mehdi jafari

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6 years, 122 days

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These are answers submitted by mehdi jafari

There is an equivalent form of variation of a functional in form of differentiation. 
You can use this form to find variation of a functional with the corresponding boundary conditions.
but in this form, you should use physics package "diff" since you should differentiate with respect to another function.
see https://en.wikipedia.org/wiki/Euler–Lagrange_equation
 


 

restart:

P:=proc(f,n,a,b) local i,V;
V:=LinearAlgebra:-RandomVector[row](n,generator=a..b,outputoptions=[datatype=float[8]]):
seq(eval(f,x=(V[i])),i=1..n);
end proc

proc (f, n, a, b) local i, V; V := LinearAlgebra:-RandomVector[row](n, generator = a .. b, outputoptions = [datatype = float[8]]); seq(eval(f, x = V[i]), i = 1 .. n) end proc

(1)

P(x^2+sin(x),5,1,10)

HFloat(49.65698659871879), HFloat(49.65698659871879), HFloat(9.141120008059866), HFloat(35.72058450180107), HFloat(24.04107572533686)

(2)

 


 

Download proce.mw

for using plot3d you shouldn't assign x1, and it should remain varibale to get a range. when u assing x1, you can just use plot

 

restart

f2:=3.579167+3.537500*x1-2.645833*x2-0.250000*x1*x1-0.012500*x2*x1+0.175000*x2*x2;

3.579167+3.537500*x1-2.645833*x2-.250000*x1^2-0.12500e-1*x2*x1+.175000*x2^2

(1)

 plot3d (f2,x1=3..7 ,x2=3..7, axes=boxed);

 

x1:=5;
f1:=3.579167+3.537500*x1-2.645833*x2-0.250000*x1*x1-0.012500*x2*x1+0.175000*x2*x2;

5

 

15.016667-2.708333*x2+.175000*x2^2

(2)

 plot (f1, x2=3..7, labels=[grape, preference], axes=boxed);

 

 


 

Download plot.mw

i think the problem is due to maple can not determine the varibales sign in the "abs" function. so use assumptions to solve it :


 

restart

f:=sqrt(x)*ln(y);evalc(Im(f))

x^(1/2)*ln(y)

 

(1/2)*abs(x)^(1/2)*(1-signum(x))*ln(abs(y))+(1/2)*abs(x)^(1/2)*(1+signum(x))*(1/2-(1/2)*signum(y))*Pi

(1)

solve(evalc(Im(f)), {x,y}) assuming x>0

{x = 0, y = y}, {x = x, 0 <= y}

(2)

solve(evalc(Im(f)), {x,y}) assuming y>0

{x = x, y = 1}, {y = y, 0 <= x}, {x = 0, 0 < y}

(3)

g:=x*ln(y);evalc(Im(g))

x*ln(y)

 

x*(1/2-(1/2)*signum(y))*Pi

(4)

solve(evalc(Im(g)), {x,y})

{x = 0, y = y}, {x = x, 0 <= y}

(5)

 


 

Download assum.mw


 

restart

A:=[
y[a0]-y[d0],
k[d1]*y[a1]-k[d2]*y[d2],
k[d1]*y[a1]*x[1]-k[d2]*y[d2]*x[2]];

[y[a0]-y[d0], k[d1]*y[a1]-k[d2]*y[d2], k[d1]*x[1]*y[a1]-k[d2]*x[2]*y[d2]]

(1)

select(has,A,x);

[k[d1]*x[1]*y[a1]-k[d2]*x[2]*y[d2]]

(2)

remove(has,A,x);

[y[a0]-y[d0], k[d1]*y[a1]-k[d2]*y[d2]]

(3)

#or
select(has,A,[seq](x[i],i=1..2));

[k[d1]*x[1]*y[a1]-k[d2]*x[2]*y[d2]]

(4)

 


 

Download select.mw

it was better to upload the worksheet here, i think you should change " := " to " = " or bring the right hand side to the left with minus sign and without any " := " . i think it will work.


 

restart:printlevel:=2:

M:=5:

for i to M do
if i<=M then
N[i,0](u):=1;N[0,i](u):=0;
fi;od;  

1

 

0

 

1

 

0

 

1

 

0

 

1

 

0

 

1

 

0

(1)

 


 

Download another_way.mw

here is a numerical solution based on the procedure on this link :
https://www.mapleprimes.com/questions/123970-Finding-All-Roots-For-An-ODE-In-A-Given-Interval
 

restart:

allRoots := proc(f :: procedure, rng :: range, $)
local result, root;
  result := Vector(0, 'datatype = float');
  root := RootFinding:-NextZero(f, lhs(rng), 'maxdistance' = rhs(rng) - lhs(rng));
  while root <> FAIL and root <= rhs(rng) do
    result(numelems(result) + 1) := root;
    root := RootFinding:-NextZero(f, root, 'maxdistance' = rhs(rng) - root);
  end do;
  return result;
end proc:

f:=proc(x) sin(sqrt(x));end proc

proc (x) sin(sqrt(x)) end proc

(1)

allRoots(f,0..500)

Vector(7, {(1) = 9.86960440100000, (2) = 39.4784176000000, (3) = 88.8264396000000, (4) = 157.913670400000, (5) = 246.740110000000, (6) = 355.305758400000, (7) = 483.610615600000})

(2)

 

NULL


 

Download allroots.mw

@Preben Alsholm 
 

restart:

EQ:=diff(C(t,r),t$2)=(Z(r)^2-1)-(C(t,r)^2)/(4*alpha)*(1+((1+(16*(alpha)^2/C(t,r)^4)*(Z(r)^2-1)^2-(4*alpha*c/3)+(8*alpha*axi/C(t,r)^4)))^(1/2));

diff(diff(C(t, r), t), t) = Z(r)^2-1-(1/4)*C(t, r)^2*(1+(1+16*alpha^2*(Z(r)^2-1)^2/C(t, r)^4-(4/3)*alpha*c+8*alpha*axi/C(t, r)^4)^(1/2))/alpha

(1)

pdsolve(EQ)

C(t, r) = RootOf(-6*Intat(alpha/(-6*alpha*(-12*Z(r)^2*_g*alpha+3^(1/2)*(Int(((-4*alpha*c*_g^4+48*Z(r)^4*alpha^2+3*_g^4-96*Z(r)^2*alpha^2+48*alpha^2+24*alpha*axi)/_g^4)^(1/2)*_g^2, _g))+12*_g*alpha-6*_F1(r)*alpha+_g^3))^(1/2), _g = _Z)+t+_F2(r))

(2)

 

 


 

Download pdsolve.mw

dear saba, is this what you want ?


 

restart:

seq((rand(-0.5..0.5))(),i=1..100);

.3775318970, -0.816018978e-1, .4069399814, 0.436850085e-1, .3400491380, -.2479170987, -.3512207408, .3520884908, .3284431967, .4848657575, -.1791386899, -.1232575041, -.2524655778, 0.415600955e-1, -.2497371105, -.4648512003, .2807139719, -.2520269759, .1736016048, .3769801486, -.4414281014, -.3251280474, .4242775386, .3820156214, -.1894856664, 0.355971810e-1, -0.876596969e-1, .4631586324, .2262350965, -.4987583014, -0.577514277e-1, -.1300704008, -.2113775594, -.3423466445, -.3155519987, .3500474044, .4412388256, .4804309386, -.1620749471, .2995389112, .2310788109, .1395464705, .1070027478, -.2336220760, -.3235718624, -0.116933206e-1, .3600757475, .3613973235, -0.251119420e-1, -.1103399691, .2885438090, .2936223970, .4041439521, .2022338125, -.4142977244, 0.48001182e-2, -.1128453058, .3110457143, -.2925728264, -.1011149341, .3811756267, -.3500983792, .4594186023, -.4020483728, -.3367579919, -.4805910478, .2398511034, -.3562160995, .2713482470, -.4823600714, .4997302326, -0.543323339e-1, -.1878650770, .2789945410, -0.715776470e-1, .1761052304, .4313688588, -.2796250715, -.1844289845, .4848339962, .4762345853, .3604187411, -.2616383954, .3661844308, -.2666432152, .2765591826, -.2967571547, -0.776444908e-1, .3793445976, -.1646914707, .2421038691, -0.146469119e-1, -.4513732251, -0.575143578e-1, 0.376307310e-1, .4210504051, .2949097933, -.2470575237, -.4927796779, 0.465061082e-1

(1)

 

 


 

Download random.mw


 

restart:

eq1:=diff(A1(t),t$2)+2*diff(A2(t),t$2)+3*diff(A3(t),t$2)-4*diff(A1(t),t)+diff(A1(t),t)*diff(A2(t),t)+3*diff(A3(t),t);

diff(diff(A1(t), t), t)+2*(diff(diff(A2(t), t), t))+3*(diff(diff(A3(t), t), t))-4*(diff(A1(t), t))+(diff(A1(t), t))*(diff(A2(t), t))+3*(diff(A3(t), t))

(1)

eq2:=2*diff(A1(t),t$2)+3*diff(A2(t),t$2)-3*diff(A1(t),t)+2*diff(A2(t),t)+3*(A1(t));

2*(diff(diff(A1(t), t), t))+3*(diff(diff(A2(t), t), t))-3*(diff(A1(t), t))+2*(diff(A2(t), t))+3*A1(t)

(2)

eq3:=2*diff(A2(t),t$2)+3*diff(A3(t),t$2)-4*diff(A2(t),t)+2*diff(A3(t),t)+3*A1(t)+A2(t)+A3(t);

2*(diff(diff(A2(t), t), t))+3*(diff(diff(A3(t), t), t))-4*(diff(A2(t), t))+2*(diff(A3(t), t))+3*A1(t)+A2(t)+A3(t)

(3)

dsolve({seq}(eq||i,i=1..3));

[{A1(t) = ODESolStruc(_a, [{(diff(diff(diff(diff(_b(_a), _a), _a), _a), _a))*_b(_a)^4+(1/9)*(3604*_b(_a)-243*(diff(diff(_b(_a), _a), _a))*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^5*(diff(_b(_a), _a))+486*(diff(diff(_b(_a), _a), _a))*_b(_a)^3*(diff(_b(_a), _a))^2*_a+486*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^4*(diff(_b(_a), _a))*_a+1404*(diff(diff(_b(_a), _a), _a))*_b(_a)^2*(diff(_b(_a), _a))*_a-477*_a+3180*(diff(diff(_b(_a), _a), _a))*_b(_a)^2+3180*(diff(_b(_a), _a))^2*_b(_a)+2332*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^3-27*_b(_a)^2*_a+240*_b(_a)*_a+3286*(diff(_b(_a), _a))*_b(_a)-1179*(diff(_b(_a), _a))^3*_b(_a)^3-4943*(diff(_b(_a), _a))^2*_b(_a)^2+477*(diff(_b(_a), _a))^2*_b(_a)^3-99*(diff(_b(_a), _a))*_b(_a)^3+2548*_b(_a)^2*(diff(_b(_a), _a))-3339*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^3*(diff(_b(_a), _a))-5247*(diff(diff(_b(_a), _a), _a))*(diff(_b(_a), _a))^2*_b(_a)^2+9328*(diff(diff(_b(_a), _a), _a))*_b(_a)^2*(diff(_b(_a), _a))-54*(diff(_b(_a), _a))^2*_b(_a)^2*_a+273*(diff(_b(_a), _a))*_b(_a)*_a+972*(diff(diff(_b(_a), _a), _a))^3*_b(_a)^5-243*(diff(diff(diff(_b(_a), _a), _a), _a))^2*_b(_a)^6-648*_b(_a)^3*(diff(_b(_a), _a))^5+756*(diff(diff(_b(_a), _a), _a))^2*_b(_a)^5+1188*_b(_a)^3*(diff(_b(_a), _a))^4+162*_b(_a)^2*(diff(_b(_a), _a))^5-2196*(diff(diff(_b(_a), _a), _a))^2*_b(_a)^4+189*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^5-1008*_b(_a)^2*(diff(_b(_a), _a))^4-1908*(diff(diff(_b(_a), _a), _a))^2*_b(_a)^3+45*(diff(diff(_b(_a), _a), _a))*_b(_a)^4+1434*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^4+2718*_b(_a)^2*(diff(_b(_a), _a))^3-477*_b(_a)*(diff(_b(_a), _a))^4-5080*(diff(diff(_b(_a), _a), _a))*_b(_a)^3+2332*_b(_a)*(diff(_b(_a), _a))^3-27*_b(_a)^3-453*_b(_a)^2-972*(diff(diff(_b(_a), _a), _a))^2*_b(_a)^5*(diff(_b(_a), _a))-243*(diff(diff(_b(_a), _a), _a))^2*_b(_a)^4*(diff(_b(_a), _a))^2-648*(diff(diff(_b(_a), _a), _a))*_b(_a)^4*(diff(_b(_a), _a))^3+972*(diff(diff(_b(_a), _a), _a))*_b(_a)^3*(diff(_b(_a), _a))^4+324*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^5*(diff(_b(_a), _a))^2+1215*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^4*(diff(_b(_a), _a))^3+648*(diff(diff(_b(_a), _a), _a))^2*_b(_a)^4*(diff(_b(_a), _a))-729*(diff(diff(_b(_a), _a), _a))^2*_b(_a)^4*_a+324*(diff(diff(_b(_a), _a), _a))*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^5+81*(diff(diff(_b(_a), _a), _a))*_b(_a)^4*(diff(_b(_a), _a))^2-162*(diff(diff(_b(_a), _a), _a))*_b(_a)^3*(diff(_b(_a), _a))^3-621*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^5*(diff(_b(_a), _a))-810*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^4*(diff(_b(_a), _a))^2-243*_b(_a)^2*(diff(_b(_a), _a))^4*_a-306*(diff(diff(_b(_a), _a), _a))*_b(_a)^4*(diff(_b(_a), _a))+1575*(diff(diff(_b(_a), _a), _a))*_b(_a)^3*(diff(_b(_a), _a))^2+1998*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^4*(diff(_b(_a), _a))+7020*(diff(diff(_b(_a), _a), _a))*_b(_a)^3*(diff(_b(_a), _a))-324*(diff(diff(_b(_a), _a), _a))*_b(_a)^3*_a+351*(diff(diff(diff(_b(_a), _a), _a), _a))*_b(_a)^3*_a+351*_b(_a)*(diff(_b(_a), _a))^3*_a+1440*(diff(diff(_b(_a), _a), _a))*_b(_a)^2*_a+1440*_b(_a)*(diff(_b(_a), _a))^2*_a)/(27*(diff(diff(_b(_a), _a), _a))*_b(_a)^2+27*(diff(_b(_a), _a))^2*_b(_a)-18*(diff(_b(_a), _a))*_b(_a)+9*_b(_a)-53) = 0}, {_a = A1(t), _b(_a) = diff(A1(t), t)}, {t = Int(1/_b(_a), _a)+_C2, A1(t) = _a}])}, {A2(t) = Int((18*(diff(A1(t), t))*(diff(diff(diff(A1(t), t), t), t))+36*(diff(diff(A1(t), t), t))^2-81*(diff(A1(t), t))*(diff(diff(A1(t), t), t))+54*(diff(diff(A1(t), t), t))*A1(t)+27*(diff(A1(t), t))^2-27*(diff(diff(diff(diff(A1(t), t), t), t), t))+150*(diff(diff(diff(A1(t), t), t), t))+80*(diff(diff(A1(t), t), t))+168*(diff(A1(t), t))+39*A1(t))/(27*(diff(diff(diff(A1(t), t), t), t))-18*(diff(diff(A1(t), t), t))+9*(diff(A1(t), t))-53), t)+_C1}, {A3(t) = (2043*A1(t)+318*A2(t)+3415*(diff(A1(t), t))+1871*(diff(diff(A1(t), t), t))-162*A2(t)*(diff(diff(diff(A1(t), t), t), t))+108*A2(t)*(diff(diff(A1(t), t), t))-54*A2(t)*(diff(A1(t), t))-27*(diff(A1(t), t))^2*A1(t)+135*(diff(diff(diff(A1(t), t), t), t))*(diff(A1(t), t))^2-99*(diff(diff(A1(t), t), t))^2*(diff(A1(t), t))+162*(diff(diff(A1(t), t), t))^2*A1(t)-234*(diff(diff(A1(t), t), t))*(diff(A1(t), t))^2-81*(diff(diff(diff(diff(A1(t), t), t), t), t))*(diff(diff(A1(t), t), t))+27*(diff(diff(A1(t), t), t))*(diff(diff(diff(A1(t), t), t), t))-81*(diff(A1(t), t))*(diff(diff(diff(diff(A1(t), t), t), t), t))-756*(diff(diff(diff(A1(t), t), t), t))*A1(t)+1395*(diff(diff(A1(t), t), t))*A1(t)+24*(diff(A1(t), t))*A1(t)+222*(diff(A1(t), t))*(diff(diff(diff(A1(t), t), t), t))-92*(diff(A1(t), t))*(diff(diff(A1(t), t), t))+1991*(diff(diff(diff(A1(t), t), t), t))-387*(diff(diff(diff(diff(A1(t), t), t), t), t))+108*(diff(A1(t), t))^3+108*(diff(diff(A1(t), t), t))^3+81*(diff(diff(diff(A1(t), t), t), t))^2+1002*(diff(diff(A1(t), t), t))^2+561*(diff(A1(t), t))^2+216*(diff(diff(A1(t), t), t))*(diff(A1(t), t))*A1(t)-81*(diff(diff(diff(A1(t), t), t), t))*(diff(A1(t), t))*A1(t))/(162*(diff(diff(diff(A1(t), t), t), t))-108*(diff(diff(A1(t), t), t))+54*(diff(A1(t), t))-318)}]

(4)

 


 

Download dsolve.mw

please upload your equations, i think you have syntax problem,
you should solve like this :
fsolve({eq1,eq2,eq3,eq4,eq5},{x1,x2,x3,x4,x5},x=0..1};

as an example :

restart:

eq1:=a1+3*a2+5*a3;eq2:=6*a1-12*a2+20;eq3:=8*a1-53*a2+23*a3;

a1+3*a2+5*a3

 

6*a1-12*a2+20

 

8*a1-53*a2+23*a3

(1)

fsolve({seq(eq||i,i=1..3)},{a1,a2,a3},a1=-5..0)

{a1 = -3.711111111, a2 = -.1888888889, a3 = .8555555556}

(2)

 

 

 

 


 

Download fsolve.mw


restart;

 asal := proc(n)
local asal,m,s,i;
 if n<2 then
 m :=n/2+1;
 s:=m;
 print(m);
 else
 for i from 2 to n do
 if irem(n,i)=0 then
 print("asal değil");
 elif i = m then
 print("asaldir");
 fi;
 od;
fi;
 end proc;

proc (n) local asal, m, s, i; if n < 2 then m := (1/2)*n+1; s := m; print(m) else for i from 2 to n do if irem(n, i) = 0 then print("asal deÄŸil") elif i = m then print("asaldir") end if end do end if end proc

(1)

asal(1);

3/2

(2)

asal(5);

 

"asal deÄŸil"

(3)

 


Download corrected.mw

you can begin with maple help as Markiyan Hirnyk said or maybe you want to start with an ebook or tutorial like this link : 
www.math.mtu.edu/~msgocken/pdebook2/mapletut2.pdf
as soon as you start , you can ask your questions on this site and there are a lot of professions for answering your questions.
good luck


restart:

m:=10;# number of n[i]'s ,to put in the function #

10

(1)

P:=proc (M::list) local i ,S,f ;
for i to m do
f:=x->1+2*(1+x)^n[i];
S[i]:=plot(f(x),x=0..5);od:
return plots:-display(seq(S[i],i=1..m)):
end proc;

proc (M::list) local i, S, f; for i to m do f := proc (x) options operator, arrow; 1+2*(1+x)^n[i] end proc; S[i] := plot(f(x), x = 0 .. 5) end do; return plots:-display(seq(S[i], i = 1 .. m)) end proc

(2)

#example#

for i to m do
n[i]:=i:od:

M:=[seq](n[i],i=1..m);

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

(3)

P(M);

 

 


Download plot.mw

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