Markiyan Hirnyk

## 7228 Reputation

11 years, 334 days

## Numeric solution...

The difficulty consists in the nonstandard IC. In order to get over the difficulty, we make the change of the variables.

```PDE := (y-u(x, y))*(diff(u(x, y), x))+(u(x, y)-x)*(diff(u(x, y), y)) = x-y;:
solve({eta = y-1/x, xi = x}, {x, y});

{x = xi, y = (eta*xi+1)/xi}

with(PDEtools):
tr := {x = xi, y = eta+1/xi}:
PDE1 := normal(dchange(tr, PDE, simplify));

PDE1 := -(u(xi, eta)*(diff(u(xi, eta), xi))*xi^3-(diff(u(xi, eta), eta))*xi^3*u(xi, eta)-
(diff(u(xi, eta), xi))*eta*xi^3+(diff(u(xi, eta), eta))*xi^4+u(xi, eta)*(diff(u(xi, eta),
eta))*xi-(diff(u(xi, eta), xi))*xi^2-(diff(u(xi, eta), eta))*eta*xi-(diff(u(xi, eta), eta)))/xi^3 =
-(eta*xi-xi^2+1)/xi
```

The condition u(x,1/x)=0 transforms to u{xi,0)=0. It is clear that a symbolic solution of PDE1 with that condition has no chance. Therefore we make use of the numeric option, adding the condition u(0,eta)=0).

```ans := pdsolve({PDE1}, {u(0, eta) = 0, u(xi, 0) = 0}, numeric, time = xi, range = 0 .. 1);
ans := module() ... end module
ans:-plot3d(xi = 0 .. 1, eta = 0 .. 1);
```

pde.mw

## Likely inconsistent problem...

in view of

```restart;
PDE := (y-u(x, y))*(diff(u(x, y), x))+(u(x, y)-x)*(diff(u(x, y), y)) = x-y:
infolevel[pdsolve] := 3:
ans := pdsolve(PDE, HINT = sum);
HINT = sum
Trying HINT = sum
Warning: System is inconsistent

```

PS. or a bug in pdsolve.

## Workaround in Maple 2017.3...

The following works for me.

```abs(simplify(convert(-((-(1/20)*sqrt(5)-1/20)*cos((1/5)*Pi)-
(1/20)*sqrt(2)*sqrt(5-sqrt(5))*sin((1/5)*Pi))*24^(1/5)*5^(3/5)-

(1/5)*24^(1/5)*5^(3/5)```

The command convert/radical may be omitted, but the result is more complicated in this case.

abs.mw

## By op...

```op(1, 1490.90920124091 .. 1497.18170785000);
1490.90920124091
op(2, 1490.90920124091 .. 1497.18170785000);
1497.18170785000
```

so use

```op(1,OneSampleZTest(S,mu,sdev,confidence=0.95,output='confidenceinterval'));
op(2,OneSampleZTest(S,mu,sdev,confidence=0.95,output='confidenceinterval'));```

to this end.

## Product over empty list...

is treated as 1 (one). Therefore,

```convert(hypergeom([], [], z),elementary);
exp(z)```

## By Quantile...

The Quantile command does the job:

```restart;
with(Statistics):
X := RandomVariable(Poisson(2.6)):
Quantile(X, .95);
5.
```

## By intersectplot...

See the first example in  ?intersectplot to this end.

## Syntax...

This works for me

```int((x^2+y^2)*piecewise(y-x-1/2 <= 0 and y+2*x-2 <= 0 and y+(1/2)*x-1/2 >= 0, 1, 0),
[x = 0 .. 1, y = 0 .. 1]);
7/32```

## Complex-valued function with branch poin...

Let us consider Int(exp(LambertW(1/(-1+x))*(-1+x)), x)+1. First, the term 1 is of no importance. Second, let us consider the int(exp(LambertW(1/(-1+x))*(-1+x)), x = 1 .. t) command instead of an inert Int one. Third,

`FunctionAdvisor(branch_cuts, exp(LambertW(1/(-1+x))*(-1+x)), plot = 2.);`

[exp(LambertW(1/(-1+x))*(-1+x)), And(-exp(1)+1 < x, x < 1)]

says the integrand is a complex-valued function with branch points. Because of this,

```a := Re(int(exp(LambertW(1/(-1+x))*(-1+x)), x = 1 .. t)):
plot(a, t = 0 .. 2, numpoints = 5, thickness = 3);
```

```int(exp(LambertW(1/(-1+x))*(-1+x)), x = 1 .. 2.1);
1.670860297
int(exp(LambertW(1/(-1+x))*(-1+x)), x = 1 .. -2.1+1.*I);
-7.784875149 - 3.643815590 I

```

int_of_Lambert.mw

## By DirectSearch:-SolveEquations...

This can be done as follows (The DirectSearch package should be downloaded from http://www.maplesoft.com/applications/view.aspx?SID=101333 and installed in your Maple >=12.).

```sol4 := DirectSearch:-SolveEquations([sol1, sol2], {T = 0 .. .4}, AllSolutions, solutions = 4);
sol4 := Matrix(1, 4, [[0.1040897473e-19,
Vector[column](2, [-0.6873203012e-10, -0.7539816154e-10]),
[T = .321611763404719, W = 29.4643511745074], 1013]])```

The plot produced by

`plots:-implicitplot([sol1, sol2], T = 0 .. 5, W = 0 .. 50, gridrefine = 2, color = [blue, red]);`

system.mw

## DiscreteUniform...

This can be done in such a way:

```with(Statistics):
X := RandomVariable(DiscreteUniform(i-x, i+x)):
Mean(X);
i
Variance(X);

(1/12)*(i-x)^2+(1/12)*(i+x)^2+(1/3)*x-(1/6*(i-x))*(i+x)
i := 6:x := 11:
Sample(X,6);
Vector[row](6, [13., 15., -3., 16., 9., -3.])```

## Two-dimensional case...

This is not so simple. Here is one way. We start from plotting

`restart; plots:-implicitplot(t^8+s^6+s*t^5 = 1, s = -10 .. 10, t = -10 .. 10, gridrefine = 2);`

We see two almost vertical pieces. This means that a small change of s implies a relatively big change of t.

Now we find the range of t by

```Optimization:-Maximize(t, {t^8+s^6+s*t^5 = 1})[1];
1.08827028359556088
```

Then we solve

```sol := solve(t^8+s^6+s*t^5 = 1);
sol := {s = RootOf(t^8+_Z^6+_Z*t^5-1), t = t}
allvalues(sol);
{s = RootOf(t^8+_Z^6+_Z*t^5-1, index = 1), t = t},
{s = RootOf(t^8+_Z^6+_Z*t^5-1, index = 2), t = t},
{s = RootOf(t^8+_Z^6+_Z*t^5-1, index = 3), t = t},
{s = RootOf(t^8+_Z^6+_Z*t^5-1, index = 4), t = t},
{s = RootOf(t^8+_Z^6+_Z*t^5-1, index = 5), t = t},
{s = RootOf(t^8+_Z^6+_Z*t^5-1, index = 6), t = t}
```

and build the parametric plots, substituting the above in

```x := t*cos(t^2+s); y := s^3-sin(t);
Digits := 15; [seq(plot([eval(x, allvalues(sol)[j]), eval(y, allvalues(sol)[j]),
t = -1.08827028359556088 .. 1.08827028359556088], thickness = 2), j = 1 .. 6)];
plots:-display(%);
```

The result is not satisfactory for me: we obtain two warnings

Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct
Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct
and a somewhat srange plot

which includes a superfluous interval and two breaks. I think the first one is caused by Indexed RootOf (especially by s = RootOf(t^8+_Z^6+_Z*t^5-1, index = 1) ) and the second one is caused by roundoff errors which produce complex numbers (Hope Maple developers will shed light on it.).

In oder to overcome the first deficiency, I do the following (The problem with sol[1] and t = -1 were found through trial-and-error method.).

```a := plot([eval(x, allvalues(sol)[1]), eval(y, allvalues(sol)[1]),
t = -1.08827028359556088 .. -1], thickness = 2, numpoints = 500):
b := plot([eval(x, allvalues(sol)[1]), eval(y, allvalues(sol)[1]),
t = -.99 .. 1.08827028359556088], thickness = 2, numpoints = 500):
plots:-display([seq(plot([eval(x, allvalues(sol)[j]), eval(y, allvalues(sol)[j]),
t = -1.08827028359556088 .. 1.08827028359556088], thickness = 2, numpoints = 500),
j = 2 .. 6), a, b]);
```

and

Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct
Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct
parametric_plot.mw

## All solutions...

can be obtained in such a way.

```expr := 1/(u+v)^2+4*u*v-1:
normal(expr);
(4*u^3*v+8*u^2*v^2+4*u*v^3-u^2-2*u*v-v^2+1)/(u+v)^2
```

Let us plot it in the positive quadrant:

`plots:-implicitplot(numer(normal(expr)) = 0, u = 0 .. 8, v = 0 .. 8, gridrefine = 2, thickness = 2);`

Now

```A := RealDomain:-solve(numer(normal(expr)) = 0, v);
(1/12)*(16*u^4-8*(64*u^6+48*u^4+12*sqrt(-192*u^6-144*u^4+288*u^2-3)*u-204*u^2+1)^(1/3)*u^2+(64*u^6+48*u^4+12*sqrt(-192*u^6-144*u^4+288*u^2-3)*u-204*u^2+1)^(2/3)+8*u^2+(64*u^6+48*u^4+12*sqrt(-192*u^6-144*u^4+288*u^2-3)*u-204*u^2+1)^(1/3)+1)/(u*(64*u^6+48*u^4+12*sqrt(-192*u^6-144*u^4+288*u^2-3)*u-204*u^2+1)^(1/3))                                                                                              ```

For example,

```eval(A, u = 2);
(1/24)*(289-31*(4049+24*sqrt(-13443))^(1/3)+(4049+24*sqrt(-13443))^(2/3))/(4049+24*sqrt(-13443))^(1/3)
evalf(%);

0.09656230030 + 2.101279539 10 ^(-10)   I
```

Let us chek it by plotting

`plot(A, u = 1 .. 8);`

Because expr is symmetric in u and v, the rest is clear.

## By solve...

One can solve this system analitically by the command of Maple 2016.2

`solve({-5 <= 4*y1+2*y2+3*y3, -2 <= 3*y1+5*y2+2*y3, -1 <= y1+2*y2+y3, 0 <= y2, y1 <= 0}, {y1, y2, y3});`

{y2 = 0, y3 = -(3/2)*y1-1, y1 < -2}, {y2 = 0, y1 < -2, -(3/2)*y1-1 < y3}, {y1 = -2, y2 = 0, 2 <= y3}, {y2 = 0, y3 = -(3/2)*y1-1, -2 < y1, y1 < 0}, {y2 = 0, -2 < y1, y1 < 0, -(3/2)*y1-1 < y3}, {y1 = 0, y2 = 0, -1 <= y3}, {y3 = -1-(3/2)*y1-(5/2)*y2, 0 < y2, y1 < -2+4*y2, y2 < 4/11}, {0 < y2, y1 < -2+4*y2, y2 < 4/11, -1-(3/2)*y1-(5/2)*y2 < y3}, {y1 = -2+4*y2, y3 = 2-(17/2)*y2, 0 < y2, y2 < 4/11}, {y1 = -2+4*y2, 0 < y2, y2 < 4/11, 2-(17/2)*y2 < y3}, {y3 = -1-(3/2)*y1-(5/2)*y2, 0 < y2, y1 < -y2, y2 < 4/11, -2+4*y2 < y1}, {0 < y2, y1 < -y2, y2 < 4/11, -2+4*y2 < y1, -1-(3/2)*y1-(5/2)*y2 < y3}, {y1 = -y2, y3 = -y2-1, 0 < y2, y2 < 4/11}, {y1 = -y2, 0 < y2, y2 < 4/11, -y2-1 < y3}, {y3 = -1-y1-2*y2, 0 < y2, y1 < 0, y2 < 4/11, -y2 < y1}, {0 < y2, y1 < 0, y2 < 4/11, -y2 < y1, -1-y1-2*y2 < y3}, {y1 = 0, y3 = -2*y2-1, 0 < y2, y2 < 4/11}, {y1 = 0, 0 < y2, y2 < 4/11, -2*y2-1 < y3}, {y2 = 4/11, y3 = -21/11-(3/2)*y1, y1 < -6/11}, {y2 = 4/11, y1 < -6/11, -21/11-(3/2)*y1 < y3}, {y1 = -6/11, y2 = 4/11, -12/11 <= y3}, {y2 = 4/11, y3 = -21/11-(3/2)*y1, -6/11 < y1, y1 < -4/11}, {y2 = 4/11, -6/11 < y1, y1 < -4/11, -21/11-(3/2)*y1 < y3}, {y1 = -4/11, y2 = 4/11, -15/11 <= y3}, {y2 = 4/11, y3 = -19/11-y1, -4/11 < y1, y1 < 0}, {y2 = 4/11, -4/11 < y1, y1 < 0, -19/11-y1 < y3}, {y1 = 0, y2 = 4/11, -19/11 <= y3}, {y3 = -1-(3/2)*y1-(5/2)*y2, 4/11 < y2, y1 < -2+4*y2, y2 < 2/5}, {4/11 < y2, y1 < -2+4*y2, y2 < 2/5, -1-(3/2)*y1-(5/2)*y2 < y3}, {y1 = -2+4*y2, y3 = 2-(17/2)*y2, 4/11 < y2, y2 < 2/5}, {y1 = -2+4*y2, 4/11 < y2, y2 < 2/5, 2-(17/2)*y2 < y3}, {y3 = -1-(3/2)*y1-(5/2)*y2, 4/11 < y2, y1 < -y2, y2 < 2/5, -2+4*y2 < y1}, {4/11 < y2, y1 < -y2, y2 < 2/5, -2+4*y2 < y1, -1-(3/2)*y1-(5/2)*y2 < y3}, {y1 = -y2, y3 = -y2-1, 4/11 < y2, y2 < 2/5}, {y1 = -y2, 4/11 < y2, y2 < 2/5, -y2-1 < y3}, {y3 = -1-y1-2*y2, 4/11 < y2, y1 < 4-11*y2, y2 < 2/5, -y2 < y1}, {4/11 < y2, y1 < 4-11*y2, y2 < 2/5, -y2 < y1, -1-y1-2*y2 < y3}, {y1 = 4-11*y2, y3 = -5+9*y2, 4/11 < y2, y2 < 2/5}, {y1 = 4-11*y2, 4/11 < y2, y2 < 2/5, -5+9*y2 < y3}, {y3 = -1-y1-2*y2, 4/11 < y2, y1 < 0, y2 < 2/5, 4-11*y2 < y1}, {4/11 < y2, y1 < 0, y2 < 2/5, 4-11*y2 < y1, -1-y1-2*y2 < y3}, {y1 = 0, y3 = -2*y2-1, 4/11 < y2, y2 < 2/5}, {y1 = 0, 4/11 < y2, y2 < 2/5, -2*y2-1 < y3}, {y2 = 2/5, y3 = -2-(3/2)*y1, y1 < -2/5}, {y2 = 2/5, y1 < -2/5, -2-(3/2)*y1 < y3}, {y1 = -2/5, y2 = 2/5, -7/5 <= y3}, {y2 = 2/5, y3 = -9/5-y1, -2/5 < y1, y1 < 0}, {y2 = 2/5, -2/5 < y1, y1 < 0, -9/5-y1 < y3}, {y1 = 0, y2 = 2/5, -9/5 <= y3}, {y3 = -1-(3/2)*y1-(5/2)*y2, 2/5 < y2, y1 < 4-11*y2, y2 < 1/2}, {2/5 < y2, y1 < 4-11*y2, y2 < 1/2, -1-(3/2)*y1-(5/2)*y2 < y3}, {y1 = 4-11*y2, y3 = 14*y2-7, 2/5 < y2, y2 < 1/2}, {y1 = 4-11*y2, 2/5 < y2, y2 < 1/2, 14*y2-7 < y3}, {y3 = -5/3-(4/3)*y1-(2/3)*y2, 2/5 < y2, y1 < -y2, y2 < 1/2, 4-11*y2 < y1}, {2/5 < y2, y1 < -y2, y2 < 1/2, 4-11*y2 < y1, -5/3-(4/3)*y1-(2/3)*y2 < y3}, {y1 = -y2, y3 = -5/3+(2/3)*y2, 2/5 < y2, y2 < 1/2}, {y1 = -y2, 2/5 < y2, y2 < 1/2, -5/3+(2/3)*y2 < y3}, {y3 = -5/3-(4/3)*y1-(2/3)*y2, 2/5 < y2, y1 < -2+4*y2, y2 < 1/2, -y2 < y1}, {2/5 < y2, y1 < -2+4*y2, y2 < 1/2, -y2 < y1, -5/3-(4/3)*y1-(2/3)*y2 < y3}, {y1 = -2+4*y2, y3 = -6*y2+1, 2/5 < y2, y2 < 1/2}, {y1 = -2+4*y2, 2/5 < y2, y2 < 1/2, -6*y2+1 < y3}, {y3 = -1-y1-2*y2, 2/5 < y2, y1 < 0, y2 < 1/2, -2+4*y2 < y1}, {2/5 < y2, y1 < 0, y2 < 1/2, -2+4*y2 < y1, -1-y1-2*y2 < y3}, {y1 = 0, y3 = -2*y2-1, 2/5 < y2, y2 < 1/2}, {y1 = 0, 2/5 < y2, y2 < 1/2, -2*y2-1 < y3}, {y2 = 1/2, y3 = -9/4-(3/2)*y1, y1 < -3/2}, {y2 = 1/2, y1 < -3/2, -9/4-(3/2)*y1 < y3}, {y1 = -3/2, y2 = 1/2, 0 <= y3}, {y2 = 1/2, y3 = -2-(4/3)*y1, -3/2 < y1, y1 < -1/2}, {y2 = 1/2, -3/2 < y1, y1 < -1/2, -2-(4/3)*y1 < y3}, {y1 = -1/2, y2 = 1/2, -4/3 <= y3}, {y2 = 1/2, y3 = -2-(4/3)*y1, -1/2 < y1, y1 < 0}, {y2 = 1/2, -1/2 < y1, y1 < 0, -2-(4/3)*y1 < y3}, {y1 = 0, y2 = 1/2, -2 <= y3}, {y3 = -1-(3/2)*y1-(5/2)*y2, 1/2 < y2, y1 < 4-11*y2}, {1/2 < y2, y1 < 4-11*y2, -1-(3/2)*y1-(5/2)*y2 < y3}, {y1 = 4-11*y2, y3 = 14*y2-7, 1/2 < y2}, {y1 = 4-11*y2, 1/2 < y2, 14*y2-7 < y3}, {y3 = -5/3-(4/3)*y1-(2/3)*y2, 1/2 < y2, y1 < -y2, 4-11*y2 < y1}, {1/2 < y2, y1 < -y2, 4-11*y2 < y1, -5/3-(4/3)*y1-(2/3)*y2 < y3}, {y1 = -y2, y3 = -5/3+(2/3)*y2, 1/2 < y2}, {y1 = -y2, 1/2 < y2, -5/3+(2/3)*y2 < y3}, {y3 = -5/3-(4/3)*y1-(2/3)*y2, 1/2 < y2, y1 < 0, -y2 < y1}, {1/2 < y2, y1 < 0, -y2 < y1, -5/3-(4/3)*y1-(2/3)*y2 < y3}, {y1 = 0, y3 = -(2/3)*y2-5/3, 1/2 < y2}, {y1 = 0, 1/2 < y2, -(2/3)*y2-5/3 < y3}

This approach works in higher dimensions too.

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