)*f(y)*(diff(f(y), y, y))+(1/2)*(diff(f(y), y))^2-g(y), diff(g(y), y, y) = -(1/4)*P[r](f(y)*(diff(g(y), y))*n-4*n*(diff(f(y), y))*g(y)+3*f(y)*(diff(g(y), y)))

alpha*(diff(diff(diff(f(y), y), y), y)) = -(1/4)*f(y)*(diff(diff(f(y), y), y))*n+(1/2)*(diff(f(y), y))^2*n-(3/4)*f(y)*(diff(diff(f(y), y), y))+(1/2)*(diff(f(y), y))^2-g(y), diff(diff(g(y), y), y) = -(1/4)*P[r](f(y)*(diff(g(y), y))*n-4*n*(diff(f(y), y))*g(y)+3*f(y)*(diff(g(y), y)))

(1)

bc := (D(f))(y)

(D(f))(y)

(2)

gg := convert(bc, diff)

diff(f(y), y)

(3)

gg1 := limit(gg, y = infinity)

limit(diff(f(y), y), y = infinity)

(4)

gg2 := convert(gg1, D) = 0

limit((D(f))(y), y = infinity) = 0

(5)

gg2

(D(f))(y) = 0

(6)

bc1 := eval(f(y), y = 0) = 0, eval((D(f))(y), y = 0) = 0, convert(gg1, D) = 0, limit(g(y), y = infinity) = 0, eval(g(y), y = 0) = 1

f(0) = 0, (D(f))(0) = 0, limit((D(f))(y), y = infinity) = 0, limit(g(y), y = infinity) = 0, g(0) = 1

(7)

``

odesub1 := eval(eval(sol1, n = .6), P[r] = 0)

alpha*(diff(diff(diff(f(y), y), y), y)) = -.9000000000*f(y)*(diff(diff(f(y), y), y))+.8000000000*(diff(f(y), y))^2-g(y), diff(diff(g(y), y), y) = 0

(8)

sol2 := dsolve({bc1, odesub1}, numeric, [f(y), g(y)], output = operator, continuation = alpha, initmesh = 2400, method = bvp[midrich])

Error, (in dsolve/numeric/bvp/convertsys) unable to convert to an explicit first-order system

 

``


 

 

 

 

 

 

 

 

 

 

Without my knowingly changing any Maple parameters, the execution time values of variables appearing in the window displayed when a DEBUG statement is executed have recently started to be listed at the end of my worksheet.

If I have been doing extensive debugging this list can be quite lengthy and a nuisance to erase.

Why did this listing suddenly begin and how can I stop it?

I have the following expression

((4*(N-i+2))*((N-i-2)*(-(N-i-4)*(N+i+2)*(N+2)*(N+4)+N^4+4*N^3+4*N^2+16*N-40)-(4*(N-1))*(2*N+3)*(N+5))+(8*(N+5))*(N^2+8*N+6))/((N-i+1)*(N-i+3)*((N-i-2)*(i+3)*(N+2)*(N+4)-(8*(N+5))*(N-1)))

The parameters i and N are nonnegative integers and i is less than or equal to N. The purpose is to make it as short as possible. Based on my experience, it could be expressed as a small binomial expression or as a sum of two or three binomials. However, by Maple commands the conversion does not give me binomials or any smaller expression.

Is there any way for the conversion to binomials or any other conversion to shorten the expression?

I appreciate any help.

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