Mariusz Iwaniuk

1491 Reputation

14 Badges

8 years, 260 days

Social Networks and Content at Maplesoft.com

MaplePrimes Activity


These are answers submitted by Mariusz Iwaniuk


 

restart

ODE := ((D@@2)(x))(tau)+(1+a^2*x(tau)^2)*x(tau) = 0

((D@@2)(x))(tau)+(1+a^2*x(tau)^2)*x(tau) = 0

(1)

sol := dsolve(ODE, Lie)

Intat(-2/(-2*_a^4*a^2-4*_a^2+4*_C1)^(1/2), _a = x(tau))-tau-_C2 = 0, Intat(2/(-2*_a^4*a^2-4*_a^2+4*_C1)^(1/2), _a = x(tau))-tau-_C2 = 0

(2)

ans := sol[2]

Intat(2/(-2*_a^4*a^2-4*_a^2+4*_C1)^(1/2), _a = x(tau))-tau-_C2 = 0

(3)

eq := diff(ans, tau)

2*(diff(x(tau), tau))/(-2*a^2*x(tau)^4-4*x(tau)^2+4*_C1)^(1/2)-1 = 0

(4)

eq1 := solve(eq, diff(x(tau), tau))

(1/2)*(-2*a^2*x(tau)^4-4*x(tau)^2+4*_C1)^(1/2)

(5)

eq2 := eval(eq1, x(tau) = y)

(1/2)*(-2*a^2*y^4-4*y^2+4*_C1)^(1/2)

(6)

eq3 := eval(eq2, y = A)

(1/2)*(-2*A^4*a^2-4*A^2+4*_C1)^(1/2)

(7)

_C1 := solve(eq3, _C1)

(1/2)*A^4*a^2+A^2

(8)

eq4 := eq2

(1/2)*(2*A^4*a^2-2*a^2*y^4+4*A^2-4*y^2)^(1/2)

(9)

T := `assuming`([4*(int(1/eq4, y = 0 .. A))/omega], [a > 0, A > 0, x > 0, y > 0, y < A, x < A])

4*2^(1/2)*EllipticK(A*a/(2*A^2*a^2+2)^(1/2))/(omega*(2*A^2*a^2+2)^(1/2))

(10)

k := op([5, 1], T)

A*a/(2*A^2*a^2+2)^(1/2)

(11)

tau := `assuming`([omega*t = int(1/eq4, y = x .. A)], [a > 0, A > 0, x > 0, y > 0, y < A, x < A])

omega*t = 2^(1/2)*EllipticF((A^2-x^2)^(1/2)/A, A*a/(2*A^2*a^2+2)^(1/2))/(2*A^2*a^2+2)^(1/2)

(12)

eq5 := solve(tau, x)

(1-JacobiSN((1/2)*omega*t*(2*A^2*a^2+2)^(1/2)*2^(1/2), A*a/(2*A^2*a^2+2)^(1/2))^2)^(1/2)*A, -(1-JacobiSN((1/2)*omega*t*(2*A^2*a^2+2)^(1/2)*2^(1/2), A*a/(2*A^2*a^2+2)^(1/2))^2)^(1/2)*A

(13)

X := eq5[1]

(1-JacobiSN((1/2)*omega*t*(2*A^2*a^2+2)^(1/2)*2^(1/2), A*a/(2*A^2*a^2+2)^(1/2))^2)^(1/2)*A

(14)

u := (1/2)*omega*t*sqrt(2*A^2*a^2+2)*sqrt(2)NULL

(1/2)*omega*t*(2*A^2*a^2+2)^(1/2)*2^(1/2)

(15)

XX := eval(X, JacobiSN(u, k)^2 = 1-JacobiCN(u, k)^2)

(JacobiCN((1/2)*omega*t*(2*A^2*a^2+2)^(1/2)*2^(1/2), A*a/(2*A^2*a^2+2)^(1/2))^2)^(1/2)*A

(16)

XXX := simplify(XX, sqrt, symbolic)

JacobiCN((1/2)*omega*t*(2*A^2*a^2+2)^(1/2)*2^(1/2), A*a/(2*A^2*a^2+2)^(1/2))*A

(17)

NULL


 

Download The_Hard_Spring_by_Maple_2018.mw

sol := `assuming`([dsolve(Eq2 = 0)], [phi(xi) > 0]);
with(DEtools);
odeadvisor(op([2, 2], sol)[1, 1]);

dsolve can't solve symbolical Abel's equation. Try numerically.

 I invert that laplace transform numerically,but answer is different than :(section 4.1 page 13629, data used for FIG. 3 )


 

restart

Digits := 100

100

(1)

CK := .3; Z := 10; L := 1; alpha := .95; ZetaR := 10

.3

 

10

 

1

 

.95

 

10

(2)

r1 := proc (s) options operator, arrow; (1/2)*(L*s^2+sqrt(L^2*s^4+4*s^(-alpha+3)*(CK+(CK*Z+1)*s+Z*s^2)*(1+s+(1/2)*s^2)))/(s^(-alpha+1)*(CK+(CK*Z+1)*s+Z*s^2)) end proc

proc (s) options operator, arrow; (1/2)*(L*s^2+sqrt(L^2*s^4+4*s^(-alpha+3)*(CK+(CK*Z+1)*s+Z*s^2)*(1+s+(1/2)*s^2)))/(s^(-alpha+1)*(CK+(CK*Z+1)*s+Z*s^2)) end proc

(3)

r2 := proc (s) options operator, arrow; (1/2)*(L*s^2-sqrt(L^2*s^4+4*s^(-alpha+3)*(CK+(CK*Z+1)*s+Z*s^2)*(1+s+(1/2)*s^2)))/(s^(-alpha+1)*(CK+(CK*Z+1)*s+Z*s^2)) end proc

proc (s) options operator, arrow; (1/2)*(L*s^2-sqrt(L^2*s^4+4*s^(-alpha+3)*(CK+(CK*Z+1)*s+Z*s^2)*(1+s+(1/2)*s^2)))/(s^(-alpha+1)*(CK+(CK*Z+1)*s+Z*s^2)) end proc

(4)

c := proc (n, i) options operator, arrow; (-1)^(i+(1/2)*n)*(sum(evalf(k^((1/2)*n)*factorial(2*k)/(factorial((1/2)*n-k)*factorial(k)*factorial(k-1)*factorial(i-k)*factorial(2*k-i))), k = floor((1/2)*i+1/2) .. min(i, (1/2)*n))) end proc

INVLAP := proc (f, s, t, n) if type(n, even) then return evalf(ln(2)*(sum(c(n, i)*(eval(f, s = i*ln(2)/t)), i = 1 .. n))/t) else return 0 end if end proc

R1 := proc (beta) options operator, arrow; INVLAP(r1(s), s, beta, 20) end proc

proc (beta) options operator, arrow; INVLAP(r1(s), s, beta, 20) end proc

(5)

R1(1)

-0.1541362927375609459212764576340201403535836302397525888029243229676698002949846272721795130378111302e-1

(6)

R2 := proc (beta) options operator, arrow; INVLAP(r2(s), s, beta, 20) end proc

proc (beta) options operator, arrow; INVLAP(r2(s), s, beta, 20) end proc

(7)

R2(1)

-0.9553862141072868790424645051586718244791894211562423732209924901761210733190720403974860481626895380e-2

(8)

theta := proc (Zeta, beta) options operator, arrow; exp(R2(beta)*ZetaR)*exp(R1(beta)*Zeta)/(exp(R2(beta)*ZetaR)-exp(R1(beta)*ZetaR))-exp(R1(beta)*ZetaR)*exp(R2(beta)*Zeta)/(exp(R2(beta)*ZetaR)-exp(R1(beta)*ZetaR)) end proc

proc (Zeta, beta) options operator, arrow; exp(R2(beta)*ZetaR)*exp(R1(beta)*Zeta)/(exp(R2(beta)*ZetaR)-exp(R1(beta)*ZetaR))-exp(R1(beta)*ZetaR)*exp(R2(beta)*Zeta)/(exp(R2(beta)*ZetaR)-exp(R1(beta)*ZetaR)) end proc

(9)

theta(2, 1)

.78023310949683032910207875270495938180505912081434286871303196902181168839514493744039058044012316

(10)

plot(theta(Zeta, 1), Zeta = 0 .. 10)

 

NULL

``

``

``


 

Download FracPDE_vers_2.mw

 

You can restrict T  using assumptions.

plot([Re(sqrt(sin(x))), Im(sqrt(sin(x)))], x = 0 .. 4*Pi, legend = [typeset("Curve: ", Re(sqrt(sin(x)))), typeset("Curve: ", Im(sqrt(sin(x))))]);

for: 0 < T < Pi

simplify(int(sqrt(sin(x)), x = 0 .. T)) assuming 0 < T < Pi
#-sqrt(2)*(EllipticF(sqrt(2)*sqrt(sin(T))/sqrt(sin(T)+1), (1/2)*sqrt(2))-EllipticPi(sqrt(2)*sqrt(sin(T))/sqrt(sin(T)+1), 1/2, (1/2)*sqrt(2)))

and for: Pi < T < 2*Pi

simplify(int(sqrt(sin(x)), x = 0 .. T)) assuming Pi < T < 2*Pi
#sqrt(2)*(I*EllipticF(2*sqrt(sin((1/2)*T))*sqrt(-cos((1/2)*T))/(cos((1/2)*T)-sin((1/2)*T)), (1/2)*sqrt(2))-I*EllipticPi(2*sqrt(sin((1/2)*T))*sqrt(-cos((1/2)*T))/(cos((1/2)*T)-sin((1/2)*T)), 1/2, (1/2)*sqrt(2))+4*EllipticE((1/2)*sqrt(2))-2*EllipticK((1/2)*sqrt(2)))

EDITED:

Generally my method works if Maple does not have errors(Bugs).

Workaround:

f := proc (T) options operator, arrow; int(sqrt(sin(x)), x = 0 .. T) end proc;
[f((1/4)*Pi), f((1/2)*Pi), f(3*Pi*(1/4)), f(Pi)];

#[-sqrt(2)*EllipticF(sqrt(2)*sqrt(sqrt(2)/(2+sqrt(2))), (1/2)*sqrt(2))+sqrt(2)*EllipticPi(sqrt(2)*sqrt(sqrt(2)/(2+sqrt(2))), 1/2, (1/2)*sqrt(2)), -sqrt(2)*EllipticK((1/2)*sqrt(2))+2*sqrt(2)*EllipticE((1/2)*sqrt(2)), -sqrt(2)*EllipticK((1/2)*sqrt(2))+2*sqrt(2)*EllipticE((1/2)*sqrt(2))-sqrt(2)*EllipticF(cos(3*Pi*(1/8))*sqrt(2), (1/2)*sqrt(2))+2*sqrt(2)*EllipticE(cos(3*Pi*(1/8))*sqrt(2), (1/2)*sqrt(2)), -2*sqrt(2)*EllipticK((1/2)*sqrt(2))+4*sqrt(2)*EllipticE((1/2)*sqrt(2))]


Regards,MI

Well Maple is very weak for solving symbolic sum and can't solve your example.

`assuming`([sum((d*k+a)^r, k = 1 .. n)], [n::posint, r::posint, r > 0, a > 0, d > 0]);

# returns unevaluated

Mathematica 11.2 solution:

Translated code form MMA to Maple:

Sum((d*k+a)^r, k = 1 .. n) = d^r*(Zeta(0, -r, (a+d)/d)-Zeta(0, -r, (d*n+a+d)/d));

Regards,MI

 

sol:=identify(evalf(2*ln(3)-3*ln(2)));

# arcsinh(17/144)

simplify(convert(sol, arccoth), symbolic);

# 2*arctanh(1/17)

Only for inverse hyperbolic function.

 

with(gfun);
l := [8, 32, 128, 512, 2048];
rec := listtorec(l, u(n), [ogf]);

# rec := [{u(1+n)-4*u(n), u(0) = 8}, ogf]

You must only change to:  u(1) = 8

sol:=rsolve({u(1+n)-4*u(n), u(1) = 8}, u(n));

#Answer is: 2*4^n
#or:
sol:=rsolve({eval(op([1, 2], rec), 0 = 1), op([1, 1], rec) = 0}, u(n))

[seq(sol, n = 1 .. 6)]; #Check:

#[8, 32, 128, 512, 2048, 8192]

I don't know how to change: 2*4^n = 2*2^(2 n) = 2^(2n+1) in Maple!

simplify(2*4^n, power);# Dosen't work!

 

For first example:

restart: 
Digits := 30:
convert((k-2)*(k^2+5)*(k^3-k^2+7*k+8)/(6*k*(k^2-3*k+8)), parfrac, k);

and factor expression k^2-3k+8  by:

(1/6)*k^3+k-1/2-5/(3*k)+convert((-3*k-6)/identify(factor(denom(op(4, sol)), complex)), parfrac, k);

 and last 2 expression:

identify(factor(k^3+3*k^2+11*k-3, complex));
identify(factor(k^2+2*k+9, complex))

 

f := proc (x, z) options operator, arrow; ln(x^z) end proc;
simplify(f(x, 2), symbolic);
collect(expand(simplify(f(x, 1/2+I*y), symbolic)), ln(x));

Note: When the symbolic option is specified, any branch of a multi-valued function can be chosen during the simplification process. The result of such an operation is in general not valid over the whole complex plane and can lead to incorrect results if you assume the expressions represent analytical functions.
 

eq16 := r(t) = d[vol]*V/(KUS*V^2+L*tau);
ex := InertForm:-MakeInert(convert(rhs(eq16), fullparfrac, KUS)):
eq17 := r(t) = op(1, ex)/expand(simplify(op(2, ex)));

 

 

restart:
with(Physics[Vectors]):
Setup(mathematicalnotation = true):
eq := r(t) = 2*t^2*_i+16*_j+(10*t-12)*_k;
v := diff(rhs(eq), t);
V := eval(v, t = 10);
simplify(Norm(V));

 

restart;
with(Physics[Vectors]);
Setup(mathematicalnotation = true);
eq := r(t) = 3*cos(5*t)*_i+sin(5*t)*_j+3*sin(5*t)*_k;
Norm(rhs(eq));#Calculate Norm
plots:-spacecurve([Component(rhs(eq), 1), Component(rhs(eq), 2), Component(rhs(eq), 3)], t = 0 .. 4*Pi, color = pink);

Have fun!

 

restart;
q1 := 9045.084972*(diff(z[1](t), t$2))+863728.7570*z[1](t) = -1963.525491562420*sin(20*t);
q2 := 3454.915028*(diff(z[2](t), t$2))+2.261271243*10^6*z[2](t) = -286.4745084375789*sin(20*t); icy := seq([z[i](0) = 0, (D(z[i]))(0) = 0], i = 1 .. 2, 1);

so := dsolve({q1, q2, seq(icy[i][], i = 1 .. 2), A(t) = diff(z[1](t), t$2), B(t) = diff(z[2](t), t$2)}, numeric); so(1);

plots:-odeplot(so, [[t, A(t)], [t, B(t)]], t = 0 .. 1, color = [red, blue], legend = ["diff(z[1](t), t$2)", "diff(z[2](t), t$2)"], legendstyle = [font = [times, bold, 20]]);

Have fun!

w := proc (x, y) options operator, arrow; piecewise(y <= .5, -2*tanh(y-.25), .5 < y, 2*tanh(.75-y)) end proc;
Matrix([seq([seq(w(x, y), x = 0 .. 10)], y = 0 .. 10)]);

#or;

Matrix([seq([seq(w(x, y), y = 0 .. 10)], x = 0 .. 10)]);

 

 

You don't gives values of constans,so I assuming.

restart;
Digits := 20;
Theta := (1/3)*Pi; Upsilon := 1/10;#assume!
eq := ((D@@2)(u))(r) = (-(D(u))(r)^2*u(r)+((Upsilon-1)*(1/2))*(1-u(r)^2-(D(u))(r)^2)*((D(u))(r)*cot(Theta)+2*u(r)))/((D(u))(r)^2-((Upsilon-1)*(1/2))*(1-u(r)^2-(D(u))(r)^2));
sol := dsolve({eq, u(13.75) = .7787, (D(u))(13.75) = .344037}, numeric, abserr = 1.*10^(-16));
plots:-odeplot(sol, [[r, u(r)], [r, (D(u))(r)]], r = 0 .. 13.75, legend = [typeset("Curve: ", u(r)), typeset("Curve: ", (D(u))(r))]);

 

 

First 13 14 15 16 17 18 19 Page 15 of 19