Mariusz Iwaniuk

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9 years, 129 days

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These are answers submitted by Mariusz Iwaniuk

Y := proc (delta, b, n) 
if n = 0 then 1/(b^2+1) 
elif n = 1 then arctan(1/b) 
elif n = 2 then 1+(1/2)*ln(b^2+1)-b*arctan(1/b) 
elif n = 3 then delta-(3/2)*b-(1/2)*b*ln(b^2+1)+(1/2)*arctan(1/b)*(b^2-1) 
elif n = 4 then (1/2)*delta^2-b*delta+(11/12)*b^2-11/36+(1/12)*ln(b^2+1)*(3*b^2-1)+(1/6)*b*arctan(1/b)*(3-b^2) 
elif n = 5 then (1/3)*delta^3-(1/2)*delta^2*b+(1/6)*delta*(3*b^2-1)+(25/72)*b-(25/72)*b^3+(1/12)*b*log(b^2+1)*(1-b^2)+(1/24)*arctan(1/b)*(1-6*b^2+b^4) 
elif n = 6 then (1/4)*delta^4-(1/3)*delta^2*b+(1/12)*delta^2*(3*b^2-1)+(1/6)*b*delta*(1-b^2)+(137/1440)*b^4-(137/720)*b^2+137/7200+(1/240)*log(b^2+1)*(5*b^4-10*b^2+1)+(1/120)*b*arctan(1/b)*(-5+10*b^2-b^4) 
elif n = 7 then (1/5)*delta^5-(1/4)*delta^4*b+(1/18)*delta^3*(3*b^2-1)+(1/12)*delta^2*b*(1-b^2)+(1/120)*delta*(5*b^4-10*b^2+1)-(49/2400)*b+(49/720)*b^3-(49/2400)*b^5+(1/720)*b*log(b^2+1)*(-3*b^4+10*b^2-3)+(1/720)*arctan(1/b)*(-1+15*b^2-15*b^4+b^6) 
end if 
end proc:

plot(Y(5, b, 7), b = 0 .. 10); # Example of use

 

u1 := proc (x, y) options operator, arrow; 1-exp(a*x)*cos(2*Pi*y) end proc;

a := -0.39323780;

evalf(int(u1(x, y)^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

#5.493248990

#OR:

u := unapply(1-exp(a*x)*cos(2*Pi*y), [x, y]);

a := -0.39323780;

evalf(int(u(x, y)^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

#OR:

u := 1-exp(a*x)*cos(2*Pi*y);

a := -0.39323780;

evalf(int(u^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

Use  ExcelTools for import file.

 


 

Download testxlsx.mw

 

I don't have Maple V (This is an obsolete version,you need upgrade).

integral := Int(2*(sin(theta)/cos(theta))^(2*p-1), theta = 0 .. (1/2)*Pi);

with(IntegrationTools):

integral2 := Change(convert(integral, tan), tan(theta) = sqrt(t));

`assuming`([value(integral2)], [0 < p and p < 1]);

#Answer is: Pi*csc(Pi*p)

integral.mw

Executed in Maple 2018

Maple can't simplify better, probably V is not a solution of pde[1].

See attached file:

solution_ver_3.mw

BVP := [4*(diff(u(x, t), t))-9*(diff(u(x, t), x, x))-5*u(x, t) = 0, u(0, t) = 0, u(6, t) = 0, u(x, 0) = sin((1/6)*Pi*x)^2];

sol := pdsolve(BVP);

plot3d(eval(rhs(sol), infinity = 10), x = 0 .. 6, t = 0 .. 4);

OR:

plot3d(subs(infinity = 10, rhs(sol)), x = 0 .. 6, t = 0 .. 4);

OR:

plot3d(subs(infinity = 10, op(2, sol)), x = 0 .. 6, t = 0 .. 4)

 I am no expert in special functions,but adding comand "_EnvLegendreCut"

_EnvLegendreCut := 1 .. infinity;

plot(LegendreQ((1/2)*sqrt(5)-1/2, x), x = -1 .. 1);

for more info execute command:

?LegendreQ;

restart;

with(ListTools):

l := seq(n, n = -10 .. 20);

[l];

l1 := Flatten([Transpose([[seq(n, n = -10 .. 20)], [seq(n, n = 1 .. 31)]])]);

l2 := Transpose([[seq(n, n = -10 .. 20)], [seq(n, n = 1 .. 31)]]);

l3 := select(type, [seq(n, n = 115 .. 231)], 'odd');

[l3];

In yours first example Maple gives incorect answer.It's a Bug.!!!

 

evalf[10](sum(2^n*floor(2^n), n = 1 .. infinity));

#-1.333333333

evalf[10](Sum(2^n*floor(2^n), n = 1 .. infinity));# Big 'S' in Sum

#-1.333333333

`assuming`([sum(2^n*floor(2^n), n = 1 .. m)], [m > 0]);

#(1/3)*2^(m+1)*floor(2^(m+1))-4/3

limit(%, m = infinity)

#infinity

Second one give a correct answer:

evalf[10](sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. infinity));

#Float(infinity)

evalf[10](Sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. infinity));# Big 'S' in Sum

#If Maple dosen't know the answer then: Returns unevaluated,not infinity in this case.

`assuming`([sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. m)], [m > 0]);

#Returns unevaluated

 

Executed in Maple 2018.

Using CauchyPrincipalValue =true:

`assuming`([int(exp(I*x*t)/((x-a)*(x+a)), x = -infinity .. infinity, CauchyPrincipalValue = true)], [a > 0, t > 0])

#(1/2*I)*Pi*(exp(I*a*t)-exp(-I*a*t))/a

int.mw

Exectuted in Maple 2018.

remove[flatten](x-> x = 0, [seq(sin((1/4)*k*Pi), k = 1 .. 8)]);

The function RootOf is a placeholder for representing all the roots of an equation in one variable.  In particular, it is the standard representation for Maple algebraic numbers, algebraic functions.

For more info execute this code in Maple:

?RootOf
sol := solve({x^2+y^2 = 3, x^2+2*y^2 = 3}, {x, y});
allvalues(sol);
evalf(%);

#{x = sqrt(3), y = 0}, {x = -sqrt(3), y = 0}
#{x = 1.732050808, y = 0.}, {x = -1.732050808, y = 0.}

 


 

I don't know way it's happens,but if we convert to rational(exact) numbers then works:

 

   

eq := [190.0315457*d^6-7.601261828*d^4+344.0*c^2*d^2-2.677685950*c^4, 0.6830134554e-2*c^4+2084.317025*d^8-166.7453620*d^6+3.334907240*d^4-c^2*d^2];
eliminate(convert(eq, rational, exact), c);

#[{c = -(1/14265)*sqrt(13071106675-17435*sqrt(9667390500*d^2+561670891405))*d}, {121*d^4*(582104796124362625620*d^4-46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)-1272306087125807465)}], [{c = (1/14265)*sqrt(13071106675-17435*sqrt(9667390500*d^2+561670891405))*d}, {121*d^4*(582104796124362625620*d^4-46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)-1272306087125807465)}], [{c = -(1/14265)*sqrt(13071106675+17435*sqrt(9667390500*d^2+561670891405))*d}, {-121*d^4*(-582104796124362625620*d^4+46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)+1272306087125807465)}], [{c = (1/14265)*sqrt(13071106675+17435*sqrt(9667390500*d^2+561670891405))*d}, {-121*d^4*(-582104796124362625620*d^4+46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)+1272306087125807465)}]

 


 

NULL

latex(a[p, q] = 'sum(cos(2*Pi*(p+1)*(n-1)/q), n = 1 .. q)')

a_{{p,q}}=\sum _{n=1}^{q}\cos \left( 2\,{\frac {\pi\, \left( p+1
 \right)  \left( n-1 \right) }{q}} \right)

 

NULL

latex(a[p, q] = Sum(cos(2*Pi*(p+1)*(n-1)/q), n = 1 .. q))

a_{{p,q}}=\sum _{n=1}^{q}\cos \left( 2\,{\frac {\pi\, \left( p+1
 \right)  \left( n-1 \right) }{q}} \right)

 

NULL

latex(a[p, q] = Sum(cos(2*Pi*(p+1)*(n-1)/q), n = 1 .. q))

a_{{p,q}}=\sum _{n=1}^{q}\cos \left( 2\,{\frac {\pi\, \left( p+1
 \right)  \left( n-1 \right) }{q}} \right)

 


 

Download Latex_Sum.mw

pdsolve([diff(f2(lam, mu, l, m), m)-(diff(f1(lam, mu, l, m), l))-(diff(f4(lam, mu, l, m), mu))+diff(f3(lam, mu, l, m), lam) = 0, diff(f1(lam, mu, l, m), m)+diff(f2(lam, mu, l, m), l)-(diff(f3(lam, mu, l, m), mu))-(diff(f4(lam, mu, l, m), lam)) = 0], HINT = `+`);

#A simple solutions.

pdsolve([diff(f2(lam, mu, l, m), m)-(diff(f1(lam, mu, l, m), l))-(diff(f4(lam, mu, l, m), mu))+diff(f3(lam, mu, l, m), lam) = 0, diff(f1(lam, mu, l, m), m)+diff(f2(lam, mu, l, m), l)-(diff(f3(lam, mu, l, m), mu))-(diff(f4(lam, mu, l, m), lam)) = 0], HINT = TWS(sin));

#Not simple solution.For more info. execute this code below?

?pdsolve

 

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