I am solving a hybrid nanofluid flow problem in a bifurcated artery using Maple. The governing equations for velocity and temperature are solved using dsolve(..., numeric, method=bvp[midrich]).

My Maple code successfully produces for both the artery  parentartery_and_daughter_artery_error.mw.

The velocity profiles are obtained correctly using odeplot.

However, I want to compute additional physical quantities and generate plots similar to the velocity profiles.

Specifically I want to plot:

1. Flow rate Q versus axial distance z

2. Impedance (flow resistance) λ versus z

3. Wall shear stress τ versus z

for different values of Hartmann number Ha.

The formulas I am using are

Flow rate:

Q=2π(R∫01ηw(η) dη+R2∫01w(η) dη)Q = 2\pi \left( R \int_0^1 \eta w(\eta)\,d\eta + R_2 \int_0^1 w(\eta)\,d\eta \right)Q=2π(R∫01​ηw(η)dη+R2​∫01​w(η)dη)

Wall shear stress:

τ=μ∣dwdr∣\tau = \mu \left|\frac{dw}{dr}\right|τ=μ​drdw​​

Impedance:

λ=∣dp/dz∣Q\lambda = \frac{|dp/dz|}{Q}λ=Q∣dp/dz∣​
Please help me to solve this question.

For decades, Maple has been built around one of the world’s most powerful mathematics engines—helping students, educators, engineers, and researchers explore ideas, solve complex problems, and communicate mathematics clearly.

Maple 2026 builds on that foundation with major advances in the math engine, expanding the kinds of problems Maple can solve while improving reliability and performance.

At the same time, Maple 2026 introduces new AI-powered tools that help you work faster—finding commands, generating visualizations, explaining concepts, and helping you explore ideas. The key difference is that these tools sit on top of Maple’s math engine, so the results are grounded in real computation rather than guesswork.

If you’ve been following along with our recent Mathy teaser videos and sneak peek posts, you may already have seen hints of some of these features. Now I’m excited to finally share them in full.

One of the most exciting additions in Maple 2026 is the new AI Assistant.

AI tools are incredibly useful for exploring ideas, writing code, and learning new topics. But when the mathematics becomes more involved, relying on AI alone can be risky. The Maple AI Assistant brings those productivity benefits into Maple while keeping the mathematics grounded in Maple’s trusted computation engine.

You can ask the AI Assistant questions in natural language and have it help you:

• find Maple commands or formulas
• generate Maple code
• create visualizations
• explain mathematical concepts
• draft examples, worksheets, or reports

Because Maple performs the underlying computations where appropriate, the results are grounded in Maple’s powerful math engine. The AI Assistant becomes a productivity partner that helps you accomplish tasks in Maple faster and more easily, combining the flexibility of AI with mathematics you can trust.

Watch the AI Assistant in action.

Another feature I’m particularly excited about is Document Import.

Many of us have years of mathematical content stored in PDFs, lecture notes, journal articles, slides, or even handwritten pages. Traditionally these documents are static—you can read them, but you can’t interact with the mathematics inside them.

With Maple 2026, that changes.

Document Import allows Maple to convert many document formats—including PDFs, DOCX files, and presentations—into Maple worksheets where the mathematics becomes live and executable. 

The image below illustrates the transformation.

On the left (“Before”), scribbled handwritten notes from a Calculus III lecture were saved in a Word document. The notes include hand-drawn sketches, formulas, and written explanations.

After importing the document into Maple (“After”), the mathematical expressions were recognized and converted into live, editable Maple mathematics. The text was preserved, and the hand-drawn sketches were retained as images. The resulting worksheet supports evaluation, editing, and further computation.

Once imported, you can:

• evaluate expressions
• modify formulas
• extend derivations
• add visualizations
• explore variations of the mathematics

Instead of recreating examples from scratch, you can bring existing material directly into Maple and start exploring.

While the new AI features are exciting, the heart of Maple has always been its mathematics engine—and Maple 2026 delivers significant advances here.

One particularly notable improvement is Maple’s expanded ability to solve linear recurrence equations. Through improvements to the rsolve command and major extensions to the LREtools package, Maple can now solve dramatically more recurrence relations than before, including many third- and fourth-order cases that were previously beyond reach.

In fact, Maple can now fully solve over 94% of the 55,979 entries in the Online Encyclopedia of Integer Sequences (OEIS) that that can be shown to satisfy a linear recurrence relation. These advances reflect ongoing research into linear difference equations and their algorithmic implementation in Maple, continuing Maple’s long tradition of advancing the state of computer algebra.

Beyond recurrence solving, Maple 2026 includes many improvements across its core symbolic and numeric algorithms. Maple’s assumption system has been strengthened to improve reasoning under mathematical assumptions, and enhancements to the simplify, combine, and evalc commands allow Maple to produce more compact and mathematically natural forms for a wider range of expressions.

There are also improvements to Maple’s differential equation solvers, polynomial system solving, and numerical solving routines such as fsolve, along with updates to other foundational parts of the math library used throughout the system.

Taken together, these improvements expand the range of problems Maple can solve and improve the robustness, correctness, and efficiency of the results.

Maple has always offered extensive control over plotting options, but achieving consistent visual styling across multiple plots could require specifying many settings each time.

Maple 2026 introduces Plotting Themes, which allow you to define a plotting style once and apply it across many plots with a single option.

Themes make it easy to maintain consistent visual styles in worksheets, teaching materials, reports, and publications, while still allowing individual plots to override specific options when needed.

The image below shows an example of creating and applying a custom plotting theme. 

Maple continues to be widely used in classrooms around the world, and Maple 2026 includes several improvements designed to support teaching and learning.

The Check My Work system has been enhanced so Maple can recognize a wider variety of valid student solution steps and provide more accurate feedback.

Maple 2026 also improves the generation of similar practice problems, making it easier to create variations of a problem while preserving its mathematical structure.

In addition, Maple’s step-by-step solutions have been expanded to support more types of expressions, helping students better understand the reasoning behind the mathematics they’re learning.

Maple 2026 also introduces improvements for developers building advanced applications, along with performance enhancements across the system.

One particularly interesting addition is the new VectorSearch package, which implements a vector database directly inside Maple.

If you’re not familiar with vector databases, one way to think about them is through recommendation systems like Netflix or Spotify. Each movie or song can be represented by a vector containing thousands of numbers describing its characteristics—things like genre, pacing, or mood. When you watch something, the system finds other items whose vectors are closest to it, which is how recommendations are generated.

With the new VectorSearch package, Maple can store thousands (or more) of vectors and efficiently find the ones most similar to a given vector. This makes it easier to build applications involving machine learning, data analysis, and modern AI workflows directly in Maple.

Maple 2026 also delivers significant performance improvements. For example, operations involving quantities with units have been greatly optimized—some computations now run over 90 times faster, making Maple even more efficient for engineering and scientific workflows.

Maple 2026 also expands the benefits available through the Maplesoft Elite Maintenance Program (EMP). The new benefits include access to additional Maplesoft products and services:

• Maple Learn, the online environment for teaching and learning mathematics
• Maple Calculator Premium, bringing the power of Maple to your phone with full access to features like Solution Steps and Check My Work
• Maple MCP, which allows you to connect Maple’s math engine to external AI tools so they can produce mathematical results you can trust

These additions extend Maple beyond the desktop, giving users powerful tools for learning, teaching, and exploring mathematics across web and mobile platforms, as well as through integrations with external AI tools.

This post only scratches the surface of what’s new in Maple 2026. There are many more improvements across the math library, programming tools, and performance.

To learn more about all the new features and enhancements in Maple 2026, visit the What’s New in Maple page on our website.

I am studying a nonlinear wave equation and trying to reproduce the energy balance method shown in a research paper. First, the original partial differential equation is reduced to an ordinary differential equation using a traveling wave transformation. After obtaining the reduced equation, the paper rewrites it in a form suitable for the energy balance method and derives the corresponding variational principle and Hamiltonian invariant. Then a trial periodic solution in cosine form is assumed. Using the Hamiltonian invariant and some initial conditions, the parameters of the trial function are determined and a periodic solution is obtained.

I would like to know how to implement this procedure in Maple. Specifically,  compute the Hamiltonian invariant from the equation, substitute the cosine trial function, and determine the unknown parameter in the trial solution using the energy balance method. I will attach images from the paper that show the derivation steps I am trying to reproduce. Any guidance on how to perform these symbolic steps in Maple would be very helpful.

f-s.mw

Hi,

For a pedagogical purpose, I am trying to illustrate the orthogonal projection H of a point A on to a plane P1

I constructed the line l1​ passing through A and perpendicular to the plane P1
However, in the graphical visualization, the line does not appear to be perpendicular to the plane. Visually, it gives the impression that the line is not orthogonal to the plane.

Do you have any idea what might cause this effect?

Thank you for your help.

Q_Espace.mw

While teaching a linear programming course I put together a worksheet to illustrate finding the largest disk inside a convex polygon as in section 2.6 of Understanding and Using Linear Programming by Jiří Matoušek and Bernd Gärtner.  I used both the Optimization[LPSolve] and the simplex[maximize] routines with the same objective and the same constraints.  Optimization[LPSolve] gives the correct answer but simplex[maximize] does not.  Is this a bug or did I do something wrong?

Below is the worksheet.

Download inscribe_circle.mw

I have generated the regional plot, but the boundaries between the regions are not clearly marked. How can I highlight the boundaries using a black line or a black dashed line, similar to the example image? What syntax should I use? I have attached my generated image file.

Regional_new.mw

how we solve x^(4)-12*x-12=0, 

mapple sent after solving: RootOf(_Z^4 - 12*_Z - 12, index = 1), RootOf(_Z^4 - 12*_Z - 12, index = 2), RootOf(_Z^4 - 12*_Z - 12, index = 3), RootOf(_Z^4 - 12*_Z - 12, index = 4)

sol:=ln( (y-1)^(1/3)* (y^2+y+1)^(1/3) ) - ln(y) = 2/5* ln(t^2+1)+_C1;
solve( sol,y);

in real domain is fine also. But all my attempts failed. I waited 3-4 minutes each time and stopped it.

Any one can find a trick? Below worksheet showing my attempts and also solution by Mathematica which took 0.3 seconds

Make sure to save all your work first. This problem is known to crash Maple !

Download solve_problem_march_7_2026.mw

How to solve the following system of equations?

restart;
el1 := a[0] = 1;
el2 := a[0]+a[1]+a[2]+a[3]+a[4]+a[5]+a[6]+a[7]+a[8]+a[9]+a[10]+a[11]+a[12]+a[13]+a[14]+a[15] = 3/2+sinh(1);
el3 := a[1] = 1;
el4 := a[1]+2*a[2]+3*a[3]+4*a[4]+5*a[5]+6*a[6]+7*a[7]+8*a[8]+9*a[9]+10*a[10]+11*a[11]+12*a[12]+13*a[13]+14*a[14]+15*a[15] = 1+cosh(1);
el5 := 24*a[4]-(2*(1+c))*a[2]+c*a[0] = -1;
el6 := 24*a[4]+120*a[5]+360*a[6]+840*a[7]+1680*a[8]+3024*a[9]+5040*a[10]+7920*a[11]+11880*a[12]+17160*a[13]+24024*a[14]+32760*a[15]-(1+c)*(2*a[2]+6*a[3]+12*a[4]+20*a[5]+30*a[6]+42*a[7]+56*a[8]+72*a[9]+90*a[10]+110*a[11]+132*a[12]+156*a[13]+182*a[14]+210*a[15])+c*(a[0]+a[1]+a[2]+a[3]+a[4]+a[5]+a[6]+a[7]+a[8]+a[9]+a[10]+a[11]+a[12]+a[13]+a[14]+a[15])-(1/2)*c = -1;
el7 := 120*a[5]-(6*(1+c))*a[3]+c*a[1] = 0;
el8 := 120*a[5]+720*a[6]+2520*a[7]+6720*a[8]+15120*a[9]+30240*a[10]+55440*a[11]+95040*a[12]+154440*a[13]+240240*a[14]+360360*a[15]-(1+c)*(6*a[3]+24*a[4]+60*a[5]+120*a[6]+210*a[7]+336*a[8]+504*a[9]+720*a[10]+990*a[11]+1320*a[12]+1716*a[13]+2184*a[14]+2730*a[15])+c*(a[1]+2*a[2]+3*a[3]+4*a[4]+5*a[5]+6*a[6]+7*a[7]+8*a[8]+9*a[9]+10*a[10]+11*a[11]+12*a[12]+13*a[13]+14*a[14]+15*a[15])-c = 0;
el9 := 720*a[6]-(24*(1+c))*a[4]+2*c*a[2]-c = 0;
el10 := 720*a[6]+5040*a[7]+20160*a[8]+60480*a[9]+151200*a[10]+332640*a[11]+665280*a[12]+1235520*a[13]+2162160*a[14]+3603600*a[15]-(1+c)*(24*a[4]+120*a[5]+360*a[6]+840*a[7]+1680*a[8]+3024*a[9]+5040*a[10]+7920*a[11]+11880*a[12]+17160*a[13]+24024*a[14]+32760*a[15])+c*(2*a[2]+6*a[3]+12*a[4]+20*a[5]+30*a[6]+42*a[7]+56*a[8]+72*a[9]+90*a[10]+110*a[11]+132*a[12]+156*a[13]+182*a[14]+210*a[15])-c = 0;
el11 := 5040*a[7]-(120*(1+c))*a[5]+6*c*a[3] = 0;
el12 := 5040*a[7]+40320*a[8]+181440*a[9]+604800*a[10]+1663200*a[11]+3991680*a[12]+8648640*a[13]+17297280*a[14]+32432400*a[15]-(1+c)*(120*a[5]+720*a[6]+2520*a[7]+6720*a[8]+15120*a[9]+30240*a[10]+55440*a[11]+95040*a[12]+154440*a[13]+240240*a[14]+360360*a[15])+c*(6*a[3]+24*a[4]+60*a[5]+120*a[6]+210*a[7]+336*a[8]+504*a[9]+720*a[10]+990*a[11]+1320*a[12]+1716*a[13]+2184*a[14]+2730*a[15]) = 0;
el13 := 40320*a[8]-(720*(1+c))*a[6]+24*c*a[4] = 0;
el14 := 40320*a[8]+362880*a[9]+1814400*a[10]+6652800*a[11]+19958400*a[12]+51891840*a[13]+121080960*a[14]+259459200*a[15]-(1+c)*(720*a[6]+5040*a[7]+20160*a[8]+60480*a[9]+151200*a[10]+332640*a[11]+665280*a[12]+1235520*a[13]+2162160*a[14]+3603600*a[15])+c*(24*a[4]+120*a[5]+360*a[6]+840*a[7]+1680*a[8]+3024*a[9]+5040*a[10]+7920*a[11]+11880*a[12]+17160*a[13]+24024*a[14]+32760*a[15]) = 0;
el15 := 362880*a[9]-(5040*(1+c))*a[7]+120*c*a[5] = 0;
el16 := 362880*a[9]+3628800*a[10]+19958400*a[11]+79833600*a[12]+259459200*a[13]+726485760*a[14]+1816214400*a[15]-(1+c)*(5040*a[7]+40320*a[8]+181440*a[9]+604800*a[10]+1663200*a[11]+3991680*a[12]+8648640*a[13]+17297280*a[14]+32432400*a[15])+c*(120*a[5]+720*a[6]+2520*a[7]+6720*a[8]+15120*a[9]+30240*a[10]+55440*a[11]+95040*a[12]+154440*a[13]+240240*a[14]+360360*a[15]) = 0;

solve({el1, el2, el3, el4, el5, el6, el7, el8, el9, el10, el11, el12, el13, el14, el15, el16}, {a[0], a[1], a[2], a[3], a[4], a[5], a[6], a[7], a[8], a[9], a[10], a[11], a[12], a[13], a[14], a[15]});

i need it in  numerical format

I never used Maplets before.

Been learning Explore, which is OK, but the UI does not look good. Too much white spaces between sliders.  So been looking at Maplets to use instead.

Should one just use Maplets to make interactive demos with sliders, buttons, popup menus and so on or use Explore? Which is better?

I do not waste more time learning Explore more if Maplets is a better choice.

If someone here knows Maplets, here is something I just wrote in Explore.  Could this be coded in Maplets to see how it will look like. It took me 30 minutes to make it using Explore and I am no expert in Explore, so hopefully it should not take someone who knows Maplets much time to produce same thing as Maplet. I wanted to see if the UI will look better or same issues with too wasted spaces between sliders.  I assume with Maplets, there is same concept as Explore, with range of variables, and initial values and so on...

I understand one can run Maplet using Maple viewer without needing to have Maple installed on the PC, so this is an advantage.

Download lotka_volterra.mw

Screen shot of the UI

How would the above look using Maplets instead of Explore?

I am not sure whether this should be in "create a post" or "ask a question." Let me know if this is more appropriate in "ask a question"

The standing wave equation is given by:

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

IBC := {u(x, 0) = A0*cos(Pi*x/L), D[1](u)(0, t) = 0, D[1](u)(L, t) = 0, D[2](u)(x, 0) = 0}

pdsolve(PDE,IBC,numeric)

In my problem, the material is a magneto-elastic material where c, the speed of the acoustic wave, is a function of a magnetic field H.  The material is nonlinear and saturable.  I define it by a 3 segment piecewise nonlinear function of H.  The material response is a result of a sinusoidal H field.  I am interested in solving u(x,t).  

With that, c in the PDE has to be rewritten as c(H(t)), pdsolve gives an error as PDE has to be expressed as a function of u,t, or x.  So I redefine c(H(t)) as c(t).  I ran into another error in pdsolve as the piecewise has to be based on t or x and not H.  

The problem is that depending on H I can go through all 3 segments and back in one cycle and I have to find the corresponding t's for the piecewise.  Now I am driving the material with 100 cycles, I have to find and list all those piecewise transition points which is hardly practical.

Is there any other ways to approach and solve this problem?

Hello Friends.

I have used Maple to create a Polynomial Regression model.  The model is called "PRModel."  It works fine.  The independent variable is "X" and the dependent variable is "Y."  Both X and Y are of the vector variety.  The model is as follows:

PRModel:=PolynomialFit(10, X, Y, summarize = true);

I would like an estimate of Y for each value of X.  I would like the estimates of Y to be in a variable called "estY."   I have not been successful with this.  I have tried many different variations of the following, but have not been successful.  

estY := eval(PRModel, X);

Any suggestions as to how I can capture the estimates of Y? 

Thank you.

I am very pleased to announce that Volume 6 Number 1 of Maple Transactions has been published.  This is a Special Issue on Matrices and Polynomials in Computer Algebra, and the Guest Editors (our first ever!) were Marc Moreno Maza and Tomas Recio.  There are still some papers that are expected to be added to the issue when they come in, but at this moment there are 8 papers there for you to read (and a description of the issue in the Front Matter section, by the Guest Editors).

A link to this Special Issue

On my journey of discovery through the Maple world, I now want to try out Maple's convenient features in the complex plane, something that used to be laboriously worked out and demonstrated on the blackboard with chalk. I couldn't find a suitable introduction in the help text. I'm interested in whether a package needs to be loaded and how to handle polynomials, series, and line integrals (I have a reasonable understanding only of the theory).

once i founded but  i lost the technique and i can't remember how i can reach the point how to find thus parameter and find the solution of pde

t1.mw