Mariusz Iwaniuk

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These are answers submitted by Mariusz Iwaniuk

restart;
ODE := diff(W(x), x $ 2) + (A*ohm^2*ro - kd)*W(x)/T = p;
sol1 := dsolve(ODE, W(x));
assign(sol1);

subs(_C1 = 0, W(x));

#or:

eval(W(x), _C1 = 0);


#sin(sqrt(A*ohm^2*ro - kd)*x/sqrt(T))*_C2 + p*T/(A*ohm^2*ro - kd)

 

 

int(exp(-2*r)*cos(theta)^3*r^2*sin(theta), phi = 0 .. 2*Pi, r = 0 .. infinity, theta = 0 .. Pi);
#O

 

 

We can find solution with series representation:

restart;
odeSystem := {diff(y1(x), x) = -x*y2(x) - (1 + x)*y3(x), diff(y2(x), x) = -x*y1(x) - (1 + x)*y4(x), diff(y3(x), x) = -x*y1(x) - (1 + x)*y4(x) - 5*x*cos(1/2*x^2), diff(y4(x), x) = -x*y2(x) - (1 + x)*y3(x) + 5*x*sin(1/2*x^2), y1(0) = 5, y2(0) = 1, y3(0) = -1, y4(0) = 0};
Order := 10;
systemSol := dsolve(odeSystem, [y1(x), y2(x), y3(x), y4(x)], series);
F := convert(systemSol, polynom);
plot([rhs(F[1]), rhs(F[2]), rhs(F[3]), rhs(F[4])], x = 0 .. 1, color = [red, blue, green, gold], legend = [y1(x), y2(x), y3(x), y4(x)]);

A workaround with Mathematica:

f[k_] := Sum[(-1)^i (k - i + 1)^(2 k + 4)/((i!)*((2 k - i + 2)!)), {i,0, k}];
L = Table[f[k], {k, 0, 1000}];
FindSequenceFunction[L, k]

(*1/12 (k + 3 k^2 + 2 k^3)*)

With Maple:

restart;
with(gfun):
f := k -> sum((-1)^i*(k - i + 1)^(2*k + 4)/(i!*(2*k - i + 2)!), i = 0 .. k);
l := [seq(f(k), k = 0 .. 1000)];
rec := listtorec(l, u(k), [ogf]);
rsolve(op(1, rec), u(k));

#1/6*k^3 + 3/4*k^2 + 13/12*k + 1/2

 

See attached file.

integral.mw

Sum((-1)^k*sin(k*x)/k, k = -infinity .. infinity) = evalc(sum((-1)^k*sin(k*x)/k, k = -infinity .. infinity, formal))

 

f := x -> convert(sin(theta), x);
map(f, [cos, tan, cot, csc, sec]);

#[-cos(theta + Pi/2), 2*tan(theta/2)/(1 + tan(theta/2)^2), 2*cot(theta/2)/(cot(theta/2)^2 + 1), 1/csc(theta), -1/sec(theta + Pi/2)]

sol := dsolve({diff(f(x), x) = f(1/x), f(0) = 0}, numeric, delaymax = 1.0, delaypts = 2000);
plots:-odeplot(sol, [x, f(x)], x = -1 .. 1);

Maybe you can try with another initial condition f(0)=1 ?

evalf[30](subs(z = 2 + 3*I, Zeta(-z) + 2*z!*sin(Pi*z/2)*Zeta(z + 1)/(2*Pi)^(z + 1)));

rtable([seq(evalf[2^m](subs(z = 2 + 3*I, Zeta(-z) + 2*z!*sin(Pi*z/2)*Zeta(z + 1)/(2*Pi)^(z + 1))), m = 1 .. 10)], subtype = Vector[column]);#Depending of the precise

I have another sum with Laguerre polynomials to  give exp(z).

Sum((m + 1)!*z^k*LaguerreL(k, k + m, z)/(k + m + 1)!, k = 0 .. infinity) = exp(z)

We can check:

restart;
f := (z, m) -> sum((m + 1)!*z^k*LaguerreL(k, k + m, z)/(k + m + 1)!, k = 0 .. infinity)
interface(rtablesize = 100);
rtable([seq([m, evalf(f(1, m))], m = 0 .. 20)], subtype = Vector[column]);

And for your example:

Digits := 30;
f := (x, m) -> sum(m!*x^(m - n)*LaguerreL(m, n - m, x)^2/n!, n = 0 .. infinity);
interface(rtablesize = 100);
rtable([seq([m, evalf(f(1, m))], m = 0 .. 20)], subtype = Vector[column]);

From this book on page 152 I borrowed the code.

See attached files.

fracdiff.mw

fracdiff_for_alpha=1_test.mw

For first question:

Int(exp(-(abs(x - mu)/sigma)^beta), x = -infinity .. s) = piecewise(Or(s = mu, s <= 0), sigma*GAMMA(1 + 1/beta), And(mu < s, 0 < s), ((-s + mu)*Ei(-(1 + beta)/beta, (s - mu)^beta*sigma^(-beta)) + 2*sigma*GAMMA(1/beta))/beta, sigma*GAMMA(1/beta, (-s + mu)^beta*sigma^(-beta))/beta)

 

 

Try this:

eq := 2*exp(-2*t) + 4*t = 127;
solve([eq, 0 < t], {t});
limit(LambertW(-exp(-x))/2 + 127/4, x = infinity);
interface(rtablesize = 100);
rtable([evalf[100](seq(limit(LambertW(-exp(-x))/2 + 127/4, x = m), m = 2 .. 200, 5))], subtype = Vector[column]);

 

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

 


 

 

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

 

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