Posts - MaplePrimes

Linkage mechanisms

by: one man 1360 Product: Maple 15

July 10 2016

1 5

        General description of the method of solving underdetermined systems of equations. As a particular application of the idea proposed a universal method kinematic analysis for all kinds of spatial and planar link mechanisms with any number degrees of freedom.  The method can be used for powerful CAD linkages.
        http://www.maplesoft.com/applications/view.aspx?SID=154228

Points on the coordinate plane

by: Alsu 445 Product: Maple

July 08 2016

2 0

Points on the coordinate plane

(Guidance manual for the 6th class)

Changing the initial coordinates and going through the entire program first, we get a new picture-task

Point_on_co-plane.mws

And Another Coordinate_plane.mws

Coordinate axis - for use at lessons

by: Alsu 445 Product: Maple

July 08 2016

3 0

Coordinate axis

6th class (in Russia)

Guidance manual for use at lessons (at school)

Coordinate_line_lesson.mws

(It is possible bad English)

Aggregate Statistics on DataFrames

by: Daniel Skoog 1771 Product: Maple 2016

July 08 2016

3 2

Aggregate statistics are calculated by splitting the rows of a DataFrame by each factor in a given column into subsets and computing summary statistics for each of these subsets.

The following is a short example of how the Aggregate command is used to compute aggregate statistics for a DataFrame with housing data:

To begin, we construct a DataFrame with housing data: The first column has number of bedrooms, the second has the area in square feet, the third has price.

bedrooms := <3, 4, 2, 4, 3, 2, 2, 3, 4, 4, 2, 4, 4, 3, 3>:
area := <1130, 1123, 1049, 1527, 907, 580, 878, 1075, 
         1040, 1295, 1100, 995, 908, 853, 856>:
price := <114700, 125200, 81600, 127400, 88500, 59500, 96500, 113300, 
          104400, 136600, 80100, 128000, 115700, 94700, 89400>:
HouseSalesData := DataFrame([bedrooms, area, price], columns = [Bedrooms, Area, Price]);

Note that the Bedrooms column has three distinct levels: 2, 3, and 4.

convert(HouseSalesData[Bedrooms], set);

The following returns the mean of all other columns for each distinct level in the column, Bedrooms:

Aggregate(HouseSalesData, Bedrooms);

Adding the columns option controls which columns are returned.

Aggregate(HouseSalesData, Bedrooms, columns = [Price])

Additionally, the tally option returns a tally for each of the levels.

Aggregate(HouseSalesData, Bedrooms, tally)

The function option allows for the specification of any command that can be applied to a DataSeries. For example, the Statistics:-Median command computes the median for each of the levels of Bedrooms.

Aggregate(HouseSalesData, Bedrooms, function = Statistics:-Median);

By default, Aggregate uses the SplitByColumn command to creates a separate sub-DataFrame for every discrete level in the column given by bycolumn.

with(Statistics);

ByRooms := SplitByColumn(HouseSalesData, Bedrooms);

We can create box plots of the price for subgroups of sales defined by number of bedrooms.

BoxPlot( map( (m)->m[Price], ByRooms), 
             deciles=false, 
             datasetlabels=["2 bdrms", "3 bdrms", "4 bdrms"], 
             color=["Red", "Purple", "Blue"]);

I have recorded a short video that walks through this example here:

The worksheet for this example can be downloaded here: Aggregate.mw

The definition of the derivative of the function

by: Alsu 445 Product: Maple

July 06 2016

5 11

diff_first.mws

New version of the Multivariate Calculus Study...

by: eithne 2072 Products: Maple , Maple Toolboxes

June 29 2016

2 0

We have just released a new version of the Multivariate Calculus Study Guide. It provides a new section on Vector Calculus, with over 100 additional worked problems, and makes extensive use of Maple’s Clickable Math tools as well as commands.

Existing study guide customers can get the new content via a free update, available through the Check for Updates system or from our website. See Multivariate Calculus Study Guide 2016 Update for details.

For more information about this guide, including a full table of contents, visit Multivariate Calculus Study Guide.

eithne

PDE solutions: when are they "general"?

by: ecterrab 14640 Product: Maple

June 27 2016

4 0

Hi,
The latest update to the differential equations Maple libraries (this week, can be downloaded from the Maplesoft R&D webpage for Differential Equations and Mathematical functions) includes new functionality in pdsolve, regarding whether the solution for a PDE or PDE system is or not a general solution.

In brief, a general solution of a PDE in 1 unknown, that has differential order N, and where the unknown depends on M independent variables, involves N arbitrary functions of M-1 arguments. It is not entirely evident how to extend this definition in the case of a coupled, possibly nonlinear PDE system. However, using differential algebra techniques (automatically used by pdsolve when tackling a PDE system), that extension to define a general solution for a DE system is possible, and also when the system involves ODEs and PDEs, and/or algebraic (that is, non-differential) equations, and/or inequations of the form involving the unknowns, and all of this in the presence of mathematical functions (based on the use of Maple's PDEtools:-dpolyform). This is a very nice case were many different advanced developments come together to naturally solve a problem that otherwise would be rather difficult.

The issues at the center of this Maple development/post are then:

a) How do you know whether a PDE or PDE system solution returned is a general solution?

b) How could you indicate to pdsolve that you are only interested in a general PDE or PDE system solution?

The answer to a) is now always present in the last line of the userinfo. So input infolevel[pdsolve] := 3 before calling pdsolve, and check what the last line of the userinfo displayed tells.

The answer to b) is a new option, generalsolution, implemented in pdsolve so that it either returns a general solution or otherwise it returns NULL. If you do not use this new option, then pdsolve works as always: first it tries to compute a general solution and if it fails in doing that it tries to compute a particular solution by separating the variables in different ways, or computing a traveling wave solution or etc. (a number of other well known methods).

The examples that follow are from the help page pdsolve,system, and show both the new userinfo telling whether the solution returned is a general one and the option generalsolution at work.The examples are all of differential equation systems but the same userinfos and generalsolution option work as well in the case of a single PDE.

Example 1.

Solve the determining PDE system for the infinitesimals of the symmetry generator of example 11 from Kamke's book . Tell whether the solution computed is or not a general solution.

>

(1.1)

The PDE system satisfied by the symmetries of Kamke's ODE example number 11 is

>

sys__1 := [diff(xi(x, y), y, y) = 0, diff(eta(x, y), y, y)-2*(diff(xi(x, y), y, x)) = 0, 3*x^r*y^n*(diff(xi(x, y), y))*a+2*(diff(eta(x, y), y, x))-(diff(xi(x, y), x, x)) = 0, 2*(diff(xi(x, y), x))*x^r*y^n*a-x^r*y^n*(diff(eta(x, y), y))*a+eta(x, y)*a*x^r*y^n*n/y+xi(x, y)*a*x^r*r*y^n/x+diff(eta(x, y), x, x) = 0]

This is a second order linear PDE system, with two unknowns and four equations. Its general solution is given by the following, where we now can tell that the solution is a general one by reading the last line of the userinfo. Note that because the system is overdetermined, a general solution in this case does not involve any arbitrary function

>

-> Solving ordering for the dependent variables of the PDE system: [xi(x,y), eta(x,y)]

-> Solving ordering for the independent variables (can be changed using the ivars option): [x, y]
tackling triangularized subsystem with respect to xi(x,y)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to eta(x,y)
<- Returning a *general* solution

${eta(x, y) = -_C1*y*(r+2)/(n-1), xi(x, y) = _C1*x}$

(1.2)

Next we indicate to pdsolve that and are parameters of the problem, and that we want a solution for , making more difficult to identify by eye whether the solution returned is or not a general one. Again the last line of the userinfo tells that pdsolve's solution is indeed a general one

>

[diff(diff(xi(x, y), y), y) = 0, diff(diff(eta(x, y), y), y)-2*(diff(diff(xi(x, y), x), y)) = 0, 3*x^r*y^n*(diff(xi(x, y), y))*a+2*(diff(diff(eta(x, y), x), y))-(diff(diff(xi(x, y), x), x)) = 0, 2*(diff(xi(x, y), x))*x^r*y^n*a-x^r*y^n*(diff(eta(x, y), y))*a+eta(x, y)*a*x^r*y^n*n/y+xi(x, y)*a*x^r*r*y^n/x+diff(diff(eta(x, y), x), x) = 0, n <> 1]

(1.3)

>	$`sol__1.1` := pdsolve(`sys__1.1`, parameters = {n, r})$

-> Solving ordering for the dependent variables of the PDE system: [r, n, xi(x,y), eta(x,y)]

-> Solving ordering for the independent variables (can be changed using the ivars option): [x, y]
tackling triangularized subsystem with respect to r
tackling triangularized subsystem with respect to n
tackling triangularized subsystem with respect to xi(x,y)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to eta(x,y)
tackling triangularized subsystem with respect to r
tackling triangularized subsystem with respect to n
tackling triangularized subsystem with respect to xi(x,y)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to eta(x,y)
tackling triangularized subsystem with respect to r
tackling triangularized subsystem with respect to n
tackling triangularized subsystem with respect to xi(x,y)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to eta(x,y)
tackling triangularized subsystem with respect to n
tackling triangularized subsystem with respect to xi(x,y)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to eta(x,y)
tackling triangularized subsystem with respect to n
tackling triangularized subsystem with respect to xi(x,y)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to eta(x,y)
tackling triangularized subsystem with respect to xi(x,y)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to eta(x,y)
<- Returning a *general* solution

${n = 2, r = -5, eta(x, y) = y*(_C1*x+3*_C2), xi(x, y) = x*(_C1*x+_C2)}, {n = 2, r = -20/7, eta(x, y) = -(2/343)*(-6*_C1*x^2-98*x^(8/7)*_C1*a*y-147*_C2*a*x*y)/(x*a), xi(x, y) = _C1*x^(8/7)+_C2*x}, {n = 2, r = -15/7, eta(x, y) = -(1/343)*(-49*_C2*a*x*y-147*x^(6/7)*_C1*a*y+12*_C1*x)/(x*a), xi(x, y) = _C1*x^(6/7)+_C2*x}, {n = 2, r = r, eta(x, y) = -_C1*y*(r+2), xi(x, y) = _C1*x}, {n = -r-3, r = r, eta(x, y) = ((_C1*x+_C2)*r+4*_C1*x+2*_C2)*y/(r+4), xi(x, y) = x*(_C1*x+_C2)}, {n = n, r = r, eta(x, y) = -_C1*y*(r+2)/(n-1), xi(x, y) = _C1*x}$

(1.4)

>

(1.5)

Example 2.

Compute the solution of the following (linear) overdetermined system involving two PDEs, three unknown functions, one of which depends on 2 variables and the other two depend on only 1 variable.

>

sys__2 := [-(diff(F(r, s), r, r))+diff(F(r, s), s, s)+diff(H(r), r)+diff(G(s), s)+s = 0, diff(F(r, s), r, r)+2*(diff(F(r, s), r, s))+diff(F(r, s), s, s)-(diff(H(r), r))+diff(G(s), s)-r = 0]

The solution for the unknowns G, H, is given by the following expression, were again determining whether this solution, that depends on 3 arbitrary functions, , is or not a general solution, is non-obvious.

>

-> Solving ordering for the dependent variables of the PDE system: [F(r,s), H(r), G(s)]

-> Solving ordering for the independent variables (can be changed using the ivars option): [r, s]
tackling triangularized subsystem with respect to F(r,s)
First set of solution methods (general or quasi general solution)
Trying differential factorization for linear PDEs ...
differential factorization successful.
First set of solution methods successful
tackling triangularized subsystem with respect to H(r)
tackling triangularized subsystem with respect to G(s)
<- Returning a *general* solution

${F(r, s) = _F1(s)+_F2(r)+_F3(s-r)-(1/12)*r^2*(r-3*s), G(s) = -(diff(_F1(s), s))-(1/4)*s^2+_C2, H(r) = diff(_F2(r), r)-(1/4)*r^2+_C1}$

(1.6)

>

(1.7)

Example 3.

Compute the solution of the following nonlinear system, consisting of Burger's equation and a possible potential.

>

sys__3 := [diff(u(x, t), t)+2*u(x, t)*(diff(u(x, t), x))-(diff(u(x, t), x, x)) = 0, diff(v(x, t), t) = -v(x, t)*(diff(u(x, t), x))+v(x, t)*u(x, t)^2, diff(v(x, t), x) = -u(x, t)*v(x, t)]

We see that in this case the solution returned is not a general solution but two particular ones; again the information is in the last line of the userinfo displayed

>

-> Solving ordering for the dependent variables of the PDE system: [v(x,t), u(x,t)]

-> Solving ordering for the independent variables (can be changed using the ivars option): [x, t]
tackling triangularized subsystem with respect to v(x,t)
tackling triangularized subsystem with respect to u(x,t)
First set of solution methods (general or quasi general solution)
Second set of solution methods (complete solutions)
Trying methods for second order PDEs
Third set of solution methods (simple HINTs for separating variables)
PDE linear in highest derivatives - trying a separation of variables by *
HINT = *
Fourth set of solution methods
Trying methods for second order linear PDEs
Preparing a solution HINT ...
Trying HINT = _F1(x)*_F2(t)
Fourth set of solution methods
Preparing a solution HINT ...
Trying HINT = _F1(x)+_F2(t)
Trying travelling wave solutions as power series in tanh ...
* Using tau = tanh(t*C[2]+x*C[1]+C[0])
* Equivalent ODE system: {C[1]^2*(tau^2-1)^2*diff(diff(u(tau),tau),tau)+(2*C[1]^2*(tau^2-1)*tau+2*u(tau)*C[1]*(tau^2-1)+C[2]*(tau^2-1))*diff(u(tau),tau)}
* Ordering for functions: [u(tau)]
* Cases for the upper bounds: [[n[1] = 1]]
* Power series solution [1]: {u(tau) = tau*A[1,1]+A[1,0]}
* Solution [1] for {A[i, j], C[k]}: [[A[1,1] = 0], [A[1,0] = -1/2*C[2]/C[1], A[1,1] = -C[1]]]
travelling wave solutions successful.
tackling triangularized subsystem with respect to v(x,t)
First set of solution methods (general or quasi general solution)
Trying differential factorization for linear PDEs ...
Trying methods for PDEs "missing the dependent variable" ...
Second set of solution methods (complete solutions)
Trying methods for second order PDEs
Third set of solution methods (simple HINTs for separating variables)
PDE linear in highest derivatives - trying a separation of variables by *
HINT = *
Fourth set of solution methods
Trying methods for second order linear PDEs
Preparing a solution HINT ...
Trying HINT = _F1(x)*_F2(t)
Third set of solution methods successful
tackling triangularized subsystem with respect to u(x,t)
<- Returning a solution that *is not the most general one*

${u(x, t) = -_C2*tanh(_C2*x+_C3*t+_C1)-(1/2)*_C3/_C2, v(x, t) = 0}, {u(x, t) = -_c[1]^(1/2)*((exp(_c[1]^(1/2)*x))^2*_C1-_C2)/((exp(_c[1]^(1/2)*x))^2*_C1+_C2), v(x, t) = _C3*exp(_c[1]*t)*_C1*exp(_c[1]^(1/2)*x)+_C3*exp(_c[1]*t)*_C2/exp(_c[1]^(1/2)*x)}$

(1.8)

>

(1.9)

This example is also good for illustrating the other related new feature: one can now request to pdsolve to only compute a general solution (it will return NULL if it cannot achieve that). Turn OFF userinfos and try with this example

>

This returns NULL:

>

Example 4.

Another where the solution returned is particular, this time for a linear system, conformed by 38 PDEs, also from differential equation symmetry analysis

>

There are 38 coupled equations

>

(1.10)

When requesting a general solution pdsolve returns NULL:

>

A solution that is not a general one, is however computed by default if calling pdsolve without the generalsolution option. In this case again the last line of the userinfo tells that the solution returned is not a general solution

>

(1.11)

>

-> Solving ordering for the dependent variables of the PDE system: [eta[1](x,y,z,t,u), xi[1](x,y,z,t,u), xi[2](x,y,z,t,u), xi[3](x,y,z,t,u), xi[4](x,y,z,t,u)]

-> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, y, z, u]
tackling triangularized subsystem with respect to eta[1](x,y,z,t,u)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
-> Solving ordering for the dependent variables of the PDE system: [_F1(x,y,z,t), _F2(x,y,z,t)]
-> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, y, z, u]
tackling triangularized subsystem with respect to _F1(x,y,z,t)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
-> Solving ordering for the dependent variables of the PDE system: [_F3(x,y,z), _F4(x,y,z)]
-> Solving ordering for the independent variables (can be changed using the ivars option): [x, y, z, t]
tackling triangularized subsystem with respect to _F3(x,y,z)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to _F4(x,y,z)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
-> Solving ordering for the dependent variables of the PDE system: [_F5(y,z), _F6(y,z)]
-> Solving ordering for the independent variables (can be changed using the ivars option): [y, z, x]
tackling triangularized subsystem with respect to _F5(y,z)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to _F6(y,z)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
-> Solving ordering for the dependent variables of the PDE system: [_F7(z), _F8(z)]
-> Solving ordering for the independent variables (can be changed using the ivars option): [z, y]
tackling triangularized subsystem with respect to _F7(z)
tackling triangularized subsystem with respect to _F8(z)
tackling triangularized subsystem with respect to _F2(x,y,z,t)
First set of solution methods (general or quasi general solution)
Trying differential factorization for linear PDEs ...
Trying methods for PDEs "missing the dependent variable" ...
Second set of solution methods (complete solutions)
Third set of solution methods (simple HINTs for separating variables)
PDE linear in highest derivatives - trying a separation of variables by *
HINT = *
Fourth set of solution methods
Preparing a solution HINT ...
Trying HINT = _F3(x)*_F4(y)*_F5(z)*_F6(t)
Third set of solution methods successful

tackling triangularized subsystem with respect to xi[1](x,y,z,t,u)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
-> Solving ordering for the dependent variables of the PDE system: [_F1(x,z,t), _F2(x,z,t)]
-> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, z, y]
tackling triangularized subsystem with respect to _F1(x,z,t)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to _F2(x,z,t)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful

-> Solving ordering for the dependent variables of the PDE system: [_F3(x,t), _F4(x,t)]
-> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, z]
tackling triangularized subsystem with respect to _F3(x,t)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to _F4(x,t)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
-> Solving ordering for the dependent variables of the PDE system: [_F5(x), _F6(x)]
-> Solving ordering for the independent variables (can be changed using the ivars option): [x, t]
tackling triangularized subsystem with respect to _F5(x)
tackling triangularized subsystem with respect to _F6(x)
tackling triangularized subsystem with respect to xi[2](x,y,z,t,u)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
-> Solving ordering for the dependent variables of the PDE system: [_F1(t), _F2(t)]
-> Solving ordering for the independent variables (can be changed using the ivars option): [t, z]
tackling triangularized subsystem with respect to _F1(t)
tackling triangularized subsystem with respect to _F2(t)
tackling triangularized subsystem with respect to xi[3](x,y,z,t,u)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
tackling triangularized subsystem with respect to xi[4](x,y,z,t,u)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
First set of solution methods successful
<- Returning a solution that *is not the most general one*

${eta[1](x, y, z, t, u) = (_C13*(_C10*(exp(_c[3]^(1/2)*z))^2+_C11)*(_C8*(exp(_c[2]^(1/2)*y))^2+_C9)*(_C6*(exp(_c[1]^(1/2)*x))^2+_C7)*cos((-_c[1]-_c[2]-_c[3])^(1/2)*t)+_C12*(_C10*(exp(_c[3]^(1/2)*z))^2+_C11)*(_C8*(exp(_c[2]^(1/2)*y))^2+_C9)*(_C6*(exp(_c[1]^(1/2)*x))^2+_C7)*sin((-_c[1]-_c[2]-_c[3])^(1/2)*t)+u*exp(_c[1]^(1/2)*x)*exp(_c[2]^(1/2)*y)*exp(_c[3]^(1/2)*z)*(_C1*t+_C2*x+_C3*y+_C4*z+_C5))/(exp(_c[1]^(1/2)*x)*exp(_c[2]^(1/2)*y)*exp(_c[3]^(1/2)*z)), xi[1](x, y, z, t, u) = -(1/2)*_C2*x^2+(1/2)*(-2*_C1*t-2*_C3*y-2*_C4*z+2*_C17)*x+(1/2)*(-t^2+y^2+z^2)*_C2+_C16*t+_C15*z+_C14*y+_C18, xi[2](x, y, z, t, u) = -(1/2)*_C3*y^2+(1/2)*(-2*_C1*t-2*_C2*x-2*_C4*z+2*_C17)*y+(1/2)*(-t^2+x^2+z^2)*_C3+_C20*t+_C19*z-_C14*x+_C21, xi[3](x, y, z, t, u) = -(1/2)*_C4*z^2+(1/2)*(-2*_C1*t-2*_C2*x-2*_C3*y+2*_C17)*z+(1/2)*(-t^2+x^2+y^2)*_C4+_C22*t-_C19*y-_C15*x+_C23, xi[4](x, y, z, t, u) = -(1/2)*_C1*t^2+(1/2)*(-2*_C2*x-2*_C3*y-2*_C4*z+2*_C17)*t+(1/2)*(-x^2-y^2-z^2)*_C1+_C20*y+_C22*z+_C16*x+_C24}$

(1.12)

>

(1.13)

Example 5.

Finally, the new userinfos also tell whether a solution is or not a general solution when working with PDEs that involve anticommutative variables set using the Physics package

>

(1.14)

Set first and as suffixes for variables of type/anticommutative (see Setup )

>

$[anticommutativeprefix = {Q, _lambda, theta}]$

(1.15)

A PDE system example with two unknown anticommutative functions of four variables, two commutative and two anticommutative; to avoid redundant typing in the input that follows and redundant display of information on the screen let's use PDEtools:-diff_table PDEtools:-declare

>

(1.16)

>

(1.17)

Consider the system formed by these two PDEs (because of the q diff_table just defined, we can enter derivatives directly using the function's name indexed by the differentiation variables)

>

(1.18)

>

(1.19)

The solution returned for this system is indeed a general solution

>

-> Solving ordering for the dependent variables of the PDE system: [_F4(x,y), _F2(x,y), _F3(x,y)]

-> Solving ordering for the independent variables (can be changed using the ivars option): [x, y]
tackling triangularized subsystem with respect to _F4(x,y)
tackling triangularized subsystem with respect to _F2(x,y)
tackling triangularized subsystem with respect to _F3(x,y)
First set of solution methods (general or quasi general solution)
Trying simple case of a single derivative.
HINT = _F6(x)+_F5(y)
Trying HINT = _F6(x)+_F5(y)
HINT is successful
First set of solution methods successful
<- Returning a *general* solution

(1.20)

>

This solution involves an anticommutative constant , analogous to the commutative constants where n is an integer.

Download PDE_general_solutions.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

New developments on exact solutions for PDEs with...

by: ecterrab 14640 Product: Maple

June 27 2016

4 3

Hi

New developments (after the release of Maple 2016) happened in the project on exact solutions for "Partial Differential Equations & Boundary Conditions". This is work in collaboration with Katherina von Bulow and the improvements are of wide range, representing a noticeable step forward in the capabilities of the Maple system for this kind of problem. As usual, these improvements can be installed in current Maple 2016 by downloading the updated library from the Maplesoft R&D webpage for Differential Equations and Mathematical functions (the update is distributed merged with the updates of the Physics package)

The improvements cover:

•	PDE&BC in semi-infinite domains for which a bounded solution is sought

•	PDE & BC problems in bounded spatial domains via eigenfunction (Fourier) expansions

•	Implementation of another algebraic method for tackling linear PDE & BC

•	Improvements in solving PDE & BC solutions by first finding the PDE's general solution.

•	Improvements in solving PDE & BC problems by using a Fourier transform.

•	PDE & BC problems that used to require the option HINT = `+` are now solved automatically

What follows is a set of examples solved now with these new developments, organized in sections according to the kind of problem. Where relevant, the sections include a subsection on "How it works step by step".

PDE&BC in semi-infinite domains for which a bounded solution is sought can now also be solved via Laplace transforms

Maple is now able to solve more PDE&BC problems via Laplace transforms.

How it works: Laplace transforms act to change derivatives with respect to one of the independent variables of the domain into multiplication operations in the transformed domain. After applying a Laplace transform to the original problem, we can simplify the problem using the transformed BC, then solve the problem in the transformed domain, and finally apply the inverse Laplace transform to arrive at the final solution. It is important to remember to give pdsolve any necessary restrictions on the variables and constants of the problem, by means of the "assuming" command.

A new feature is that we can now tell pdsolve that the dependent variable is bounded, by means of the optional argument HINT = boundedseries.

>

Consider the problem of a falling cable lying on a table that is suddenly removed (cf. David J. Logan's Applied Partial Differential Equations p.115).

>

If we ask pdsolve to solve this problem without the condition of boundedness of the solution, we obtain:

>

u(x, t) = -invlaplace(exp(s*x/c)*_F1(s), s, t)+invlaplace(exp(-s*x/c)*_F1(s), s, t)-(1/2)*g*t^2+invlaplace(exp(s*x/c)/s^3, s, t)*g

(1.1)

New: If we now ask for a bounded solution, by means of the option HINT = boundedseries, pdsolve simplifies the problem accordingly.

>

u(x, t) = (1/2)*g*(Heaviside(t-x/c)*(c*t-x)^2-c^2*t^2)/c^2

(1.2)

And we can check this answer against the original problem, if desired:

>

(1.3)

How it works, step by step

Let us see the process this problem undergoes to be solved by pdsolve, step by step.

First, the Laplace transform is applied to the PDE:

>

s^2*laplace(u(x, t), t, s)-(D[2](u))(x, 0)-s*u(x, 0)-c^2*(diff(diff(laplace(u(x, t), t, s), x), x))+g/s

(1.1.1)

and the result is simplified using the initial conditions:

>	$simplified_transformed_PDE := eval(transformed_PDE, {iv[1]})$

s^2*laplace(u(x, t), t, s)-c^2*(diff(diff(laplace(u(x, t), t, s), x), x))+g/s

(1.1.2)

Next, we call the function "laplace(u(x,t),t,s)" by the new name U:

>

(1.1.3)

And this equation, which is really an ODE, is solved:

>

U(x, s) = exp(-s*x/c)*_F2(s)+exp(s*x/c)*_F1(s)-g/s^3

(1.1.4)

Now, since we want a BOUNDED solution, the term with the positive exponential must be zero, and we are left with:

>

U(x, s) = exp(-s*x/c)*_F2(s)-g/s^3

(1.1.5)

Now, the initial solution must also be satisfied. Here it is, in the transformed domain:

>

(1.1.6)

Or, in the new variable U,

>

(1.1.7)

And by applying it to bounded_solution_U, we find the relationship

>

(_F2(s)*s^3-g)/s^3 = 0

(1.1.8)

>

(1.1.9)

so that our solution now becomes

>

U(x, s) = exp(-s*x/c)*g/s^3-g/s^3

(1.1.10)

to which we now apply the inverse Laplace transform to obtain the solution to the problem:

>

u(x, t) = (1/2)*g*(-t^2+Heaviside(t-x/c)*(c*t-x)^2/c^2)

(1.1.11)

Four other related examples

A few other examples:

>

pde[2] := diff(u(x, t), t, t) = c^2*(diff(u(x, t), x, x))

>

(1.2.1)

>

(1.2.2)

>

(1.2.3)

>

(1.2.4)

>

(1.2.5)

>

(1.2.6)

The following is an example from page 76 in Logan's book:

>

pde[5] := diff(u(x, t), t) = diff(u(x, t), x, x)

>

u(x, t) = (1/2)*x*(int(f(_U1)*exp(-x^2/(4*t-4*_U1))/(t-_U1)^(3/2), _U1 = 0 .. t))/Pi^(1/2)

(1.2.7)

More PDE&BC problems in bounded spatial domains can now be solved via eigenfunction (Fourier) expansions

The code for solving PDE&BC problems in bounded spatial domains has been expanded. The method works by separating the variables by product, so that the problem is transformed into an ODE system (with initial and/or boundary conditions) problem, one of which is a Sturm-Liouville problem (a type of eigenvalue problem) which has infinitely many solutions - hence the infinite series representation of the solutions.

>

Here is a simple example for the heat equation:

>

pde__6 := diff(u(x, t), t) = k*(diff(u(x, t), x, x)); iv__6 := u(0, t) = 0, u(l, t) = 0

>

u(x, t) = Sum(_C1*sin(_Z1*Pi*x/l)*exp(-k*Pi^2*_Z1^2*t/l^2), _Z1 = 1 .. infinity)

(2.1)

>

(2.2)

Now, consider the displacements of a string governed by the wave equation, where c is a constant (cf. Logan p.28).

>

pde__7 := diff(u(x, t), t, t) = c^2*(diff(u(x, t), x, x))

>

u(x, t) = Sum(sin(_Z2*Pi*x/l)*(sin(c*_Z2*Pi*t/l)*_C1+cos(c*_Z2*Pi*t/l)*_C5), _Z2 = 1 .. infinity)

(2.3)

>

(2.4)

Another wave equation problem (cf. Logan p.130):

>

pde__8 := diff(u(x, t), t, t)-c^2*(diff(u(x, t), x, x)) = 0; iv__8 := u(0, t) = 0, (D[2](u))(x, 0) = 0, (D[1](u))(l, t) = 0, u(x, 0) = f(x)

>

u(x, t) = Sum(2*(Int(f(x)*sin((1/2)*Pi*(2*_Z3+1)*x/l), x = 0 .. l))*sin((1/2)*Pi*(2*_Z3+1)*x/l)*cos((1/2)*c*Pi*(2*_Z3+1)*t/l)/l, _Z3 = 1 .. infinity)

(2.5)

>

(2.6)

Here is a problem with periodic boundary conditions (cf. Logan p.131). The function stands for the concentration of a chemical dissolved in water within a tubular ring of circumference . The initial concentration is given by , and the variable is the arc-length parameter that varies from 0 to .

>

u(x, t) = (1/2)*_C8+Sum(((Int(f(x)*sin(_Z4*Pi*x/l), x = 0 .. 2*l))*sin(_Z4*Pi*x/l)+(Int(f(x)*cos(_Z4*Pi*x/l), x = 0 .. 2*l))*cos(_Z4*Pi*x/l))*exp(-M*Pi^2*_Z4^2*t/l^2)/l, _Z4 = 1 .. infinity)

(2.7)

>

(2.8)

The following problem is for heat flow with both boundaries insulated (cf. Logan p.166, 3rd edition)

>

u(x, t) = Sum(2*(Int(f(x)*cos(_Z6*Pi*x/l), x = 0 .. l))*cos(_Z6*Pi*x/l)*exp(-k*Pi^2*_Z6^2*t/l^2)/l, _Z6 = 1 .. infinity)

(2.9)

>

(2.10)

This is a problem in a bounded domain with the presence of a source. A source term represents an outside influence in the system and leads to an inhomogeneous PDE (cf. Logan p.149):

>

u(x, t) = Int(Sum(2*(Int(p(x, tau1)*sin(_Z7*x), x = 0 .. Pi))*sin(_Z7*x)*sin(c*_Z7*(t-tau1))/(Pi*_Z7*c), _Z7 = 1 .. infinity), tau1 = 0 .. t)

(2.11)

Current pdetest is unable to verify that this solution cancells the mainly because it currently fails in identifying that there is a fourier expansion in it, but its subroutines for testing the boundary conditions work well with this problem

>

>	$pdetest_BC({ans__11}, [iv__11], [u(x, t)])$

(2.12)

Consider a heat absorption-radiation problem in the bounded domain :

>

pde__12 := diff(u(x, t), t) = diff(u(x, t), x, x)

>

u(x, t) = (1/2)*_C8+Sum(((Int(f(x)*cos((1/2)*_Z8*Pi*x), x = 0 .. 2))*cos((1/2)*_Z8*Pi*x)+(Int(f(x)*sin((1/2)*_Z8*Pi*x), x = 0 .. 2))*sin((1/2)*_Z8*Pi*x))*exp(-(1/4)*Pi^2*_Z8^2*t), _Z8 = 1 .. infinity)

(2.13)

>

(2.14)

Consider the nonhomogeneous wave equation problem (cf. Logan p.213, 3rd edition):

>

pde__13 := diff(u(x, t), t, t) = A*x+diff(u(x, t), x, x)

>

u(x, t) = Int(Sum(2*A*(Int(x*sin(Pi*_Z9*x), x = 0 .. 1))*sin(Pi*_Z9*x)*sin(Pi*_Z9*(t-tau1))/(Pi*_Z9), _Z9 = 1 .. infinity), tau1 = 0 .. t)

(2.15)

>	$pdetest_BC({ans__13}, [iv__13], [u(x, t)])$

(2.16)

Consider the following Schrödinger equation with zero potential energy (cf. Logan p.30):

>

pde__14 := I*h*(diff(f(x, t), t)) = -h^2*(diff(f(x, t), x, x))/(2*m)

>

f(x, t) = Sum(_C1*sin(_Z10*Pi*x/d)*exp(-((1/2)*I)*h*Pi^2*_Z10^2*t/(d^2*m)), _Z10 = 1 .. infinity)

(2.17)

>

(2.18)

Another method has been implemented for linear PDE&BC

This method is for problems of the form

or

", w(`x__i`,0) = f(`x__i`), (&PartialD;w)/(&PartialD;t)() ? ()|() ? (t=0) =g(`x__i`)"

where M is an arbitrary linear differential operator of any order which only depends on the spatial variables .

Here are some examples:

>

pde__15 := diff(w(x1, x2, x3, t), t)-(diff(w(x1, x2, x3, t), x2, x1))-(diff(w(x1, x2, x3, t), x3, x1))-(diff(w(x1, x2, x3, t), x3, x3))+diff(w(x1, x2, x3, t), x3, x2) = 0

>

(3.1)

>

(3.2)

Here are two examples for which the derivative with respect to t is of the second order, and two initial conditions are given:

>

pde__16 := diff(w(x1, x2, x3, t), t, t) = diff(w(x1, x2, x3, t), x2, x1)+diff(w(x1, x2, x3, t), x3, x1)+diff(w(x1, x2, x3, t), x3, x3)-(diff(w(x1, x2, x3, t), x3, x2))

>

(3.3)

>

(3.4)

>

pde__17 := diff(w(x1, x2, x3, t), t, t) = diff(w(x1, x2, x3, t), x2, x1)+diff(w(x1, x2, x3, t), x3, x1)+diff(w(x1, x2, x3, t), x3, x3)-(diff(w(x1, x2, x3, t), x3, x2))

>

w(x1, x2, x3, t) = (1/2)*t^4*x1+t^2*x1^3+3*t^2*x1^2*x3+x1^3*x3^2+(1/6)*t^3-t*x2*x3+cos(x1)*t+sin(x1)

(3.5)

>

(3.6)

More PDE&BC problems are now solved via first finding the PDE's general solution.

The following are examples of PDE&BC problems for which pdsolve is successful in first calculating the PDE's general solution, and then fitting the initial or boundary condition to it.

>

pde__18 := diff(u(x, y), x, x)+diff(u(x, y), y, y) = 0

If we ask pdsolve to solve the problem, we get:

>

(4.1)

and we can check this answer by using pdetest:

>

(4.2)

How it works, step by step:

The general solution for just the PDE is:

>

(4.1.1)

Substituting in the condition , we get:

(4.1.2)

>

(4.1.3)

We then isolate one of the functions above (we can choose either one, in this case), convert it into a function operator, and then apply it to gensol

>

(4.1.4)

>

(4.1.5)

Three other related examples

>

pde__19 := diff(u(x, y), x, x)+(1/2)*(diff(u(x, y), y, y)) = 0

>

u(x, y) = (2*sin(-y+((1/2)*I)*2^(1/2)*x)+(-I*2^(1/2)*x+2*y)*_F2(y-((1/2)*I)*2^(1/2)*x)+(I*2^(1/2)*x-2*y)*_F2(y+((1/2)*I)*2^(1/2)*x))/(I*2^(1/2)*x-2*y)

(4.2.1)

>

(4.2.2)

>

pde__20 := diff(u(x, y), x, x)+(1/2)*(diff(u(x, y), y, y)) = 0

>

u(x, y) = (sinh((1/2)*(I*2^(1/2)*x-2*y)*2^(1/2))*2^(1/2)-(I*2^(1/2)*x-2*y)*(_F2(-y+((1/2)*I)*2^(1/2)*x)-_F2(y+((1/2)*I)*2^(1/2)*x)))/(I*2^(1/2)*x-2*y)

(4.2.3)

>

(4.2.4)

>

pde__21 := diff(u(r, t), r, r)+(diff(u(r, t), r))/r+(diff(u(r, t), t, t))/r^2 = 0

>

(4.2.5)

>

(4.2.6)

More PDE&BC problems are now solved by using a Fourier transform.

>

Consider the following problem with an initial condition:

>

pdsolve can solve this problem directly:

>

(5.1)

And we can check this answer against the original problem, if desired:

>

(5.2)

How it works, step by step

Similarly to the Laplace transform method, we start the solution process by first applying the Fourier transform to the PDE:

>

(5.1.1)

Next, we call the function "fourier(u(x,t),x,s1)" by the new name U:

>

(5.1.2)

And this equation, which is really an ODE, is solved:

>

(5.1.3)

Now, we apply the Fourier transform to the initial condition :

(5.1.4)

>

(5.1.5)

Or, in the new variable U,

>

(5.1.6)

Now, we evaluate solution_U at t = 0:

>

(5.1.7)

and substitute the transformed initial condition into it:

>	$eval(solution_U_at_IC, {trasnformed_IC_U})$

(5.1.8)

Putting this into our solution_U, we get

>	$eval(solution_U, {(rhs = lhs)(IPi(Dirac(s+1)-Dirac(s-1)) = _F1(s))})$

(5.1.9)

Finally, we apply the inverse Fourier transformation to this,

>

(5.1.10)

PDE&BC problems that used to require the option HINT = `+` to be solved are now solved automatically

The following two PDE&BC problems used to require the option HINT = `+` in order to be solved. This is now done automatically within pdsolve.

>

pde__23 := diff(u(r, t), r, r)+(diff(u(r, t), r))/r+(diff(u(r, t), t, t))/r^2 = 0

>

(6.1)

>

(6.2)

>

pde__24 := diff(u(x, y), y, y)+diff(u(x, y), x, x) = 6*x-6*y

iv__24 := u(x, 0) = x^3+11*x+1, u(x, 2) = x^3+11*x-7, u(0, y) = -y^3+1, u(4, y) = -y^3+109

>

(6.3)

>

(6.4)

>

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Thermal Engineering Application Collection

by: Samir Khan 2176 Product: Maple 2016

June 22 2016

6 3

We've added a collection of thermal engineering applications to the Application Center. You could think of it as an e-book.

This collection has a few features that I think are pretty neat

The applications are collected together in a Workbook; a single file gives you access to 30 applications
You can navigate the contents using the Navigator or a hyperlinked table of contents
You can change working fluids and operating conditions, while still using accurate thermophysical data

If you don't have Maple 2016, you can view and navigate the applications (and interactive with a few) using the free Player.

The collection includes these applications.

Psychrometric Modeling
- Swamp Cooler
- Adiabatic Mixing of Air
- Human Comfort Zone
- Dew Point and Wet Bulb Temperature
- Interactive Psychrometric Chart
Thermodynamic Cycles
- Ideal Brayton Cycle
- Optimize a Rankine Cycle
- Efficiency of a Rankine Cycle
- Turbine Analysis
- Organic Rankine Cycle
- Isothermal Compression of Methane
- Adiabatic Compression of Methane
Refrigeration
- COP of a Refrigeration Cycle
- Flow Through an Expansion Valve
- Food Refrigeration
- Rate of Refrigerant Boiling
- Refrigeration Cycle Analysis 1
- Refrigeration Cycle Analysis 2
Miscellaneous
- Measurement Error in a Manometer
- Particle Falling Through Air
- Saturation Temperature of Fluids
- Water Fountain
- Water in Piston
Heat Transfer
- Dittus-Boelter Correlation
- Double Pipe Heat Exchanger
- Energy Needed to Vaporize Ethanol
- Heat Transfer Coefficient Across a Flat Plate
Vapor-Liquid Equilibria
- Water-Ethanol

I have a few ideas for more themed Maple application collections. Data analysis, anyone?

Equidistant curves

by: one man 1360 Product: Maple 15

June 22 2016

7 9

equidistant_curve_MP.mw Equidistant curves to the curves on the surface. (Without any sense, but real.)

Moderating posts in MaplePrimes - Guidelines

by: Bryon 1728 Product: MaplePrimes

June 20 2016

5 13

I have recently taken a few comments and questions about moderation, and thought it would be worthwhile to clarify our policies.

Although financed and hosted by Maplesoft, MaplePrimes is a community-run site. Our members provide thoughtful posts, answer thousands of questions each year, and generally keep things running smoothly by removing spam and moderating content. Moderation and management capabilities are granted automatically based on a member's reputation, which is a measure of how their content is valued by other community-members. Once someone's reputation score reaches 500, moderation privileges are granted. There are currently 69 members who have moderation capabilities.

We have had this system running for years, and by and large, it works well. There are occasional problems, of course, and these have been dealt with on an as-needed basis, and will continue to be dealt-with going forward.

What is the intent of moderation?

Quite simply, moderation capabilities are provided to maintain a high-standard of content. We want our moderators to help maintain a well organized and technically-sound site, without altering the intent or context of the original post.

Common examples of acceptable moderation scenarios include:

Removing spam or messages that violate our community guidelines
Editing the tags, categories or product(s) associated with a question or post
Correcting incorrect technical content such as bad URLs included in a post or poor formatting
Correcting a typo or improper spelling is appropriate in limited scenarios when the mistake made the original message confusing. In general practice, however, it is best to not edit someone else's grammar or spelling.

What is moderation not for?

MaplePrimes members cover a very diverse range of backgrounds, languages, technical skill with Maplesoft products, and familiarity with MaplePrimes. It is therefore normal that posts and questions within MaplePrimes reflect this diversity.

In order to respect the author, as well as to maintain the context of the original question or post, there are some scenarios where moderation should not be employed. For example:

Many members are not fluent in English, yet the context of their message/question is very clear. In cases like this, correcting improper spelling or grammar (unless requested) is disrespectful and should be avoided.
Do not edit or remove another person's code, even if it is incorrect or incomplete. Doing so can completely change the context of the message and makes the message very confusing for people who read that thread in the future.
Do not edit the context of a message to make it 'read better'. Even if your intentions are good, edits like these are can be seen as disrespectful, and in some cases, even inflammatory.

In general, I recommend following this rule: If you're not 100% certain that your edit will be helpful and appreciated by everyone, do not make it.

What can I do if I disagree with a moderator's changes?

MaplePrimes moderators are invaluable for keeping MaplePrimes running smoothly, and as mentioned above, they do a great job. From time to time, however, a change may be made that you disagree with. Often such changes are a result of a misunderstanding on the part of the moderator, or perhaps just an honest mistake.

In situations like these, my recommendation is to contact MaplePrimes administrators by flagging the post as 'other' and providing the details. We will then look into the issue and (hopefully) reach a speedy resolution. During this process, we will contact the poster and/or the moderator to discuss the changes. If needed, further actions can be taken, from further education up to and including suspending or removing moderator privileges.

I hope this has been helpful, and I'm looking forward to your thoughts and suggestions in the comments below.

Bryon

Is there a way to avoid the consequences of not...

by: one man 1360 Product: MaplePrimes

June 19 2016

4 12

I want to understand what the forum rules I break? And why do me constantly interferes Markiyan Hirnyk? A recent example: he removed my comments in the subject http://www.mapleprimes.com/posts/203796-Equidistant-Surface-

At the beginning of the theme I have provided the text of the program, and then gave a description of the algorithm in words. In removed by Hirnyk examples I have quoted only the formula, because the text of the program has not changed.

I have long known Hirnyk for Russian-speaking forums and personal correspondence. Communication with him for a long time I stopped, because it is extremely unpleasant man, and it seems very angry and envious. In addition, the level of competence is very low and is away from the content of my subject. I am convinced of this. I will not say he is constantly trying to harm me on Russian forums.

( Хочу понять, какие правила форума я нарушаю? И почему мне постоянно мешает Markiyan Hirnyk? Из последних примеров: он удалил мои комментарии в теме http://www.mapleprimes.com/posts/203796-Equidistant-Surface-
В начале темы я предоставил текст программы и потом дал описание алгоритма на словах. В удалённых Hirnyk примерах я привел только формулу, потому что текст программы не изменился.
Я давно знаю Hirnyk по русскоязычным форумам и по личной переписке. Общение я с ним давно прекратил, потому что человек он крайне неприятный, и, похоже, очень злой и завистливый. К тому же уровень его компетенции весьма невысок и находится в стороне от содержания моих тем. Я в этом убеждён. Не буду говорить, как он пытается постоянно вредить мне на русских форумах. )