hello everyone,

I am trying to optimize expressions symbolically. I need to find out the maximum value possible for an variable, so that the expressoin still have a valid solution.

For Example:

expr:=a-b/b-a^3;  # a=(0,10), b=(0,10)

In this eypression b=a^3 is the only case where undefined solution is possible, for a given interval of variables

This looks fine for simple expression. But in reality there are complex equations to solve with more than 2 varibles.

1) first thing is to find out all values of a variable resulting in undeifined output(or infinite)
2) assign a symbolic value(variable < or > some value) to the variable so that the undefined result can be eliminated.

 I need to optimize the given expression so that it does not have any undefined cases when solving. I understood that when optimizing, there always be a condition on variables(in this case variable max is the condition, maximum value the variable can take). output of an expression is always a real value

OptimizedExpr:=a-b[max]/b[max]-a^3 --> b[max]>a^3 or b[max]<a^3

(it is easy to to say b[max] is not equal to a^3 , also a^3 is the limiting value. In some case it is more resonable to just ignore other part of limiting value. Hence, I would like to optimize using greaterthan or lessthan of limiting value).

I would be very glad to know how I can find Optimized expressions. I tried using the solve function but observed that expressions are equalled to zero and solving. which is completely opposite to what I was looking. I really do not know is there any way to find out undefined cases in expressions and on what varibale at what values.

I tried to explain the situation at my best and I welcome for any suggested edits or furthur information required.

Thank you

how I can determied  this integral in figure below or compute area in figure which adressed in the following website?

https://en.m.wikipedia.org/wiki/Spherical_cap

how I can write taylor function for following example.

I want to gain an answer similar to the image result by using taylor function.

tylor.mw
 

Download tylor.mw

Hello,

I am trying the following command:

restart; with(IntegrationTools):
simplify(int(f(x), x = 0 .. L*Ts)-Split(int(f(x), x = 0 .. L*Ts), [i*Ts, i = 0 .. L]))

Clearly the output should be 0. However, maple is not able to output the correct result. Any ideas?

Download threads!.mw

Hi,

I would like to ask you a question on the following program. Where does the error come from?

Thank you for your help in advance.

som:=0:

for b1 from 10 to 20 by 1 do
for b2 from 1 to 10 by 0.1 do
for alpha from 0.5 to 0.9 by 0.1 do
for beta from 0.1 to 0.4 by 0.1 do
for c from 1 to 1 by 1 do
for f from 1 to 10 by 1 do
for g from 8 to 470 by 1 do
for lambdai from 0.2 to 0.2 by 0.1 do
for lambdaj from 0.2 to 0.2 by 0.1 do
for gammai from 0.4 to 0.4 by 0.1 do
for gammaj from 0.4 to 0.4 by 0.1 do

aiSQ:=(alpha*b1)/(alpha*b2+beta*b2+c);
ajSQ:=(beta*b1)/(alpha*b2+beta*b2+c);
UiSQ:=(1/2)*alpha*b1^2*(alpha^2*b2+2*alpha*beta*b2+c*alpha+beta^2*b2+2*beta*c)/(alpha*b2+beta*b2+c)^2;
UjSQ:=(1/2)*beta*b1^2*(alpha^2*b2+2*alpha*beta*b2+2*c*alpha+beta^2*b2+beta*c)/(alpha*b2+beta*b2+c)^2;
USQ:=(1/2)*b1^2*(alpha+beta)*(alpha*b2+beta*b2+2*c)/(alpha*b2+beta*b2+c)^2;

ai:=(c*b1*alpha+c*p*f+p*f*beta*b2-alpha*b2*p*f)/(c*(alpha*b2+beta*b2+c));
aj:=(-p*f*beta*b2+alpha*b2*p*f+c*beta*b1+c*p*f)/(c*(alpha*b2+beta*b2+c));
aineg:=(-p*f*lambdai*b2*beta+c*b1-c*b1*lambdai+b2*p*f*lambdaj+b2*alpha*p*f*lambdaj-p*f*lambdai*b2+c*b1*alpha-c*p*f*lambdai)/(c*(b2*alpha-b2*lambdai+b2*beta-b2*lambdaj+c+2*b2));
ajneg:=-(b2*alpha*p*f*lambdaj-p*f*lambdai*b2-p*f*lambdai*b2*beta+b2*p*f*lambdaj+c*p*f*lambdaj-c*b1-c*beta*b1+c*lambdaj*b1)/(c*(b2*alpha-b2*lambdai+b2*beta-b2*lambdaj+c+2*b2));
ui:=alpha*(b1*(aineg+ajneg)-(b2/2)*(aineg+ajneg)^2)-(c/2)*aineg^2-p*f*(aineg-ai);
uj:=beta*(b1*(aineg+ajneg)-(b2/2)*(aineg+ajneg)^2)-(c/2)*ajneg^2-p*f*(ajneg-aj);
u:=(b1*(aineg+ajneg)-(b2/2)*(aineg+ajneg)^2)+p*f*(aineg-ai)+p*f*(ajneg-aj);
eqti:=gammai*(u-tj-USQ)-((1-gammai)/(1-lambdai))*(ui-UiSQ);
eqtj:=gammaj*(u-ti-USQ)-((1-gammaj)/(1-lambdaj))*(uj-UjSQ);
solt:=solve({eqti, eqtj}, {ti, tj});
ti:=subs(solt,ti);
tj:=subs(solt,tj);
dai:=diff(ai,p);
daj:=diff(aj,p);
daineg:=diff(aineg,p);
dajneg:=diff(ajneg,p);
dti:=diff(ti,p);
dtj:=diff(tj,p);
eqp:=(b1-b2*(ai+aj))*(dai+daj)-dti-dtj+f*(aineg-ai)+p*f(daineg-dai)+f*(ajneg-aj)+p*f(dajneg-daj)-g*p=0;
p:=solve(eqp,p);

Uip:=alpha*(b1*(ai+aj)-(b2/2)*(ai+aj)^2)-(c/2)*ai^2-p*f*(aineg-ai)+(1-lambdai)*ti;
Ujp:=beta*(b1*(ai+aj)-(b2/2)*(ai+aj)^2)-(c/2)*aj^2-p*f*(ajneg-aj)+(1-lambdaj)*tj;
Up:=(b1*(ai+aj)-(b2/2)*(ai+aj)^2)+p*f*(aineg-ai)+p*f*(ajneg-aj)-((g^2)/2)*p-ti-tj;

CSQ:=b1-b2*(aiSQ+ajSQ);
Cabat:=b1-b2*(ai+aj);
Cneg:=b1-b2*(aineg+ajneg);

if (CSQ>0 and Cabat>0  and Cneg>0 and beta<alpha and p>0 and p<1)
then
print(b1,b2,alpha,beta,c,f,g,lambdai,lambdaj,`aiSQ=`,aiSQ,`ajSQ=`,ajSQ,`UiSQ=`,UiSQ,`UjSQ=`,UjSQ,`USQ=`,USQ,`ai=`,ai,`aj=`,aj,`aineg=`,aineg,`ajneg=`,ajneg,`ti=`,ti,`tj=`,tj,`p=`,p,`Uip=`,Uip,`Ujp=`,Ujp,`Up=`,Up);
som:=som+1;
fi;
od;od;od;od;od;od;od;od;od;od;od;
som;

10, 1, 0.5, 0.1, 1, 1, 8, 0.2, 0.2, aiSQ=, 3.12500000000000, ajSQ=, 

  0.625000000000000, UiSQ=, 10.3515625000000, UjSQ=, 2.85156250000000, USQ=, 

  30.4687500000000, ai=, 3.14314584993738, aj=, 0.667340316520551, aineg=, 

  4.06068541500626, ajneg=, 2.80826596834794, ti=, 19.2559726892491, tj=, 

  7.44890809994178, p=, 0.0483889331663441, Uip=, 25.8431760998162, Ujp=, 

  8.71735374786572, Up=, 2.73962847065372
Error, (in solve) a constant is invalid as a variable, 7.44890809994178, 19.2559726892491
                                      1

is possible to solve this pde via maple?

m1.mwm1.mw
 

Download m1.mw

This is a Wave PDE inside disk. with fixed edge of disk,  and no theta dependency. initial position and velocity  given.

When using "c^2" for the  wave propagation speed, Maple only gives solution when also using assumptions to tell that c is positive.

restart; 
Physics:-Version()[2]; 
`2018, November 28, 1:35 hours, version in the MapleCloud: 224, version installed in this computer: 224`

And the PDE is 

pde := diff(u(r, t), t$2) = c^2*( diff(u(r,t), r$2)  + (1/r)* diff(u(r,t),r)  ) ; 
ic:=u(r,0)=1, eval( diff(u(r,t),t),t=0)=r/3; 
bc:=u(1,t)=0; 
sol:=pdsolve([pde, ic,bc], u(r, t)) assuming t>0;

sol:=()

Now adding assuming c>0, or "c::positive", or "c>=0", it solves it 

restart;
pde := diff(u(r, t), t$2) = c^2*( diff(u(r,t), r$2)  + (1/r)* diff(u(r,t),r)  ) ; 
ic:=u(r,0)=1, eval( diff(u(r,t),t),t=0)=r/3; 
bc:=u(1,t)=0; 
sol:=pdsolve([pde, ic,bc], u(r, t)) assuming t>0,c>=0;

Any idea why the assumption "c>0"  is needed when the speed is given as "c^2" ? I also tried assumption "c::real", but it still did not solve it.   It seems related to Maple using Laplace transform to solve it.

It seems to me the assumption c>0 should not needed here. But if it is, I'd like to learn why.

Maple 2018.2, windows 10

Update

Thanks to comment below

   " so u(r, 0)=1 (for all r), and u(1, t)=0 (for all t). So what is u(1,0)??? "

I should have added "r>0,r<1".  The reason I did not, is that I copied the code to solve this from Mathematica. In Mathematica, it did not need this assumption to solve it. It seems to depend on the method of solution used by Mathematica vs. Maple to cause this difference.

But now I see it helps to have it there. With the above assumption, now, there is no need to do any assumption on "c" at all and it gives solution. adding c>0 or t>0 or c>=0, makes no change now.

restart;
pde := diff(u(r, t), t$2) = c^2*( diff(u(r,t), r$2)  + (1/r)* diff(u(r,t),r)  ) ; 
ic:=u(r,0)=1, eval( diff(u(r,t),t),t=0)=r/3; 
bc:=u(1,t)=0; 
sol:=pdsolve([pde, ic,bc], u(r, t)) assuming t>0,r>0,r<1;

Hi

https://www.energymatters.com.au/solar-calculators/solar-battery-calculator.php?utm_source=NewsletterMailingList&utm_medium=email&utm_campaign=EM171101CN

I am trying to find how it calculates how much solar will be exported back to the grid. I thought this could only be estimated  empirically, but apparently not.

Anyway, it depends on three factors, power consumption, power installed, and average hours of sun per day, or hours per day as a function of time of year. For C=4.5kWh/day and S=5kW and H=4 sun hrs per day, means 90% will be exported back to the grid Y. So Y(C,S,H)= a closed form expression.

How to guess the fuction? I tried array interpolation for a numeric solution, but it complains: independent coordinates must be sorted in increasing order

law.mw

I have the following inequality system : 

              [ U_1,2 + U_1,3 -1 = 0,

U_2,3 - U_1,2 = 0, 

-U_1,3 -U_2,3+1 = 0,

0 <=U_1,2,

0 <= U_1,3,

0 <= U_2,3]

I want to solve it using LinearMultivariateSystem, namely I excuted a command : 

Then the following is return : 

Error, (in Utilities:-SimpleAnd) invalid input: a string/name list is expected for sort method `lexorder`

 I tried that

My question is that how do you use LinearMultivariateSystem for varibles indexed by two params.

Thank you.

Hiya

from this

>with(combinat):
>A:=choose([CPC__h, SIZE__h,CPC__m, SIZE__m,CPC__l, SIZE__l],2): 

where h denotes high, m=medium, l=low

I want to select CPC and SIZE all the combinations w.r.t h, m and l

[[CPC__h, SIZE__h], [CPC__h, SIZE__l], [CPC__h, SIZE__m], [CPC__l, SIZE__h], [CPC__l, SIZE__l], [CPC__l, SIZE__m], [CPC__m, SIZE__h], [CPC__m, SIZE__l], [CPC__m, SIZE__m]]

>nops(A)-3

9

I want to apply the methodology to 
choose([CPC__h, SIZE__h, SH__h,CPC__m, SIZE__m, SH__m,CPC__l, SIZE__l, SH__l],3)
 

convert(...,Int) in Maple 2018.2 works for fourier, invfourier, laplace, but does not work for invlaplace.  

Why is that? Is there a workaround?

expr:=fourier(f(x), x, w):
convert(expr,Int);

expr:=invfourier(f(w), w, x):
convert(expr,Int);

expr:=laplace(f(s),s,t):
convert(expr,Int)

expr:=invlaplace(f(s),s,t):
convert(expr,Int)

Was expecting to see the Mellin's inverse formula.

Maple 2018.2 on windows 10

Hi I am trying to solve the following system of equation. I could solve it for approximate value of s(0)=0.9999 using middefer method of bvp, but as soon as I give the b.c. as s(0)=1, Maple gives me an error that there is an numeric exception. Can someone explain it what is going wrong with s(0)=1? Any sugesstion to solve the equation?
error.mw

b := 1;
r := .1;
l := 3;
a := 10; p := 1.5; ds := 100; dk := 1;

sol3 := dsolve({ds*(diff(i(x), x)) = exp(eta(x)), s(x)^3*i(x)*b*r+(1-s(x))^3*a*l*(diff(s(x), x))/s(x)^1.5 = (1-s(x))^3, diff(eta(x), x) = dk*(i(x)-1)/s(x)^p, i(0) = 0, i(1) = 1, s(0) = 1}, {eta(x), i(x), s(x)}, approxsoln = [eta(x) = .5, i(x) = .5, s(x) = 1], initmesh = 100000, type = numeric, maxmesh = 100000, range = 0 .. 1, output = listprocedure, method = bvp[middefer], abserr = 0.1e-3, adaptive = true);

Error, (in dsolve/numeric/bvp) numeric exception: division by zero

plots[odeplot](sol3, [[x, s(x)]], 0 .. 1, color = ["Green"]);

Thanks and regards,
 

Using Latest Physics updates (I am not sure when this started), pdsolve gives Error, (in PDEtools/eval/2) numeric exception: division by zero on the following problem from a HW from text book.

restart;
PackageTools:-IsPackageInstalled("Physics Updates");
                             "220"


pde:=diff(w(x,t),t)+3*t*diff(w(x,t),x)=w(x,t);
ic:=w(x,0)=f(x);
sol:=pdsolve([pde,ic],w(x,t));

Error, (in PDEtools/eval/2) numeric exception: division by zero
 

Mathematica answer btw is 

pde = D[w[x, t], t] + 3 t D[w[x, t], x] == w[x, t];
ic = w[x, 0] == f[x];
sol = Simplify[DSolve[{pde, ic}, w[x, t], {x, t}]]

This is on Maple 2018.2 on windows 10 64 bit.

Any idea what is causing this and any workaround? Do others get the same exception?

For some reason the Maple software is not evaluating the last bounds for a triple integral.

But the evalf command works.