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# Numerical Mathematics - Science topic

Explore the latest questions and answers in Numerical Mathematics, and find Numerical Mathematics experts.
Questions related to Numerical Mathematics
Question
function cmatrix = ccoeffunction(region,state)
n1 = 9;
nr = numel(region.x);
cmatrix = zeros(n1,nr);
cmatrix(1,:) = ones(1,nr);
cmatrix(2,:) = 2*ones(1,nr);
cmatrix(3,:) = 8*ones(1,nr);
cmatrix(4,:) = 1+region.x.^2 + region.y.^2;
cmatrix(5,:) = state.u(2,:)./(1 + state.u(1,:).^2 + state.u(3,:).^2);
cmatrix(6,:) = cmatrix(4,:);
cmatrix(7,:) = 5*region.subdomain;
cmatrix(8,:) = -ones(1,nr);
cmatrix(9,:) = cmatrix(7,:);
I created a function for c coefficient in PDE toolbox using the above example given in MATLAB documentation. My problem is a system of parabolic equations. I want to make sure that parabolic non linear system of PDE can be solved using PDE toolbox in MATLAB. However when I write the state.u as mentioned above error shows up: Function handle specifying a coefficient must accept two input arguments and return one output argument.
Could you please guide me, How you resolved this error finally?
"Function handle specifying a coefficient must accept two input arguments and return one output argument."
I have the same error.
Question
Hello all,
I am looking for an method / algorithm/ or logic which can help to figure out numerically whether the function is differentiable at a given point.
To give a more clear perspective, let's say while solving a fluid flow problem using CFD, I obtain some scalar field along some line with graph similar to y = |x|, ( assume x axis to be the line along which scalar field is drawn and origin is grid point, say P)
So I know that at grid point P, the function is not differentiable. But how can I check it using numeric. I thought of using directional derivative but couldn't get along which direction to compare ( the line given in example is just for explaining).
Ideally when surrounded by 8 grid points , i may be differentiable along certain direction and may not be along other. Any suggestions?
Thanks
The answer to a question about the numerical algorithms for resolving the issue of differentiability of a function is typically provided by the textbooks on experimental mathematics.
I recommend in particular: Chapter 5: “Exploring Strange Functions on the Computer” in the book: “Experimental Mathematic in Action”.
For the review please see
You can also get a copy of the text in a form of a preprint from
Judging by the quote placed in the beginning of Chapter 5, the issue of investigation of the “strange functions” was equally challenging i 1850s as it is 170 years later:
“It appears to me that the Metaphysics of Weierstrass’s function
still hides many riddles and I cannot help thinking that enter-
ing deeper into the matter will finally lead us to a limit of our
intellect, similar to the bound drawn by the concepts of force
and matter in Mechanics. These functions seem to me, to say
it briefly, to impose separations, not, like the rational numbers”
(Paul du Bois-Reymond, , 1875)
The situation described in your question is even more complicated because the function is represented only by a few values on a rectangular grid and it is additionally assumed that the function is not differentiable at a certain point. In this situation I can suggest to use the techniques employed in the theory of generalized functions (distributions).
For a very practical example you can consult a blog: “How to differentiate a non-differentiable function”:
In order to answer your question completely I would like to know what is the equation, boundary conditions and the numerical scheme used to obtain a set of the grid point values mentioned in the question.
Question
If corresponding factorials and triangular numbers are added, the results form the sequence of numbers, {2, 5, 12, 34, 135, 741, 5068, 40356, ...}, which I call factoriangular numbers. In the list of the first few factoriangular numbers, I found three Fibonacci numbers: 2, 5 and 34. Aside from these three, are there other Fibonacci numbers in the sequence of factoriangular numbers?
Now it is a a complete useful answer.
Question
NMPC- Nonlinear model predictive control
X_dot=f(x,u)
Y=C*x
objective function: min J = (Y-Ys)^2+du^2+u^2
w.r.t u
constraints are :
0<u<10
-0.2<du<0.2
0<Y<8 here Y is a nonlinear constraint and a vector
Hi,
Here is a complete workshop on how to implement MPC and MHE in MATALB.
The workshop shows a complete explanation of the implementation with coding examples. the codes are also provided
Question
"Holistic discretisation" also aims to provide a rigorous methodology for preserving properties of discretisations of PDEs. So I wonder what you might view as any connection. See perhaps the recent (2018). “Smooth subgrid fields under-pin rigorous closure in spatial discretisation of reaction-advection- diffusion PDEs”. In: Applied Numerical Mathematics 132, pp. 91– 110. doi: 10.1016/j.apnum.2018.05.011 or the original (2001). “Holistic discretisation ensures fidelity to Burgers’ equation”. In: Applied Numerical Modelling 37, pp. 371–396. doi: 10.1016/S0168-9274(00)00053-2.
Hi Tony -- I echo MIsha's answer. In the 1970s, Misha was developing support operator methods while I was developing compatible differencing. These turned out to be very similar (Misha and I wrote some joint papers, and settled on mimetic methods as a nomenclature). On the other hand, the discrete calculus of Mattiussi, of Hirani, Desbrun, etc. is based on differential forms and is quite different both in structure and results. Other efforts that are perhaps less closely related are the nonstandard difference forms of Ron Mickens and the nonlocal calculus of Max Gunzburger,
After our previous email conversation, I have looked at your suggested papers on holistic discretization. Your starting point is very different and original, but I cannot readily see how it relates to other work. Perhaps you can lead this conversation toward identifying similarities and differences?
Question
Up to my knowledge, I find that the simulation with regard to the simulate of two-dimensional mixed Volterra-Fredholm equatio is confined to be in a closed subset of R^n and there are several wonderful methods. However, I happen to get a integral equation in the following form (please see the attached picture). The only difference with the existing ones is that the integration with respect to x is from 0 to infinity and the boundary condition available is f(x,T)=h(x), with h() given, which is also quite weird. I desire to do a rough simulation in the last part to make a brief illustration but I get no idea and have no inspirations from the literature. Also, since I get little knowledge and experience in simulation of solutions to equations like that, I find it's hard for me to think out a practical method to do the simulation. I'm asking if I can get some useful inspirations from you, who may be experienced in numerical simulation of integral equations. Thanks a lot for your generouse help, your attention and your precious time.
Of course you can compactify: just change variables in x! And if you know h(x), you know how it behaves at infinity, in particular, so you know how f(x,T) behaves.
Question
I have an 8D nonlinear ODE system and I would like to find all the fixed points(that is, dy/dt=0). Since the dimension is high, it is not pratical to plot the nullclines and observe the fixed points. If I use MATLAB to solve the equations, it seems that MATLAB can only help me find the fixed points near the initial value given by me. Then I need to adjust the initial value to find all the possible fixed points. Due to the high dimension, it seems arduous to use things like for-loop to confirm that all initial values have been tested. And due to the nonlinearity and the high dimension, it is hard for me to find the fixed points analytically.
Are there clever approches to find all the fixed points? Thanks!
Could you construct some coordinate transformations to reduce the problem DoF, without any loss of generality, which might be recovered later as per your real need? If possible, conventional stability analyses could be applicable.
Question
i want to solve a DAE and one of the variables has limits (e.g. x(1)<2) , dont know how to define this into the ode15s?
is there any way? or i have to use another function?
You could have a slack variable s and require s^2 = 2 - x(1). But funny things will happen when you hit x(1) == 2. In fact, normally there will not be any solution. Take for example, dx/dt = -x, x_0 = 1 with the constraint x >= 1/2. The solution is x(t) = exp(-t) *until* x(t^*) = 1/2, beyond which there is no solution. Something has to be brought in to prevent x(t) going below 1/2.
What is supposed to happen when the limit is reached? Is that the end of the simulation (in which case integrate the ODE as normal, but tell it to stop when the limit is reached -- this is available in Matlab solvers)? Or does something else happen, as occurs in contact mechanics for example?
Question
I think mesh-based methods can be completely replaced by mesh-free methods, but maybe mesh-based methods such as FEMs have some useful properties that mesh-free methods don't have them.
Its a trade off between matrix dimension and sparsity. Often, FEM methods have large but sparse matrices, whilst RBFs tend to involve smaller but less well conditioned metric matrices. Progress on ill-conditioning of RBF kernel matrices is partially overcome by Mercer expansion of the GRBF. See Fornberg & co, as well as Fasshauer and McCourt.
Question
How I can get the inverse Laplace transform of
L-1((1/(s+a)^2)*F(s))
where F(s) is variable function (we can say it is discrete and random)
OR how I solve this first order non-homogeneous differential equation,
y'+y = f
where f is variable function (we can say it is discrete and random)
Nasser
Dear Nasser
There are some questions posed by your affirmation (discrete and random).
1 - In a first approach and looking only at the the expression L-1((1/(s+a)^2)*F(s))
what I would do was a) See which is the Region of convergence of F(s). I assume it is, at least, Re(s) > 0. Fix the integration path at some Re(s) = c.
b) Perform a bilinear transformation s --> z that transforms the straight line Re(s) = c into the circle centred at z=0 and with radius 1. Put z=e^{i(2pi/N)k} with k=0, 1, 2, ... N-1. Its is convenient to use a high N= 2^K, K positive integer. c) use the inverse FFT.
Yo can find these things in any book on Signals and Systems.
2 - You say that the function is discrete. In this case, it would be better to use a discrete-time formulation using a difference equation and the Z transform.
If you want to "talk" directly, send a mail to mdo@fct.unl.pt
Question
that is:
How to find the potential within the area between the two circles that are not concentric, for example in the case of :
C'x2 + y2 = 1,   and   C:   (x-1) 2 + y2 = 9,
if the potential on the lower (inner) circle of radius 1 is equal to 1, and the higher  (outer) circle of radius 3, equal 2
In other words: solve the problem:
Laplace u = 0  within the domain  G  between the two circles C' and C
with boundary conditions:
u(x,y) = 1   on the circle  C'    and   u(x,y) = 2   on the circle
Dear colleagues,
Yes, this question is very interesting. Conform mappings are not only interesting as mathematical item, but also very useful
for engineers and and physicists, almost all technical sciences.
I am very happy that my question sparked so much interest and comments.
Mirjana
Question
I have partial differential equation, where Dirac delta-function is the only source (other terms are diffusion, convection and non-stationary term):
dC/dt+U*dC/dx=d/dz(K(z,t)*dC/dz)+delta(x-x0)*delta(z-z0)
Here x and z are spatial coordinates, t - temporal, C - dependent virable - gas concentration, U - wind speed, K(z,t) - diffusion coefficient, delta - Dirac delta function, x0 and z0 - constants.This equation is solved using usual finite-difference method (Crank–Nicolson method) on a mesh with two spatial coordinate axes and one temporal.
How I should write Dirac delta-function in numerical scheme? As the source in a single grid cell (with coordinates x0, z0) with integral of unity over this cell OR as source distributed over several cells near (x0, z0)? If second option is correct, what function and on what number of cells I should use?
According to Gianluca, the Dirac function can only be approximated on a grid of finite size. This approximation is largely used for example in the immersed boundary method on structured grids, I strongly suggest to read the way C. Peskin proposed in a series of papers. You can get an example here http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.453.2398&rep=rep1&type=pdf
Question
Some of students ask me to give the applications of mathematics in real life. What are some of the interesting applications of mathematics in real life? Could you please share your knowledge about this issue.
Thank you very much in advance for your cooperation.
You are asking about millions of mathematical application in our life, like calculation of area, volume, velocity, acceleration ....etc.
Question
Consider a continuous Lipschitz function f of the single variable and an interior point x* in a compact interval of its domain. How to use the fundamental theorem of calculus to describe f(x*) = 0? (The old version of this question was posed for C2 functions. In the current RG project this assumption was relaxed to continuous Lipschitz functions.)
There exist primal and dual characterizations of roots.
Question
In the set of prime numbers, the equation p-q = 2 has multiple solutions. There is a hypothesis that this equation has infinitely many solutions. What happens if 2 is replaced by 2k?
It is called: The twin prime conjecture. You can see this book:
Question
CH equation is an important model equation in simulating the shallow water problems. as what I have learned, it was mostly derived by the variational method. I want to learn something about it from the primary control equations by using method of perturbations, such as the Gardner-Morikawa transformation method, multi-scale method et.al. Could anyone tell me some informations or references about the derivation of Camassa-Holm equation for shallow water problems by the method of only perturbation expansions or multi-scale expansion method, but not the variational approach?
attaching a few here...including my thesis..
, Global Weak Solutions for a Shallow Water Equation, Indiana Univ. Math.
J. 47 (1998), 1527{1545.
Degasperis A., Holm D., and Hone A., A New Integrable Equation with Peakon
Solutions, Theoret. and Math. Phys. 133 (2002), 1463{1474.
Dullin H., Gottwald G., and Holm D., An Integrable Shallow Water Equation with
Linear and Nonlinear Dispersion, Phys. Rev. Lett. 87 (2001), 194501.
Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent
equations for shallow water waves, Fluid Dynam. Res. 33 (2003), 73{95.
On asymptotically equivalent shallow water wave equations, Phys. D 190
(2004), 1{14.
Question
Let us give  the power series
f(x) = c0 + c1 x + c2 xx  + c3 xxx +....
From this define a modified series with capital F
F(x) = c0 + c1 x  +c2  x(x+h)  + c3 x(x+h)(x+2h) +...
This is really a function of two variables, but concéntrate on the x variable.
The derivative of the original series is Df
Df = c1 + 2 c2 x +...
We call now the transformation of the derivative series  TDF
The point of interest is now
F(x+h) - F(x) = h TDF(x+h)
We get the right hand side by first calculating the transformation of the derivative function Df, to get TDF and then this series
is evaluated not at x but at x+h.
These operations are not  commutative.
This shows the connection between the continium and the discrete in
concrete form. Hopefully the result is correct.
If h tends to zero one just gets Df=Df
OK, Thanks. Let me suggest the following notation of the involved two operations on formal series, which may make the formulas for iterations more readable:
D(f)(x) := a1 + 2 a2 x + 3 a3 xx + ... if f=a0 + a1 x + a2 xx + ...
Th(g)(x) := b0 + b1 x + b2 x(x+h) +... if g=b0 + b1 x + b2 xx + ...
Then your propositions read that  Th  tends to the identity and
(Th(f)(x+h) - Th(f)(x)) / h= Th(D(f))(x+h),  which tends to D(f)(x) as h\to 0 .
To the best of my knowledge this formula is a new one with the use of divided differences (here, of order 1). Therefore, the orbits of the iterations [Th \circ D] should be somehow related to divided differences of higher orders. Nothing more comes now to my memory:(
Best, Joachim
Question
I have trained to define the heat flux from the integral of one-half order, from an equation of linear heat,
I am looking for how can we define or is there a relation between the flux of heat and the integral of fractional order in the nonlinear case.
Dear Luis - Felipe Velázquez - Leónm
I'm so thankful for providing me all of these informations, and references, i'll check them now.
Thank's again.
RAGHIB TAHA
Question
I want to plot dispersion equation by Matlab software, and this equation has complex function like Bessel. I could not drive the roots?- it is a little bit difficult for me!
Does any one know how to find the roots?- Frequency vs. K0.
Thank you in advance.
Thanks a lot for your response.
My question already solved.
Regards,
Hesam
Question
The function exp(x^2)*erfc(x) has several series approximations, like for example, the asymptotic expansion. However, this expansion is valid only for large values of x, and therefore, it cannot be used for a general analytical solution. Any ideas?
I like the procedure of Yew-Chung Chak, but also you can find the solution using wolfram alpha
∫ {exp(x2)*erfc(x)} dx
= ∫ {exp(x2)*[1 − erf(x)]} dx
= ∫ {exp(x2) − exp(x2)*erf(x)} dx
= ∫ {exp(x2) dx − ∫ {exp(x2)*erf(x)} dx
=(Sqrt[Pi] Erfi[x])/2 - (x^2 _2F2(1, 1; 3/2, 2; x2)/Sqrt[Pi]
Question
variational iteration method matlab code for BVP?
Question
Sir I asked a question before on your e mail but there was no reply. I ask here sir
Sir you paper in
''collocation method for solving fractional Riccati differential equation''
Sir how can i et the value of matrix bar(A) matrix.
also matrix bar(T(x))
Please sir your cooperation will be help full thanks
I apologize very much, I have no experience in this area too.
Question
Mathematica 11.0.1 has a habit of expressing real valued functions in terms of combinations of functions of a complex variable. This makes it difficult to see exactly how the function depends on the real valued variable for which it is defined.
An example that I have in mind is the function f(T) which is defined below. f is a real valued function and T is a dimensionless time which is also real.
f(T) is defined as Ei[2(Eulergamma) - i(pi) + ln[1/(4T)]] + Ei[2(Eulergamma) + i(pi) + ln[1/(4T)]].
Eulergamma is Euler's second constant which is approximately 0.5772, i is the complex number i, ln is the natural logarithm, and Ei is the Exponential Integral Ei function which is defined by Mathematica 11.0.1 as
Ei(z) = - Integral from (-z) to infinity of (e^(-t))/t dt.
If it can be shown how the above function f(T) can be written only in terms of real valued variables, I would be very grateful.
Thanks very much for your generous help,
Ron Zamir
e^(-t+i*pi))=-e^(-t)
e^(-t-i*pi))=-e^(-t)
Ei(z) = - Integral from (-z) to infinity of (e^(-t+i*pi))/(t-i*pi) dt=
= Integral from (-z) to infinity of (e^(-t))*(t+i*pi)/(t^2+pi^2) dt=
=Integral from (-z) to infinity of (e^(-t))*t/(t^2+pi^2) dt+
+i*Integral from (-z) to infinity of (e^(-t))*pi/(t^2+pi^2) dt
Question
By dynamical analysis, the PDE will have a periodic traveling wave, under the parameters a_0 and a_1, because the reduced ode have a periodic solution under wave speed equal to 1,  I just can simulate the ode have a limit cycle, but I can not plot the PDE. Many thanks for any friends who can give me suggestion and help. Here, 0<u<1, see the PDE in the figure attached.
Question
Dear Scientists , Collegues and researchers in Applied mathematics and fractional calculus
Actually , we have used optimal homotopy asymptotic method to solve fractional
Riccati equations , but our results are completely diffrent with results published in the attached paper.Would you please help and guide us about this item ?
At the first step , we have D alfa u0=1 , but even by using J operator , we can not obtain the u0 solution mentioned in the attached paper.
With Best Regards
Dr Hamed Daei Kasmaei
Associate Editor in chief International Electronic Engineering
Mathematical Society ( IEEMS)
Phd in Applied mathematics-Numerical analysis and computational
mathematics field -IAUCTB Lecturer at Faculty of Science and
Engineering at IAUCTB.
My Personal emails : hamedelectroj@gmail.com
Skype : hamed-daei
Whatsapp and imo : +989123937613
I am also ready to collaborate. My e-mail address is: msenol@nevsehir.edu.tr
Question
Hello everyone.
I am preparing an undergrads course on Applied Seismology and I wanted to introduce them with an exercise on 2D Travel Time Tomography (using the simplest least square method), but I am lacking information of how to approach the problem numerically.
Any papers, books, presentations, class notes, codes etc I can use to build a simple code to do this?
I am looking for the basic description of the steps to take, the matrixes to build and how to find the solution via least square. I want them to be able to do something like in the figure
Hi Mariano, codes and application examples for 2D tomography can be found at the link http://www.ivan-art.com/science/PROFIT/index.html
Some of these information could be useful for your course.
Question
We noticed recently that Forward Euler is pretty bad for simulating the *pendulum* equation. We feel that the pendulum example is too small of an example for the results to be so poor.
Gauss methods are implicit and therefore by the definition of simple - not simple.
Use Verlet:
x = position (or angle)
v = velocity (or angular velocity)
a(x) = acceleration (or angular acceleration)
(1) v_(n+1/2) = v_n + (dt/2)*a(x_n)      explicit
(2) x_(n+1) = x_n + dt*v_(n+1/2)         explicit
(3) v_(n+1) = v_(n+1/2) + (dt/2)*a(x_(n+1))  explicit
Simple
Stable for any timestep up to the period/pi (which is huge).
Conserves mass, momentum, angular momentum, energy within a small bound.
2nd order accurate
Works for any complicated coupled nonlinear dynamics you can dream up.
You can't do better than this given your criteria.
Question
To solve a given heat equation on the half line we can use the reflection method where the initial data is an odd extension (Dirichlet boundary conditions) /even extension (Neumann BCs). why do we choose an odd or even extension, is there any clear reasons for that?
The method is impractical and cannot be generalized but it is a good way to understand the physical meaning of a given PDE, therefore any clear explanation would be appreciated!
Hello!
It is connected with the fact that in the case of odd continuation, the corresponding solution will satisfy homogeneous Dirichler boundary conditions, and in the case of even continuation it will satisfy Neumann boundary condition. You can find proofs of these facts in 'Equations of Mathematical Physics' by A.N. Tikhonov, A.A. Samarskii, pages 254-257.
Question
I have two rather complicated ODEs that I need to solve, and they are of the form
x '' = f (y '', x ' , y ', x, y, t)
y '' = f (x '', x ' , y ', x, y, t)
So each equation has two 2nd order derivatives, x '' and y ''. Online documentation gives methods of decomposing n 2nd order ODEs into 2n 1st order equations, but they do not work for when there are two 2nd order derivatives in each equation.
How can I use the ODE solvers in MATLAB to solve these expressions numerically?
Another question is, since the functions on the RHS of the above equations depend on t as well, how do I define the functions to reflect the t-dependent nature of the equations?
I am completely new to the ODE parts of MATLAB, and any help would be much appreciated.
Your system is in fact implicit: if for instance you plug the first equation into the second one, the resulting equation is not explicit for y''. In theory, you can solve for y'' in function of x'',x',x,y',y (though it may be impossible to do it in close form because of nonlinearities); you then inject the expression so obtained into the first equation. This gives an equation which can in theory be solved for x''. Eventually, you get an explicit system for x" and y", and your are back to something you can handle.
If you cannot perform the above procedure in practice, you can use Matlab's ode15i solver, which can handle fully implicit differential equations. Beware that such "Differential Algebraic Equations" (DAE) are much wilder than plain explicit differential equations (check the litterature for DAE). The t-dependance is not a problem (all the Matlab ode solvers can handle time).
Question
Can anybody suggest me the best book in Generalized Fibonacci sequence and its applications in differential equations?
Dear Godase.
While not explicitly stated, the following link enunciates difference- and differential operators that yield generalized Fibonacci sequences and functions with extensions to the complex domain, and addresses an arsenal of open questions in mathematics, such as generalized Riemann Zeta functions, Gamma- and Polygamma functions, with wast applications to both mathematics and physics:
Furthermore - its terminating remark is as follows:
<<Regarding 'Ladder Ghost Operators', Riemann Zeta, Euler Gamma constant, and presently unsolved problems in mathematics:
It is not known if this constant is irrational, let alone transcendental (Wells 1986, p. 28). The famous English mathematician G. H. Hardy is alleged to have offered to give up his Savilian Chair at Oxford to anyone who proved gamma to be irrational (Havil 2003, p. 52), although no written reference for this quote seems to be known. Hilbert mentioned the irrationality of gamma as an unsolved problem that seems "unapproachable" and in front of which mathematicians stand helpless (Havil 2003, p. 97). Conway and Guy (1996) are "prepared to bet that it is transcendental," although they do not expect a proof to be achieved within their lifetimes.>>
-- sincerely yours,
Gudlaugur Kristinn.
Question
The problem occurs when the first element of the summation includes $\Gamma(0)$ in the denominator which is not defined?
Dear Walid,
I would like to make a short remark and continue the fine answer given by Professor Joachim Domsta given above. The series representation defining the two-parameter Mittag-Leffler function is given e.g. in Wikipedia
Specializing now \alpha=1 and separating the constant term 1/\Gamma(\beta) in the resulting series you will get a series representation
(1) E_{1,\beta}(z)=1/\Gamma(\beta)+\sum_{k=1}^\infty 1/\Gamma(k+\beta)z^k.
This actually proves the recurrence property given in the first answer. Now, using the property
(2) \Gamma(z+1)=z\Gamma(z)
of the Gamma function allows us to write the constant term on the right hand side of (1) in a form
\beta/\Gamma(\beta+1).
Now the both terms on the right hand side of (1) depend analytically on \beta, so you can extend the definition of the function E_{1,\beta} to cover also the case \beta=0 by defining
E_{1,0}(z)=0+\sum_{k=1}^\infty 1/\Gamma(k) z^k.
But the series is equal to z\exp(z) (since \Gamma(k)=(k-1)!) as it was emphasized in the answer of Professor Domsta. Therefore you will end up to an elementary function by using the analytic continuation of a known formula to cover a wider range of parameters.
Note that the continuation works not only for \beta=0 but also for negative \beta by using the property (2) of the Gamma function. For example for \beta=-1 separate the first two terms of the series expansion. Since the Gamma function has poles at non-positive integers, two first two terms will disappear in the limit and you will end up to an elementary function z^2\exp(z).
Hopefully you will find this procedure interesting.
Best regards, Jukka
Question
Dear all,
I want to know about the newest techniques about solving ODEs and PDEs by using uncertain methods like Interval Analysis, Affine Arithmetic, Improved of them, etc.
Did you know any better approaches or Ideas?
Thansks dear Vania Vieira Estrela,
I read about 4 books an very papers about this subject and the main reference is this one that you uploaded nicely. My major was (still is :D) on using soft computing methods for solving these kinds of systems. But now, after reading classical methods, I think they are still keep their priority in some cases.
Question
I just have real number line which start from zero and end at 1. I need to divided this real number line into equal parts and then I need to simulate points on this real line number such that every part contains just one point
I need to simulate points from uniform distribution and then I should divided line such that every part contains just one point.
Question
x(k+1)<=ax(k)+bx(k-1)+w(k)
Obviously, the sequence constructed above is convergent to a non - negative limit -point.
Sincerely, Octav
Question
Hi,
I am trying to solve the following differential inequality:
x_dot + k * sqrt(abs(x)) * sign(x) < epsilon,
where k and epsilon are positive constants.
I want to know what the upper bound on abs(x) is. (i.e., abs(x) < ?)
Could anyone let me know how to solve this?
Thank you.
Hancheol
After substitutiion   (in the case x > 0)   x = y^2  we come to the ineq.
y' - epsilon/y + K/2 < 0,
Let us take y= arctan t ,  t > 0.  We will get
1/(1 + t^2) - epsilion/arctan t + K/2 < 0, t > 0.
From here we easily get  epsilion >( K*pi)/4. Therefore the upper bound for x is
sqrt(K*pi/4).
Best regards,
Gevorg.
Question
Dear Researchers
I am solving a set of non-linear equations both numerically and mathematically. The numerical method is based on 4th order Runge-Kutta. Under some conditions (Chaos and bifurcations), the mathematical method results in multiple solutions, however, the numerical method only converges to one of the solutions. what is the way of controlling numerical method to extract all possible solutions?
@Saeed
If, the right-hand side of your non-linear ODE of first order does not depend explicitely on time and, in particular, does not contain a periodic driving force term, why you described your problem to Professor Rath as analog to the Huang Su Chen paper? (This paper deals with periodically driven problems for which all solutions depend nontrivially on time).
In the framework which you now suggest the word 'response' makes no sense, since there is no stimulus.
Does 'want to solve the diffferential equations again' mean that you vary the initial conditions of your Runge-Kutta integration? If yes, according to which plan?
I understand that you refrain from reproducing the ugly numbers from your funcC.m file. But it should be possible with limited effort to make clear on which variables the right-hand side of your ODE depends. In particular: does time come in in any other form than X_i(t), i=1,...6 ? E.g. as f_0(t) ?
Given that Andrew pointed out that finding steady-state solutions is not a differential equations problem at all, and that I indicated that 'steady-state solution' may here used in a meaning different from Andrew's interpretation, I would have expected clear words from you what your definition of steady-state solution is.
There is still a bit of hope on my side that the whole discussion may make sense in the end.
Question
I'm looking for links/references to research which have looked into determining the optimum number of choices to include per question on a multiple choice quiz. My question is directed at STEM disciplines.
Backer's answer is interesting as my understanding was that tests with 4 or fewer answers lead to too many statistical anomalies if people guess answers. I think the 3 answer stats are possibly good assuming there is no guessing - so possibly use three if you were to impose negative marking and 5 otherwise?
Question
Can one give a general formula for the product of three different finite sums? See the picture attached with this question.
Dear Feng, one possible formula in attached file
Question
How do I prove that a nth order differential equation has n linearly independent solutions?
Also, how to prove that there is no possible solution other than those covered by the linear combination of these solutions?
Please suggest a text to find out these answers.
The textbooks  that I used when I was a student are in Russian, which I afraid make them of little use for you :).  I have made a google search with keyword existence and uniqueness of solution for linear differential equation'' and have found immediately the following text that contains the exact statement of the theorem and the reference  to a texbook:
It also contains the explanation, why in the homogeneous case there are n linearly independent solutions, that I sketched in my previous post, and what happens in the non- homogeneous case.
Another reference:
D. Willett.  The Existence-Uniqueness Theorem for an nth Order Linear Ordinary Differential Equation.  The American Mathematical Monthly
Vol. 75, No. 2 (Feb., 1968), pp. 174-178
Question
As mentioned above, does UDEC(Univerisial distinct element method) have the ability to simulate the wave velocity testing?
Look at this Doc. Good luck.
Question
I have a pair of 4th order linear ODEs with non-constant coefficients that are coupled through their boundary conditions (details in the attached pdf file).
Is there any way to solve these equations either analytically or numerically?
N may be scaled out of the equations by making H1 and H2 to be proportional to N. Although the equations and most of the boundary conditions are homogeneous (noting that omega could be treated as an eigenvalue if all the equations and BCs were homogeneous, given that the equatons and BCs are linear), it is the case that one of the BCs is not homogeneous, and therefore the solution one gets will be a function of omega.
Question
How to solve huge linear system in Matlab. The A matrix is not fully sparse. Can we reorder A matrix ( with RCM ordering or so) and reduce the Bandwidth, and then solve the system ? If yes, how to reconstruct the solution according to the original ordering.
On a different note, it sounds feasible to use a Fortran routine inside matlab, for solving the linear system. Have you tried that ?
See:
1)V. G. Zverev, “Modified line-by-line method for difference elliptic equations”  Computational Mathematics and Mathematical Physics, 1998, 38:9, 1490–1498,
2)V. G. Zverev, “Implicit block iterative method for solving two-dimensional elliptic equations”, Comput. Math. Math. Phys., 40:4 (2000), 562–569
Question
Let us consider a weakly closed algebra generated by a unitary operator in a  separable Hilbert space. Generally speaking it isn't a star-algebra. Who is the author of the theorem giving a functional description of this algebra? The details concerning my question one can find in the attached file. Thank you!
Thank you! Feldman included the inverse operator as a generator of the algebra. It isn't  my case. Well, we can discuss this topic using my private e-mail str@mail.ru. You can use Russian, of course.
Question
Hi everyone,
I'm trying to find angle between 3 points (say a, b, & c with b at the canter) in an image. I've used this formula for calculating the angle
where n1 is the vector formed using points a & b while n2 is the vector formed using points c & b
The above formula corresponds to atan2(Y,X) in matlab i.e. norm(cross(n1,n2)) gives us Y and dot(n1,n2) gives us X
The formula works correct. I need the mathematical derivation of this formula. I searched about the vectors, the cross and dot products but didn't found expected result.
Hi Nikhil,
The formula follows in a straightforward manner from the geometric definition of the dot product, dot(n1,n2)=norm(n1)*norm(n2)*cos(angle), and the norm of the cross product, norm(cross(n1,n2))=norm(n1)*norm(n2)*sin(angle). Solve these for tan(angle)=sin(angle)/cos(angle), and you get the formula you are using.
Question
In some cases, learners find it easy to deal with decimal fractions than proper and improper fractions. Looking at the complex formation of fractions when adding or subtracting seems harder and almost impossible.
e.g.
0.5 + 3.3 = 3.8
1/2 + 33/10 =  38/10
Since children had an intensive experience with numbers, hence from my experience working with children in the primary level, adding decimal numbers is not a difficult concept for them to comprehend.  The challenge is just to help children extend their understand regarding place value of, one tenth, one hundreth etc... However, fraction is a relatively new concept for children to grasp.  Hence, I feel that operations with fractions should be taught later, once their are comfortable with the concept of fractions itself.
Question
Thanks
I would say you need an equation like (1- ∑| ΔV |/ (∑ |V1|+|V2|) ) * 100 .
Which if it is perfect match gives 100 and if it is anti-match (like V1=1, V2=-1) it gives 0. However, this kind of equations are not an standard way of calculating matching between two sets of numbers. The best why is to calculate the correlation function.
C=  ∑V1V2/ ∑V1 ∑V2
see
Question
I am reviewing techniques for absorbing boundary conditions for time domain simulation of wave propagation. Please attach some good references (opens source codes, If possible). My main concern is elastic wave equation but anything related would be helpful.
You will find a literature review and a novel proposal to handle absorbing boundaries in this paper
Benacchio, T., and L. Bonaventura. "Absorbing boundary conditions: a spectral collocation approach." International Journal for Numerical Methods in Fluids 72.9 (2013): 913-936.
which is available on my Researchgate page
Best regards, Luca
Question
I use usually Mathematica in my scientific work. It possesses two interesting functions Series[f,{x,x0,n}] and InverseSeries[s]. The first one generates a power series expansion for a given function f(x) about the point x=x0 to order (x-x0)^n, and the second takes the series s, and gives a series for the inverse of the function represented by s.
Of course we can implement appropriate algorithm in any mathematical software but I am looking for such programs which have a built in one as standard library (package).
I tried to find this feature in Matlab but I did not find any information on this subject in manuals?
I write for you example I expand the function sin(x+y) in taylor about (0,0) and I find the coefficient of x, y ,x^2y, xy^2,
restart:
f:=(x,y)-> sin(x+y);
e:=mtaylor(f(x,y), [x, y], 8);
coeftayl(e, [x, y] = [0, 0], [1, 0]);
coeftayl(e, [x, y] = [0, 0], [0, 1]);
coeftayl(e, [x, y] = [0, 0], [1, 1]);
coeftayl(e, [x, y] = [0, 0], [2, 1]);
coeftayl(e, [x, y] = [0, 0], [1, 2]);
coeftayl(e, [x, y] = [0, 0], [2, 2]);
Question
Dear colleagues:
What are the basic equations that can be coupled to momentum, mass, ... etc equations in cases of the effect of electric and magnetic fields on bubble dynamics?
I wait your answers, and suggestions of earlier studies and articles.
Best Regards.
Dear Esmaeel, Steffen and Hassan,
Question
I am working on numerical modelling over a wide region between 20-50 Eastern longitudes and 30-50 Northern latitudes. I have lat-lon gridded bouguer anomaly and topographical data to use for my studies. Since I need to discretize the area into equal rectangles, I need to convert the data  into metric system.
Is it possible to minimize the distortions when reprojecting? I converted the grid data to utm zone 34, It gave fake coordinates out of the zone. Both data fits on each other well on gis softwares. I am curious if I missed something here. After that I will resample it as 1km gridded surface.
Another question is, I am flattening the surface which is originally curved. This is another distortion for me because modelling program will read it as a pure flat surface. Also the rectangles I created will not be perfect rectangles in reality. How can I quantify the error?
Dear Mustafa,
As the area is rather big, the distortion have to be. The way to minimize it is to reproject by zones (each part separately) or create your own model for reprojecting which will adjust the appropriate reprojecting rule for each point depends on its location.
Anyway, for the big area the question is to find a balance between acceptable distortion and labor costs...
Hope you'll find the best solution!
Question
Let T=[A B;C D] be a real (m+n)X(m+n) stable matrix with controllable pair (D,C). Choosing such matrices T uniformly at random:
What is the probability that the nonsymmetric algebraic Riccati equation
XCX-XD+AX-B=0 has solutions ?
Coveying a lot of computer experiments, I always came up with a solution to the above mensioned nonsymmetric algebraic Riccati equation.
On the other hand, for A=-I, D=-I,C=I, B=[0 1;0 0], the matrix T is stable (with eigenvalue equals to -1) and (D,C) is controllable, but there is no solution for
X^{2}=B.
I conjecture that the measure of the set of such matrices among all matrices T as above is 0, but I cant find any way to proove it.
Can anyone suggest a good refference to such questions ?
Dear Yuri and dear Agoujil !
Many thanks for you help !
To Agoujil: the algebraic Riccati here is nonsymetric and the "Hamiltonian" matrix here is T itself. Is there any theorem of eigenvalues separation related to the nonsymmetric case? If so, could you please help me with this ?
Question
How about these coefficients are time-independent functions? Since the constant coefficients only lead to the convenience in the stability analysis.
Alternating direction implicit is a splitting method based on treating terms coming from the discretization of spatial derivatives one at the time. It does not matter whether the coefficients are constant or functions of space and time as well. The main issue, from a modern perspective, is that the computational gains which made ADI popular 30 years ago are nowadays considered to be insufficient to justify the introduction of directional splitting error that results from using this method. I do not recall any relevant paper in CFD using ADI in the last 10 or 15 years. Conclusion: use any implicit method you want for time discretization, but solve the resulting linear/nonlinear system as it is, without introducing directional splitting by trying to reduce it to one dimensional problems.
Question
What refinement indicators currently exist for hyperbolic system of PDEs, in particular, the shallow water equations? which is your favourite and why? also provide references.
We have used  in this paper
a criterion based on evaluation of the modal coefficients in a Legendre basis representation of the numerical solution. It is essentially heuristic but works well for smooth  functions.
Question
Thanks!
For periodic functions integrated over a period (as when computing the coefficients of a Fourier series) the midpoint and the trapezoidal rule coincide. These formulas can be proven to be superconvergent in this specific case, which means that for this special kind of definite integral, rather than being just second order accurate they are convergent of arbitrarily high order for functions that are C^infinity. Also for this reason, the discrete Fourier coefficients are defined by the application of the trapezoidal rule to the definite integral that defines the Fourier coefficient: there is no need of more accurate integration formulae, and on top of that Fast Fourier Transform algorithm apply for this definition of the discrete Fourier coefficients.
You can find more details in several books, see e.g.
Question
What is the name (for example, Vandermonde determinant) of the matrix or determinant showed by the following picture? I believe that it should have a name, but I am not sure.
I think there is no special name
Question
Any application field is fine.
A little math background: given a fixed curve on the 2-manifold, a velocity field over the 2-manifold, a scalar field on the 2-manifold conserved by the velocity field, and a time interval, the flux of the scalar through the curve can be determined by the distribution of the scalar at the initial time. I have worked out the proofs and am now in the stage of numerical testing. So any velocity field from a non trivial application would be great. It would be even better if you can give me the distribution of the conserved scalar at the initial time. Upon adopting your suggestion, I will surely cite your article!
Varun: The paper you pointed out is exactly what I was looking for! Currently I am not sure whether I want the counterparts in Cartesian coordinates, but I will bug you if my later numerical experiments make that necessary.
Thanks a lot!
Question
Is the piecewise finite element space subset of RT space?
No. Functions in RT-spaces are vector-valued functions whose normal components are continuous across element interfaces.
Question
What is the partition of repunit? Is It possible to find the general formula for partition of repunit?
Question
The closed graph theorem works in Banach space. Is it valid in spaces of distributions like D', E' or S' (duals of D=space of infinitly differentiable functions with compact support , E=space of infinitly differentiable functions  and S=space of infinitly differentiable functions with fastest decrease,respectively)
The closed graph theorem is valid in F-spaces, that is metrisable locally convex spaces (see Rudin Functional Analysis th.2.15, a great book).  The space of distributions D' is not metrisable because it does not hold that sequential compactness implies compactness as is true in a metric space (see https://cmouhot.files.wordpress.com/2010/02/main.pdf  where this spelled out in detail).
I am not entirely sure whether the tempered distributions and the distributions with compact support are metrisable, but I strongly doubt it.
Of course the closed graph theorem may still hold, but the F-spaces seem to be developed specifically to state a closed graph theorem in the greatest generality possible. What do you actually need the closed graph theorem for, or is this just a (good) question that came up.
Question
I'm aware that the solution to xd/dxF(x)=F(x)(1-F(x)) is (1+1/Kx)-1, where K is an arbitrary constant.  I'm looking for a similar equation with solution (1+1/Kx)-2.  It's been a long time and I'm a bit rusty with substitutions and so forth, and I thought some mathematician on RG might have the answer off the top of his or her head.  Thanks in advance for the help.
Just take the total derivative of the solution, and you will have the differential equation: F ' = -2 (1+k/x)^(-3) (-k/x^2)
Question
I am analyzing a quasi one dimensional supersonic flow field using Method of Characteristics.The medium is air. I am implementing Method of Characteristics numerically in the regions of continuous flow and jump conditions at discontinuities. Time step used is 7.5e-06 sec. During the simulation, I find that the solution becomes highly oscillatory and blows up in the regions adjoining contact discontinuity.
What kind of discontinuity it is? Before implementing numerical schemes, you should try to make sure that the equation formed is continuous in the given limits. Although, it is not possible to check in all cases, most discontinuities are removable. Explore the different methods to achieve that. You can try changing the variable through proper substitution so that the integrating variable is continuous in the new range of integration. Also, check if problem exists if you vary the time-step by different orders of magnitude. Besides, check numerical precision as well and your desired tolerance (accuracy) in the results.
Question
It is known that exact pole placement is NP-hard.
On the other hand it is known that regional pole placement in LMI regions can be solved in polynomial time.
The question is what is the complexity of regional pole placement in non-convex or unconnected regions ?
In this context, does the problem becomes harder if we are retricted to use static output feedback to place the eigenvalues instead of using dynamic feedback ?
Yet in this context, does the problem for discrete-time systems harder than for continuous-time systems ?
The  question is not fully clear to me. But I cam give some simple and standard response as follows:
For instance , if you have a controlled  plant  linear transfer function G = B/A with  polynomials B , A and n=degree A>=m=degree B ( for realizability), B and A without cancellations ( for controllability of  state space realizations of G of dimension n ,  minimal state dimension) then you can find polynomials S and R such that H (state space realizable feedback compensator)=S/R ( for instance with degree R=degree Am- degree A  > degree S) such that
AR+BS=Am
Am is a prefixed stable polynomial ( defined by the designer)  which contains the suited closed -loop poles . The above one is a diophantine equation in the ring of polynomials : for given data polynomials , you can find solution polynomials.
In that way , the closed- loop poles are assigned in ( stable: unstable also could be but non sense in applications) arbitrary positions for a linear controllable plant and this is not related to continuous or discrete( the polynomials can be in s ( Laplace op.)  or D=d/dt ( time- derivative op. )  for "continuous-time "  or in z ( z- transform) or q ( adveance discrete operator)  for "discrete-time") . It is not either related to the domain where they belong ( domain being connected or not) since Am can have zeros ( closed-loop designed poles) in any  chosen  allocation.
Question
Now I have a very complex function f(x1,x2,x3,x4) which is not concave and we also have some constrains on the range of  x1, x2, x3, x4.  I want to find the global maximum point of f  numerically. As the corresponding finding maximum funtion in matlab depond on inital point heavily and sometimes may stop in a local maximum point, so I can not use the inside function in matlab directly.  Now I try as follows, but I do not know whether it is correct theorically.   First if I fix the value of x3 and x4,   f becomes a concave function of x1 and x2 and I can find the optimizer uniquely, denoted by x1(x3, x4), x2(x3, x4). That means the optimizer depends on the value of x3 and x4 because we fix the value of x3 and x4 firt and  then can solve x1 and x2.  The maximum value of function is g(x3, x4):= f(x1(x3, x4), x2(x3, x4), x3, x4).   Next I range x3 and x4 and find the optimizer of g numerically,  denoted by x3' and x4'.  The corresponding point  (x1(x3', x4'),  x2(x3', x4'),  x3',  x4') is the optimizer of f.
Is there any problem in the steps above to find optimizer of f?
Hi,
Genetic Algorithm is on of your choices to solve this problem.
First of all you may to suppose that the chromosome is consists of 4 genes x1, x2, x3, x4.
The fitness function is your function f(x1,x2,x3,x4).
GA have so many implemenations on the internet. you cant use it in matlab or octave.
Cheers.
Question
I am still new to the topic, so I am searching for a numerical solver for fractional differential equations (FDEs) in the form of a MATLAB code or the like. I would like to test some simple FDEs at first in order to get a better understanding of the topic and then proceed with writing my own code.
So, is there any ready-made package that solves FDEs that you know of?
Question
Transient 3D advection-Diffusion equation
dT/dt+u. dT/dz =k.  (d2T/dx2+d2T/dy2+d2T/dz2)
where u and k are constants.
Depends on how strong the advection effects are. When they do not dominate too much geometric  Multigrid techniques can solve in O(N) operations if N is the total number of points. Have done this myself many times. When the advection terms increase in importance performance will reduce, but it is certainly worth a try. When the advection term dominates there are other things that work well, like downstream relaxation etc or preconditioning. Look for books on Multigrid by Briggs, Oosterlee, Wesseling, etc.
Question
Does non linear boolean function equal affine function?
Question
If   f(0) = 0  ,   f '(0) = 0  , f ' (infinite) = 1
which transformation is use full to convert the limit infinite in to some finite number?
OK, sir Can you suggest me name of books  for spline method?
Question
I have this simple system :
dx/dt = x(1-x)
I am trying to solve this system analytically and numerically using ode45 in MATLAB,  then trying to find the Fisher Information according to this formula :
I  = integral (  (1- 2*x)^2  / (x-x^2)^2) .
I wrote three different codes to find that but I am not feeling confident with the all three for many reasons: I am getting different results for the numerical and analytical, My system is stable at the carrying capacity but what I am get is oscillation or an increasing curve and infinite when I am using different time vector.
Can anyone give me any idea about how to write a code in Matlab to find that?
The analytical solution of your simple differential equation is:
x(t) = 1/(1+C*exp(-t))
and it is a sigmoid curve.
If you have x(0)=x0, then the solution is:
x(t) = 1/(1-exp(-t)*(x0-1)/x0)
Question
As I understand that for differentiable and monotone functions we can partition the period and find the total variation, but what about the case when it's not differentiable ?
For example in this article http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1083433 the authors have essentially mentioned the total variation of signum type function is 4 . But how is it done ? In general it looks to be 2.
The most intuitive way to think about the total variation is that it's the vertical distance you cover if you drew the graph of a function without taking your pen off the page.
So, for a periodic square wave, in each period you need to go from -1 to 1, and then from 1 to -1, which leads to the total variation of 4.
More precisely, the total variation of a scalar function of a single variable is defined in the first paragraph of the wikipedia article attached. To see how this definition works, consider any partition of the period of the square wave, and ensure that there is at least one point in the partition in the region where the square wave is +1, and at least one point in the region where the square wave is -1. You can check that the only contribution to the sum comes from the sections of the partition where the function goes from +1 to -1, and the other section where the function goes from -1 to +1. This gives you a value of at most 4, and hence the supremum is 4, since you can find an example which attains this value. Hope that helps!
Question
Numerical solutions to differential equations are obtainable with various methods if the parameters in the equations are known. When such parameters are not determined, they could be determined if the solution of the equation is known at a given (presumably abundant) number of points.
I'm interested to know, according to experts, what are the most reliable and most commonly used techniques used to handle this problem.
If you are also interested in the theoretical point of view, you may want to look at this paper DOI: 10.1002/cpa.21453. This work was mainly motivated by the so called hybrid inverse problems. You can find similar problems and techniques from my contribution page.
Question
I had some problems running the code at times and I wonder if I can communicate with anyone here with some experiences?
Why don't you use Monte Carlo? Much easier.
Question
ut +B ux=Cuxx
i)  u(x,o)=0
ii) Boundary cds                   u(0,t)=0    u(L,t)=1
and
iii) Boundary conditions       U(0,t)=1      u(L,t)=0
Question