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Mariusz Iwaniuk

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These are answers submitted by Mariusz Iwaniuk

Gamma......

Mariusz Iwaniuk 1401
September 23 2021
1 0

This integral, in most cases, cannot be expressed in terms of elementary functions,but we can expressed in terms of GAMMA function.

data_v1.mw
 

 

 

 

 

"D1(s,t) :=P- (alpha1-beta*S) + alpha2 + beta2 *q(t)^();"

proc (s, t) options operator, arrow, function_assign; P+beta*S-alpha1+alpha2+beta2*q(t) end proc

 (1)

"(->)"

dem

 (2)

NULL

ode1 := diff(q(t), t)+theta*q(t)/(1+N-t) = -D1(s, t)

diff(q(t), t)+theta*q(t)/(1+N-t) = -P-beta*S+alpha1-alpha2-beta2*q(t)

 (3)
> 

fn1 := q(t)

q(t)

 (4)
> 

ic1 := q(T) = 0

q(T) = 0

 (5)
> 

sol1 := simplify(dsolve({ic1, ode1}, fn1))

q(t) = (-S*beta-P+alpha1-alpha2)*(Int(exp(beta2*_z1)*(1+N-_z1)^(-theta), _z1 = T .. t))*exp(-beta2*t)*(1+N-t)^theta

 (6)

Int(exp(beta2*_z1)*(1+N-_z1)^(-theta), _z1 = T .. t) = -exp(N*beta2+beta2)*((1+N-t)^(-theta)*(beta2*(1+N-t))^theta*(theta*GAMMA(-theta)+GAMMA(1-theta, beta2*(1+N-t)))+(1+N-T)^(-theta)*(beta2*(1+N-T))^theta*(GAMMA(1-theta)-GAMMA(1-theta, beta2*(1+N-T))))/beta2

Int(exp(beta2*_z1)*(1+N-_z1)^(-theta), _z1 = T .. t) = -exp(N*beta2+beta2)*((1+N-t)^(-theta)*(beta2*(1+N-t))^theta*(theta*GAMMA(-theta)+GAMMA(1-theta, beta2*(1+N-t)))+(1+N-T)^(-theta)*(beta2*(1+N-T))^theta*(GAMMA(1-theta)-GAMMA(1-theta, beta2*(1+N-T))))/beta2

 (7)

lprint(Int(exp(beta2*_z1)*(1+N-_z1)^(-theta), _z1 = T .. t) = -exp(N*beta2+beta2)*((1+N-t)^(-theta)*(beta2*(1+N-t))^theta*(theta*GAMMA(-theta)+GAMMA(1-theta, beta2*(1+N-t)))+(1+N-T)^(-theta)*(beta2*(1+N-T))^theta*(GAMMA(1-theta)-GAMMA(1-theta, beta2*(1+N-T))))/beta2)

Int(exp(beta2*_z1)*(1+N-_z1)^(-theta),_z1 = T .. t) = -1/beta2*exp(N*beta2+
beta2)*((1+N-t)^(-theta)*(beta2*(1+N-t))^theta*(theta*GAMMA(-theta)+GAMMA(1-
theta,beta2*(1+N-t)))+(1+N-T)^(-theta)*(beta2*(1+N-T))^theta*(GAMMA(1-theta)-
GAMMA(1-theta,beta2*(1+N-T))))

  

NULL


 

Download data_v1.mw

 

InvLap....

Mariusz Iwaniuk 1401
September 20 2021
1 4 

 

 

> 

restart
> 

with(inttrans)
> 

expr := exp(a-sqrt(a^2+b*s))/s

exp(a-(a^2+b*s)^(1/2))/s

 (1)
> 

`assuming`([invlaplace(exp(a-(a^2+b*s)^(1/2))/s, s, t)], [b > 0])

(1/2)*(b/Pi)^(1/2)*(int(exp(-a^2*_U1/b+a-(1/4)*b/_U1)/_U1^(3/2), _U1 = 0 .. t))

 (2)
> 

NULL
> 

(1/2)*(b/Pi)^(1/2)*(int(exp(-a^2*_U1/b+a-(1/4)*b/_U1)/_U1^(3/2), _U1 = 0 .. t)) = (1/2)*erfc((-2*a*t+b)/(2*sqrt(b*t)))+(1/2)*exp(2*a)*erfc((2*a*t+b)/(2*sqrt(b*t)))

(1/2)*(b/Pi)^(1/2)*(int(exp(-a^2*_U1/b+a-(1/4)*b/_U1)/_U1^(3/2), _U1 = 0 .. t)) = (1/2)*erfc((1/2)*(-2*a*t+b)/(b*t)^(1/2))+(1/2)*exp(2*a)*erfc((1/2)*(2*a*t+b)/(b*t)^(1/2))

 (3)
> 

NULL
> 

a := 2; b := 5

2

  

5

 (4)
> 

evalf[20](invlaplace(exp(a-sqrt(a^2+b*s))/s, s, 1.0))

.49675487848107438114

 (5)
> 

evalf[20](eval((1/2)*erfc((-2*a*t+b)/(2*sqrt(b*t)))+(1/2)*exp(2*a)*erfc((2*a*t+b)/(2*sqrt(b*t))), t = 1.0))

.49675487848107438112

 (6)
> 

NULL


 

Download invLaplace.mw

 

Try:...

Mariusz Iwaniuk 1401
June 26 2021
1 2


Do you have any reason to think there is a closed form?

Most integrals don't have one.

Maybe the best you can do is numerical methods.

See atthached file.

ode.mw

 

See teory of First Order Differential Equations. Only one initial value problem can be not two.

Try:...

Mariusz Iwaniuk 1401
June 26 2021
0 6

See attached file:

integral.mw

 

Maybe...My answers helps...

Mariusz Iwaniuk 1401
May 27 2021
4 1

See attached file:

PDE_by_Elziki_Transform.mw

Homework.....

Mariusz Iwaniuk 1401
May 23 2021
0 0

For first question: 

 

f := x -> 36*x^6 + 2665*x^4 + 240*x - 675 + 4534*x^2 - 5836*x^3 - 516*x^5;

minimize(f(x), x = 0 .. 4, location);
#-675, {[{x = 0}, -675]}

evalf(maximize(f(x), x = 0 .. 4, location));
#703.9550742, {[{x = 3.800387934}, 703.9550742]}

 

Try...

Mariusz Iwaniuk 1401
May 20 2021
2 7

Try:

ode := diff(U(z), z $ 4) + c^2*diff(U(z), z $ 2) + k*c*diff(U(z), z $ 2) - (3*U(z)^2 + a)*diff(U(z), z $ 2) = 0;
Order := 5;dsolve(ode, U(z), type = 'series');


#U(z) = U(0) + D(U)(0)*z + 1/2*(D@@2)(U)(0)*z^2 + 1/6*(D@@3)(U)(0)*z^3 + (U(0)^2*(D@@2)(U)(0)/8 - c^2*(D@@2)(U)(0)/24 - k*c*(D@@2)(U)(0)/24 + (D@@2)(U)(0)*a/24)*z^4 + O(z^5)

With initial conditions 

Order := 5;dsolve([ode, U(A) = A1, D(U)(A) = B1, (D@@2)(U)(A) = C1], U(z), type = 'series');

#U(z) = A1 + B1*(z - A) + 1/2*C1*(z - A)^2 + 1/6*(D@@3)(U)(A)*(z - A)^3 + (1/8*A1^2*C1 - 1/24*c^2*C1 - 1/24*k*c*C1 + 1/24*C1*a)*(z - A)^4 + O((z - A)^5)

 

Workaround....

Mariusz Iwaniuk 1401
April 17 2021
0 0

As a workround using fourier transform:

(inttrans:-invfourier(int((inttrans:-fourier(sin(p*r), p, s) assuming (0 <= r))*sin(q*r)/(p*q), r = 0 .. infinity), s, p) assuming (q < p));

#-Pi*Dirac(p + q)/(2*p*q)

Try: simplify(pdetest(sol, sys)); giv...

Mariusz Iwaniuk 1401
April 05 2021
2 4

Try:

simplify(pdetest(sol, sys));

gives:

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

General formula for n...

Mariusz Iwaniuk 1401
March 14 2021
1 1

Maybe this helps, see attached file:

General_formula_.mw

 

Or try:...

Mariusz Iwaniuk 1401
March 13 2021
1 0 
simplify(diff(int(JacobiSN(x, k)^2, x), x));

#JacobiSN(x, k)^2

Answer...

Mariusz Iwaniuk 1401
March 11 2021
2 2

See attached file:

EQ_v3.mw

Only......

Mariusz Iwaniuk 1401
February 18 2021
1 0

Only solution,not  phase portrait.See atached file.

Solution.mw

A way:...

Mariusz Iwaniuk 1401
February 12 2021
2 1

One way is:

[seq(rhs(op(1, rootsq0[[n]])), n = 1 .. numelems([rootsq0]))];

 

Maybe...

Mariusz Iwaniuk 1401
February 10 2021
1 0

Maybe like this:



Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/integrals.mw .
 

Download integrals.mw

 

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