## Exact symbolic solution of a system of partial dif...

Asked by:

Hi Everybody,

I have a simple question: Does Maple solve systems of partial differential equations with boundary conditions?

Can somebody give me an example?

I have only found numerical solutions to this kind of systems but no symbolic example.

Thanks a lot for yor help.

## area under pdsolve diagram ...

Asked by:

i want to know the area under a diagram plotted by pdsolve, how can i do that? for example in below , what is the area under p1 diagram?

 > restart:k:=5;
 (1)
 > EQ:=diff(u(x,t),t)=k*diff(u(x,t),x\$2);
 (2)
 > ibc:=u(0,t)=0,u(1,t)=0, u(x,0) = x;
 (3)
 > sol:=pdsolve({EQ},{ibc},numeric);
 (4)
 > p1:=sol:-plot(u,x=0.5,t=0...10,style = line,color = "Blue",legend = "heat Plot",axes=boxed);
 > M:=op(1,op(1,p1));
 (5)
 >
 >
 >
 >

Download heat_equation_(2).mw

## plot the solution of a complex PDE equation ...

Asked by:

Hello

I solved a complex PDE equation in maple but I can not plot the output.

The manner was like bellow:

PDE := [diff(A(z, t), z)+(1/2)*alpha*A(z, t)+(I*beta[2]*(1/2))*(diff(A(z, t), t, t))-(I*beta[3]*(1/6))*(diff(A(z, t), t, t, t))-I*(GAMMA(omega[0]))(abs(A(z, t))^2*A(z, t)) = 0];
IBC := {(D[2](A))(z, 1), A(0, t) = -sin(2*Pi*t), A(z, 0) = sin(2*Pi*z), (D[2](A))(z, 0) = 2*z};
pds := pdsolve(PDE, IBC, type = numeric, time = t, range = 0 .. 1);
pds:-plot3d(A(z, t)*conjugate(A(z, t)), t = 0 .. 1, z = 0 .. 10, shading = zhue, axes = boxed, labels = ["x", "t", "A(z,t)"], labelfont = [TIMES, ROMAN, 20], orientation = [-120, 40]);

It is solved but there is an error like:

Error, (in pdsolve/numeric/plot3d) unable to compute solution for z>INFO["failtime"]:
unable to store 11.2781250000000+4390.00000040000*I when datatype=float[8]

could you please help me?

what is the problem?

## why does one need to convert piecewise function to...

Asked by:

UPDATE

Thanks for checking. I verify I get the error and made screen shots below

When I add the convert() command, the error goes away. Here is screen shot

I am using Maple 2017, student version, on windows 7, 64 bit, home edition.

Original post

This is using Maple 2017 on windows.

With the following input, Maple pdsolve gives an error

```pde:=diff(u(x,t),t)=k*diff(u(x,t),x\$2);
bc:=D[1](u)(0,t)=0,D[1](u)(L,t)=0:
assume(L>0):
ic:=u(x,0)=piecewise(0<x and x<=L/2,0,L/2<x and x<L,1):
sol:=pdsolve([pde,bc,ic],u(x,t)):```

However, if I add one line to convert the piecewise function above to piecewise, then pdsolve no longer gives an error. So the following input works

```restart;
pde:=diff(u(x,t),t)=k*diff(u(x,t),x\$2);
bc:=D[1](u)(0,t)=0,D[1](u)(L,t)=0:
assume(L>0):
ic:=u(x,0)=piecewise(0<x and x<=L/2,0,L/2<x and x<L,1):
ic:=convert(ic,piecewise,x):
sol:=pdsolve([pde,bc,ic],u(x,t)):```

Notice the extra line. Why does one have to convert piecewise to piecewise to make pdsolve accept the input?

sorry did not write down the error message and I am writing this from school library PC. But if you try the first case, you'll see the error.

## on pdsolve 2017, heat PDE, homogenous neumann B.C...

Asked by:

I am trying Maple 2017 pdsolve for heat PDE in 1D. It seems Maple can solve now  heat PDE with homogeneous dirichlet boundary conditions (good). But when I set the boundary conditions to homogeneous neumann B.C. instead, I get an answer when this B.C. is prescribed to the left side. When this B.C. is on the right side, I get an error. Which is strange.

I am newbie in Maple, so may be I am doing something wrong in the syntax?  In addition, the answer I get when homogeneous neumann B.C. is on the left side, does not match my hand solution, which I know is correct. I'll show this below.

First, here is the case where it works. homogeneous dirichlet boundary conditions on both sides:

```restart;
pde:=diff(u(x,t),t)=k*diff(u(x,t),x\$2);
bc:=u(0,t)=0,u(L,t)=0;
sol:=pdsolve([pde,bc]) assuming 0<L:
```

This answer is correct. Now when setting the right side to homogeneous neumann B.C. I get an error

```restart;
pde:=diff(u(x,t),t)=k*diff(u(x,t),x\$2);
bc:=u(0,t)=0,D[1](u)(L,t)=0;
pdsolve([pde,bc]) assuming 0<L;
```

I think may be it does not like `L` there in the B.C. But how else to tell it this B.C.? The above is the only syntax I know. And finally, when using homogeneous neumann B.C. on the left side, I get this result

```restart;
pde:=diff(u(x,t),t)=k*diff(u(x,t),x\$2);
bc:=D[1](u)(0,t)=0,u(L,t)=0;
pdsolve([pde,bc]) assuming 0<L;
```

The correct answer for this B.C. is

The answer should be series solution as well with eigenvalues. I think if I expand Maple solution in series may be I will get it to match my hand solution. I need to look at this more later.

my question is: Why do I get an error when homogeneous neumann B.C. is on the right side but not on the left side?

I suspect I am not entering the B.C. correctly? If so, How does one enter homogeneous neumann B.C. for this 1D heat PDE?

## Error, (in casesplit/K) this version of casesplit ...

Asked by:

I am learning pdsolve in Maple. When I try to solve a diffusion pde, I get this strange error message, and I do not understand what it means: This is using Maple 2017 on windows 7

```restart;
bc:=f(t,0)=0,f(t,1)=1;
ic:=f(0,x)=piecewise(x=0,1,0);
pde:=diff(f(t,x),t)=diff(f(t,x),x\$2);
pdsolve({pde,bc,ic},f(t,x));
```

The error is

Error, (in casesplit/K) this version of casesplit is not yet handling the function: piecewise

Am I writing the initial conditions (ic) wrong?  Maple help shows nothing on this. I think Maple does not like my initial conditions. But do not know how to correct it now.

What causes this error?

## How to solve a simple PDE?...

Asked by:

I want to solve the system of differential equations
sys :=
diff(x(t,s),t) = y(t,s),
diff(y(t,s),t) + x(t,s) = 0;

subject to the initial condition
ic := x(0,s) = a(s),
y(0,s) = b(s);

where a(s) and b(s) are given.

This looks like a system of PDEs but actually it is a system
of ODEs because there are no derivatives with respect to s.
It is easy to obtain the solution by hand:

x(t,s) = b(s)*sin(t) + a(s)*cos(t)
y(t,s) = b(s)*cos(t) - a(s)*sin(t)

I don't know how to get this in Maple, either through dsolve()
or pdsolve().

Actually both dsolve({sys}) and pdsolve({sys}) do return
the correct general solution, however dsolve({sys, ic})
or pdsolve({sys, ic}) produce no output.  Is there a trick
to make the latter work?

## Announcing Maple 2017

by: Maple 2017

Maple 2017 has launched!

Maple 2017 is the result of hard work by an enthusiastic team of developers and mathematicians.

As ever, we’re guided by you, our users. Many of the new features are of a result of your feedback, while others are passion projects that we feel you will find value in.

Here’s a few of my favourite enhancements. There’s far more that’s new - see What’s New in Maple 2017 to learn more.

MapleCloud Package Manager

Since it was first introduced in Maple 14, the MapleCloud has made thousands of Maple documents and interactive applications available through a web interface.

Maple 2017 completely refreshes the MapleCloud experience. Allied with a new, crisp, interface, you can now download and install user-created packages.

Simply open the MapleCloud interface from within Maple, and a mouse click later, you see a list of user-created packages, continuously updated via the Internet. Two clicks later, you’ve downloaded and installed a package.

This completely bypasses the traditional process of searching for and downloading a package, copying to the right folder, and then modifying libname in Maple. That was a laborious process, and, unless I was motivated, stopped me from installing packages.

The MapleCloud hosts a growing number of packages.

Many regular visitors to MaplePrimes are already familiar with Sergey Moiseev’s DirectSearch package for optimization, equation solving and curve fitting.

My fellow product manager, @DSkoog has written a package for grouping data into similar clusters (called ClusterAnalysis on the Package Manager)

Here’s a sample from a package I hacked together for downloading maps images using the Google Maps API (it’s called Google Maps and Geocoding on the Package Manager).

You’ll also find user-developed packages for exploring AES-based encryption, orthogonal series expansions, building Maple shell scripts and more.

Simply by making the process of finding and installing packages trivially easy, we’ve opened up a new world of functionality to users.

Maple 2017 also offers a simple method for package authors to upload workbook-based packages to the MapleCloud.

We’re engaging with many package authors to add to the growing list of packages on the MapleCloud. We’d be interested in seeing your packages, too!

Advanced Math

We’re committed to continually improving the core symbolic math routines. Here area few examples of what to expect in Maple 2017.

Resulting from enhancements to the Risch algorithm, Maple 2017 now computes symbolic integrals that were previously intractable

Groeber:-Basis uses a new implementation of the FGLM algorithm. The example below runs about 200 times faster in Maple 2017.

gcdex now uses a sparse primitive polynomial remainder sequence together.  For sparse structured problems the new routine is orders of magnitude faster. The example below was previously intractable.

The asympt and limit commands can now handle asymptotic cases of the incomplete Γ function where both arguments tend to infinity and their quotient remains finite.

Among several improvements in mathematical functions, you can now calculate and manipulate the four multi-parameter Appell functions.

Appel functions are of increasing importance in quantum mechanics, molecular physics, and general relativity.

pdsolve has seen many enhancements. For example, you can tell Maple that a dependent variable is bounded. This has the potential of simplifying the form of a solution.

Plot Builder

Plotting is probably the most common application of Maple, and for many years, you’ve been able to create these plots without using commands, if you want to.  Now, the re-designed interactive Plot Builder makes this process easier and better.

When invoked by a context menu or command on an expression or function, a panel slides out from the right-hand side of the interface.

Generating and customizing plots takes a single mouse click. You alter plot types, change formatting options on the fly and more.

To help you better learn Maple syntax, you can also display the actual plot command.

Password Protected Content

You can distribute password-protected executable content. This feature uses the workbook file format introduced with Maple 2016.

You can lock down any worksheet in a Workbook. But from any other worksheet, you can send (author-specified) parameters into the locked worksheet, and extract (author-specified) results.

Plot Annotations

You can now get information to pop up when you hover over a point or a curve on a plot.

In this application, you see the location and magnitude of an earthquake when you hover over a point

Here’s a ternary diagram of the color of gold-silver-copper alloys. If you let your mouse hover over the points, you see the composition of the points

Plot annotations may seem like a small feature, but they add an extra layer of depth to your visualizations. I’ve started using them all the time!

Engineering Portal

In my experience, if you ask an engineer how they prefer to learn, the vast majority of them will say “show me an example”. The significantly updated Maple Portal for Engineers does just that, incorporating many more examples and sample applications.  In fact, it has a whole new Application Gallery containing dozens of applications that solve concrete problems from different branches of engineering while illustrating important Maple techniques.

Designed as a starting point for engineers using Maple, the Portal also includes information on math and programming, interface features for managing your projects, data analysis and visualization tools, working with physical and scientific data, and a variety of specialized topics.

Geographic Data

You can now generate and customize world maps. This for example, is a choropleth of European fertility rates (lighter colors indicate lower fertility rates)

You can plot great circles that show the shortest path between two locations, show varying levels of detail on the map, and even experiment with map projections.

A new geographic database contains over one million locations, cross-referenced with their longitude, latitude, political designation and population.

The database is tightly linked to the mapping tools. Here, we ask Maple to plot the location of country capitals with a population of greater than 8 million and a longitude lower than 30.

There’s much more to Maple 2017. It’s a deep, rich release that has something for everyone.

Visit What’s New in Maple 2017 to learn more.

## Why the "pdsolve" of PDEtools package is not retur...

Asked by:

I am trying to solve system linear partial differential equations using command "pdsolve". I am surprised to see that the solution given by this command is not satisfying the system, instead, an additional constraint is obtained for an arbitrary function, is there something about "pdsolve" I am missing?

 >
 >
 (1)
 >
 (2)
 >
 (3)
 >
 (4)
 >
 (5)
 >

Download pdsolve_command.mw

## Is this a bug in pdsolve???...

Asked by:

Hi,

Why pdsolve is not correctly solving the following

where I get

but it is correctly solving the next one

where I get ???

I see that it is due to the fact that the variable x is multiplied by a constant, but why is not maple able to manage that?

Thanks for your help,

Javier

## Partial differential equation...

Asked by:

hi
I want to solve a pde equation:

```equa1 := diff(u(x,y), x, x)-y(1+x) = 0;

# with codition:

con:=u(0,y) = 0, (D(u[x]))(0,y) = 0;
```

the anwer must be :    u(x,y)= y(x2/2  + x3/6)
How can i solve that with maple?

Please excuse my bad English
thanks

## Parabolic PDE ...

Asked by:

I am looking for a numerical solver for a parabolic PDE (up to 2nd order derivatives but no mixed ones) on the spatio-temporal domain [X x Y x T], either as an external package or as MAPLE code.

I have coded the method of lines on the domain [X x T] and indeed also used pdsolve as a check for that case. However, pdsolve (numerical) cannot solve the PDEs on the domain [X x Y x T].  The run times and memory requirements for the latter case would of course be significantly greater.

I am about to code up the method of lines (in MAPLE) on the domain [X x Y x T], but am wondering whether there exist external FORTRAN or C code packages that would be faster if called up in MAPLE and whose results would then be post-pocessed in MAPLE.

Does anyone have any suggestions?

MRB

## how i can solve following equation?...

Asked by:

hi--how i can solve following equation?

thanks

Eq.mw

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/Eq.mw .

Download Eq.mw

## PDE numeric BVP how to solve and guess initial con...

Asked by:

Please help me on this :

 >
 >
 >
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Download untitle_2_(1).mw

## Need numerical solution of PDE...

Asked by:

Dears

Hope you would be fine. I want to solve the following PDEs by numerically for v[nf]=alpha[nf]=Ec=mu[nf]=C=1 and Pr=6.2

Eq1 := diff(u(x, t), t) = v[nf]*(diff(u(x, t), x, x));

Eq2 := diff(u(x, t), t) = alpha[nf]*(diff(theta(x, t), x, x))/Pr+Ec*mu[nf]*C*(diff(u(x, t), x))^2;

ICs := u(x, 0) = 0, theta(x, 0);

BCs := u(0, t) = 1, theta(0, t) = 1, u(10, t) = 0, theta(10, t) = 0;

and find the values of (diff(u(0, t), x))/(1-phi)^2.5 for different values of phi. Thanks in advace

With my best regards and sincerely.

Muhammad Usman

School of Mathematical Sciences
Peking University, Beijing, China

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