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Al86

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@Carl Love yes, works now just fine...

Al86 145
October 14 2015
0 0

@Carl Love yes, works now just fine.

 

@vv , it doesn't seem to be working...

Al86 145
October 14 2015
0 0

@vv , it doesn't seem to be working.

I get the following error message:

Error, `pds` does not evaluate to a module

@Carl Love here's my third try: &nb...

Al86 145
October 14 2015
0 0

@Carl Love here's my third try:

 


> 

  restart;
  PDE := diff(u(x, t), t) = diff(u(x, t), x, x)+sin(x+t)-cos(x+t);
  IBC:= D[1](u)(0,t)=sin(t),
        D[1](u)(1,t)=-0.000065*(u(1, t)+cos(1+t))^4+sin(1+t),
        u(x,0)=1-cos(x);
#
# For x=0..1, t=0..1, the solution varies only very slowly
# so I have increased the timestep/spacestep, just to speed
# up results generation for diagnostic purposes
#
  pds := pdsolve( PDE, [IBC], numeric, time = t, range = 0 .. 1,
                  spacestep = 0.1e-1, timestep = 0.1e-1,
                  errorest=true
                )

diff(u(x, t), t) = diff(diff(u(x, t), x), x)+sin(x+t)-cos(x+t)

  

int(JacobiTheta3(0, exp(-Pi^2*s))*v(1, t-s)^4, s = 0 .. t)

  

(D[1](u))(0, t) = sin(t), (D[1](u))(1, t) = -0.65e-4*(u(1, t)+cos(1+t))^4+sin(1+t), u(x, 0) = 1-cos(x)

  

_m216736768

 (1)
> 

#
# Plot the solution over the ranges x=0..1,
# time=0..1. Not a lot happens!
#
  pds:-plot(x=1, t=0..1);

 
> 

#
# Plot the estimated error over the ranges x=0..1,
# time=0..1
#
  pds:-plot( err(u(x,t)), x=1,t=0..1);

 
> 

#
# Get some numerical solution values
#
  pVal:=pds:-value(u(x,t), output=procedurelist):
  for k from 0 by 0.01 to 1 do
      pVal(1, k)[2], pVal(1, k)[3];
  od;
> 

 

t = 0., u(x, t) = Float(undefined)

  

t = 0.1e-1, u(x, t) = .468132560645211193

  

t = 0.2e-1, u(x, t) = .476620675893991042

  

t = 0.3e-1, u(x, t) = .485164343348984506

  

t = 0.4e-1, u(x, t) = .493759517499287648

  

t = 0.5e-1, u(x, t) = .502406098485052244

  

t = 0.6e-1, u(x, t) = .511102453922395727

  

t = 0.7e-1, u(x, t) = .519848081222815006

  

t = 0.8e-1, u(x, t) = .528641735121694212

  

t = 0.9e-1, u(x, t) = .537482758482757839

  

t = .10, u(x, t) = .546370042906006659

  

t = .11, u(x, t) = .555302850549631022

  

t = .12, u(x, t) = .564280135798000404

  

t = .13, u(x, t) = .573301110735468078

  

t = .14, u(x, t) = .582364762177520312

  

t = .15, u(x, t) = .591470267486582713

  

t = .16, u(x, t) = .600616631094471165

  

t = .17, u(x, t) = .609803004392364567

  

t = .18, u(x, t) = .619028401380561699

  

t = .19, u(x, t) = .628291952956824717

  

t = .20, u(x, t) = .637592677994616164

  

t = .21, u(x, t) = .646929690577315863

  

t = .22, u(x, t) = .656302011574524613

  

t = .23, u(x, t) = .665708740868090065

  

t = .24, u(x, t) = .675148899504965927

  

t = .25, u(x, t) = .684621575110758274

  

t = .26, u(x, t) = .694125787743871014

  

t = .27, u(x, t) = .703660614283566410

  

t = .28, u(x, t) = .713225073056872771

  

t = .29, u(x, t) = .722818231413405732

  

t = .30, u(x, t) = .732439105491770603

  

t = .31, u(x, t) = .742086754125860670

  

t = .32, u(x, t) = .751760191008710610

  

t = .33, u(x, t) = .761458467328929633

  

t = .34, u(x, t) = .771180594220798765

  

t = .35, u(x, t) = .780925615994519151

  

t = .36, u(x, t) = .790692541218692924

  

t = .37, u(x, t) = .800480408015832423

  

t = .38, u(x, t) = .810288222462013707

  

t = .39, u(x, t) = .820115017125833012

  

t = .40, u(x, t) = .829959795725423222

  

t = .41, u(x, t) = .839821585866029507

  

t = .42, u(x, t) = .849699389090256663

  

t = .43, u(x, t) = .859592228597243380

  

t = .44, u(x, t) = .869499103974247789

  

t = .45, u(x, t) = .879419034545566025

  

t = .46, u(x, t) = .889351018193387688

  

t = .47, u(x, t) = .899294070877883156

  

t = .48, u(x, t) = .909247189050718552

  

t = .49, u(x, t) = .919209385802187695

  

t = .50, u(x, t) = .929179656447670532

  

t = .51, u(x, t) = .939157011688529719

  

t = .52, u(x, t) = .949140446014158701

  

t = .53, u(x, t) = .959128968207002330

  

t = .54, u(x, t) = .969121572254020625

  

t = .55, u(x, t) = .979117265479617860

  

t = .56, u(x, t) = .989115041702798292

  

t = .57, u(x, t) = .999113907243194266

  

t = .58, u(x, t) = 1.00911285609529311

  

t = .59, u(x, t) = 1.01911089402067834

  

t = .60, u(x, t) = 1.02910701554026551

  

t = .61, u(x, t) = 1.03910022629866394

  

t = .62, u(x, t) = 1.04908952170027669

  

t = .63, u(x, t) = 1.05907390770885845

  

t = .64, u(x, t) = 1.06905238097441657

  

t = .65, u(x, t) = 1.07902394821156533

  

t = .66, u(x, t) = 1.08898760768213809

  

t = .67, u(x, t) = 1.09894236727932948

  

t = .68, u(x, t) = 1.10888722724634103

  

t = .69, u(x, t) = 1.11882119707896366

  

t = .70, u(x, t) = 1.12874327937397445

  

t = .71, u(x, t) = 1.13865248565033572

  

t = .72, u(x, t) = 1.14854782123258992

  

t = .73, u(x, t) = 1.15842830008022091

  

t = .74, u(x, t) = 1.16829293062122042

  

t = .75, u(x, t) = 1.17814072966983674

  

t = .76, u(x, t) = 1.18797070913415204

  

t = .77, u(x, t) = 1.19778188909438588

  

t = .78, u(x, t) = 1.20757328531600039

  

t = .79, u(x, t) = 1.21734392155340787

  

t = .80, u(x, t) = 1.22709281780686585

  

t = .81, u(x, t) = 1.23681900191025074

  

t = .82, u(x, t) = 1.24652149847591076

  

t = .83, u(x, t) = 1.25619933981954279

  

t = .84, u(x, t) = 1.26585155554231932

  

t = .85, u(x, t) = 1.27547718284112466

  

t = .86, u(x, t) = 1.28507525668193678

  

t = .87, u(x, t) = 1.29464481953920951

  

t = .88, u(x, t) = 1.30418491211861531

  

t = .89, u(x, t) = 1.31369458256549843

  

t = .90, u(x, t) = 1.32317287769875702

  

t = .91, u(x, t) = 1.33261885172483341

  

t = .92, u(x, t) = 1.34203155794780815

  

t = .93, u(x, t) = 1.35141005702225225

  

t = .94, u(x, t) = 1.36075340910752574

  

t = .95, u(x, t) = 1.37006068169009532

  

t = .96, u(x, t) = 1.37933094215266738

  

t = .97, u(x, t) = 1.38856326519400497

  

t = .98, u(x, t) = 1.39775672578593380

  

t = .99, u(x, t) = 1.40691040621648189

  

t = 1.00, u(x, t) = 1.41602338940992412

 (2)
> 

 

> 

 restart;
  PDE := diff(v(x, t), t) = diff(v(x, t), x, x);
  IBC:= D[1](v)(0,t)=0,
        D[1](v)(1,t)=-0.000065*v(1, t)^4,
        v(x,0)=1;
#
# For x=0..1, t=0..1, the solution varies only very slowly
# so I have increased the timestep/spacestep, just to speed
# up results generation for diagnostic purposes
#
  pds1 := pdsolve( PDE, [IBC], numeric, time = t, range = 0 .. 1,
                  spacestep = 0.1e-1, timestep = 0.1e-1,
                  errorest=true
                );
#
# Plot the solution over the ranges x=0..1,
# time=0..1. Not a lot happens!
#
  pds1:-plot(x=1, t=0..1);

diff(v(x, t), t) = diff(diff(v(x, t), x), x)

  

int(JacobiTheta3(0, exp(-Pi^2*s))*v(1, t-s)^4, s = 0 .. t)

  

(D[1](v))(0, t) = 0, (D[1](v))(1, t) = -0.65e-4*v(1, t)^4, v(x, 0) = 1

  

_m217228608

  

 
> 

v1 := log(v(1, t)); v2 := log(u(1, t)); plot(v1, v2)

ln(v(1, t))

  

ln(u(1, t))

  

Error, (in plot) unexpected option: ln(u(1, t))

  
> 

NULL


Download PDEcos.mw

@tomleslie here's the code:   ...

Al86 145
October 14 2015
0 0

@tomleslie here's the code:

 

Download PDE+cos.mw

@Preben Alsholm thanks. BTW do you ...

Al86 145
October 08 2015
0 0

@Preben Alsholm thanks.

BTW do you know why did the collocation code you wrote doesn't seem to work for n:=91 and T:=[seq(0.1..1,0.01)];?

 

Did you try to excute the code in your machine with the change above that I changed?

 

I appreciate your help, but I would like to use collocation method, so I am puzzled why it's get stuck on what I changed.

 

@Preben Alsholm ,I want to use 100 ...

Al86 145
October 06 2015
0 0

@Preben Alsholm ,I want to use 100 points for the precision on the accuracy of the numerical solution.

My colleague told me that 10 points of sampling isn't enough for what we want to account for.

 

@Preben Alsholm , if I want to use ...

Al86 145
October 06 2015
0 0

@Preben Alsholm , if I want to use 100 points, I need to change n:=100 and T:=[seq(0..1,0.01)];

 

I changed half an hour ago to n:=91 and T:=[seq(0.1..1,0.01)]; in the code and the code still evaluates on maple 18, is there a way to shorten the computation time?

One of my colleagues told me that something is wrong in the code if it takes that much time to evaluate?

 

@Preben Alsholm how do I find the n...

Al86 145
August 28 2015
0 0

@Preben Alsholm how do I find the numerical values of the approximated solution at the collocation points?

 

@Preben Alsholm For my problem I ca...

Al86 145
August 23 2015
0 0

@Preben Alsholm For my problem I can use the following discretization scheme:

 

If we take time steps x_i = ih, y_j = jh, then:

 

1/h^2(-u_{i,j-1}-u_{i-1,j}+4u_{i,j} - u_{i+1,j} - u_{i,j+1}) = \sqrt(u_{i,j}) + (u_{i+1,j}-u_{i-1,j})^2/(4h^2 * u_{i,j}^{3/2}

 

But how to implement this in maple?

Thanks.

 

@Preben Alsholm , how can I plot th...

Al86 145
August 09 2015
0 0

@Preben Alsholm , how can I plot the error of this approximated solution?

 

reply...

Al86 145
August 04 2015
0 0

@Carl Love u(x1,x2,x3,x4) satisfies: d^2u/dx1^2+d^2u/dx2^2 = d^2u/dx3^2+d^2u/dx4^2

and u is twice differentiable.

 

error estimate....

Al86 145
July 21 2015
0 0

@Preben Alsholm , hi, do you happen to know how can I estimate the error in this numerical solution?

 

 

response...

Al86 145
July 19 2015
0 0

@Preben Alsholm , this is another recursive integral equation which I want to solve numerically, I thought I could do it by using maple.

 

the recursion...

Al86 145
July 19 2015
0 0

@Axel Vogt t>0, we can also truncate it to t<1.

 

@tomleslie  Thanks, I appreciate yo...

Al86 145
July 13 2015
0 0

@tomleslie 

Thanks, I appreciate your time you spent on this.

Can you please provide me with the links to the algorithms you talked about? or what did you search in google?

did you search: "methods of solving 2-D inhomogeneous Volterra equations"?

 

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