Konstantin@

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14 years, 166 days
Kharkov, Ukraine
PhD in Numerical Optimization.

MaplePrimes Activity


These are questions asked by Konstantin@

Consider substutition with use of a function:

subs(F(x,y)=f(x)+g(y), F(x,y));

As expected, Maple returns the proper answer:

f(x)+g(y)

But 

subs(F(x,y)=f(x)+g(y), F(x,x));

returns 

F(x,x)

How to force Maple recognize F as a function, in order to obtain

f(x)+g(x)

in return?

The functions algsubs, applyrule work in the same way.

I want to calculate the following integral numerically with required precision.

First, the functions are defined:

G1:=-0.9445379894;
f:= (x) -> 0.9/abs(x-0.4)^(1/3)+0.1/abs(x-0.6)^(1/2);
U1 := unapply(-exp(-x)*(evalf(Int(f(t)*exp(t), t = 0 .. x))+G1)/2-exp(x)*(evalf(Int(f(t)*exp(-t), t = 0 .. x))+G1)/2, x);
U:= unapply(-exp(x)/2*(evalf(Int(f(t)*exp(-t),t=0..x))+G1)+exp(-x)/2*(evalf(Int(f(t)*exp(t),t=0..x))+G1), x);

Next, I calculate the integral in numerical form:

evalf(Int(U1(x)^2+U(x)^2-2*f(x)*U(x), x=0..1, digits=4, method = _Gquad));

If I specify digits=4, Maple return the answer -0.4291

If I use digits=5 or larger, Maple return someting like this

Is it possible to increase precision of calculation?


 

 

I try to solve numerically a boundary VP for ODE with different order of discontinuity of right part.

Say, the following BVP is given:

y''(x)+y'(x)+y(x)=F(x)

y(0)=1, y(2)=1

Let's use piecewise right part

F  := piecewise(x<=1, -x, x>1, 2*x+(x-1)^2)

plot(piecewise(x<=1, -x, x>1, 2*x+(x-1)^2), x=0..2,thickness=5)

The function

piecewise(x<=1, 1-x, x>1, (x-1)^2)

plot(piecewise(x<=1, 1-x, x>1, (x-1)^2), x=0..2, color=blue,thickness=5)

as obviuos, satisfies the BVP exclung the point x=1, where its 1st and 2nd derivatives are discontinuos.

Numerical solution

N0:=6:
As:=dsolve([diff(y(x), x$2)+diff(y(x), x)+y(x)=F,  y(0)=1, y(2)=1], y(x), type=numeric, output = Array([seq(2.0*k/N0, k=0..N0)]), 'maxmesh'=500, 'abserr'=1e-3):

provides the solution essentially different to exact one described above:

But if to use the right part

F := piecewise(x<=1, x^2+x+2, x>1, -x^2+x)

plot(piecewise(x<=1, x^2+x+2, x>1, -x^2+x), x=0..2, color=blue,thickness=5)

for which the function

piecewise(x<=1, 1-x+x^2, x>1, -1+3*x-x^2)

plot(piecewise(x<=1, 1-x+x^2, x>1, -1+3*x-x^2), x=0..2, thickness=5)

satisfies the BVP excluding x=1, where this function has discontinuity of 2nd derivative only, the corresponding numerical solution is very similar to this exact solution:

This reason of the difference between these two cases is clear. In the first case both 1st and 2nd derivatives are discontiuos, while in the second one -- 1st derivative is contiuos.

I wonder, if there are numerical methods, implemeted in Maple, for numerical solution of the first type BVP with non-smooth right part?

Dear friends!

Please, suggest me, how to build a plot loocking as the following

plot

where the values on the axis X (1, 5, 10, 50, ...) are categories, not numbers?

Dear friend,

Recently I noticed, that numerical integration returns different values for the same function.

For example the code

restart;
evalf(int((exp(x)*(4420*cos(4)*sin(4)-544*cos(4)^2+148147*exp(-1)-4225*cos(4)-215203)/(71825*exp(1)-71825*exp(-1))-exp(-x)*(4420*cos(4)*sin(4)-544*cos(4)^2+148147*exp(1)-4225*cos(4)-215203)/(71825*exp(1)-71825*exp(-1))+(32/4225)*cos(4*x)^2+(1/71825)*(4225+(2210*x-6630)*sin(4*x))*cos(4*x)+x^2+8434/4225)^2, x = 0 .. 1));

each time returns values

0.0005951015934
0.0005950850548
0.0005950974588
0.0005950960805
0.0005951297843 etc.

Maybe, evalf uses a stochastic algorithm for integration?

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