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# Numerical Mathematics - Science topic

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Questions related to Numerical Mathematics

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.

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

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?

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

"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.

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.

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!

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?

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.

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)

Thanks in advance

Nasser

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'**:

**x**, and

^{2}+ y^{2}= 1**C:**

**(x-1)**,

^{2}+ y^{2}= 9if 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

**C**

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?

**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.**

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.)

I'm looking for optimizing multivalued vector valued function.

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?

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**?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

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.

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.

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?

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

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

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.

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

Academic : ham.daeikasmaei@iauctb.ac.ir

Skype : hamed-daei

Whatsapp and imo : +989123937613

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

Thanks in advance!

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.

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!

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*2^{nd}order derivatives,**x****''**and**y****''**. Online documentation gives methods of decomposing*n*2^{nd}order ODEs into 2*n*1^{st}order equations, but they do not work for when there are two 2^{nd }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.

Can anybody suggest me the best book in Generalized Fibonacci sequence and its applications in differential equations?

The problem occurs when the first element of the summation includes $\Gamma(0)$ in the denominator which is not defined?

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?

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

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

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?

Appreciating your time, looking forward for your answers.

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.

Can one give a general formula for the product of three different finite sums? See the picture attached with this 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.

As mentioned above, does UDEC(Univerisial distinct element method) have the ability to simulate the wave velocity testing?

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?

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 ?

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!

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

angle_in_radian = atan2(norm(cross(n1,n2)),dot(n1,n2))

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.

Please help me, i need to add this point in my thesis.

Thanks in advance

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

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.

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?

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.

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?

thanks in advance.

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 ?

How about these coefficients are time-independent functions? Since the constant coefficients only lead to the convenience in the stability analysis.

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.

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.

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!

Is the piecewise finite element space subset of RT space?

What is the partition of repunit? Is It possible to find the general formula for partition of repunit?

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)

I'm aware that the solution to x

^{d}/_{dx}F(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.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.

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 ?

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?

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?

Transient 3D advection-Diffusion equation

dT/dt+u. dT/dz =k. (d

^{2}T/dx^{2}+d^{2}T/dy^{2}+d^{2}T/dz^{2})where u and k are constants.

Does non linear boolean function equal affine function?

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?

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?

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.

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.

I had some problems running the code at times and I wonder if I can communicate with anyone here with some experiences?

u

_{t }+B u_{x}=Cu_{xx }_{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}

a c 0 … 0 1

b a c 0 … 0

0 b a c 0 … 0

0 0

. a c 0

. b a c

1 0 … 0 b a