Science topic

# Differential Equations - Science topic

The study and application of differential equations in pure and applied mathematics, physics, meteorology, and engineering.

Questions related to Differential Equations

I identify as Filipino Jewish yet I have many Celtic Ancestors. Could Celts be the most privileged people?

All races are equal and to insure equal treatment it must be noted that some peoples are more privileged than others. Celts may be, beyond a reasonable doubt, the most privileged. Celts being included in Western Europeans with the Norse and the Germanics is definitely more than a geographic coincidence because all three of those groups have the highest privilege of any race. Western European may be loosely its own race given the intermixing that has occured over such abundant time between Norse, Germanics and Celts. Even contemporary power differentials between different races do not necessarily describe net privileges.

Work Cited

CIA . "Real GDP per capita ." cia.gov . www.cia.gov/the-world-factbook/field/real-gdp-per-capita/country-comparison/. Accessed 12 Sep. 2023.

CIA . " Liechtenstein - The World Factbook - CIA Central Intelligence Agency (.gov) https://www.cia.gov › liechtenstein › summaries." cia.gov . www.cia.gov/the-world-factbook/countries/liechtenstein/summaries/. Accessed 12 Sep. 2023.

Britannica, The Editors of Encyclopaedia. "Liechtenstein". Encyclopedia Britannica, 5 Jul. 2023, https://www.britannica.com/place/Liechtenstein. Accessed 12 September 2023.

Britannica, The Editors of Encyclopaedia. "Alemanni". Encyclopedia Britannica, 15 Sep. 2011, https://www.britannica.com/topic/Alemanni. Accessed 12 September 2023.

US Department of State . "Monaco (08/05) ." state.gov. 2009-2017.state.gov/outofdate/bgn/monaco/51086.htm#:~:text=Ethnic%20groups%20(2003)%3A%20French,90%25%2C%20other%2010%25. Accessed 12 Sep. 2023.

World Directory of Minorities and Indigenous Peoples. " Monaco - World Directory of Minorities & Indigenous Peoples minorityrights.org https://minorityrights.org › country › monaco." minorityrights.org . minorityrights.org/country/monaco/#:~:text=Some%2040%20per%20cent%20of,per%20cent%20of%20the%20population. Accessed 12 Sep. 2023.

CIA. " Luxembourg - The World Factbook Central Intelligence Agency (.gov) https://www.cia.gov › countries." cia.gov. 1 Sep. 2023. www.cia.gov/the-world-factbook/countries/luxembourg/. Accessed 12 Sep. 2023.

Ohnemus , Alexander . "The Antiracist Differential Equation." ResearchGate.net . www.researchgate.net/publication/373434784_The_Antiracist_Differential_Equation. Accessed 28 Aug. 2023.

PopMatters Staff. "TERRY JONES’ BARBARIANS ." popmatters.com. 19 Feb. 2008. www.popmatters.com/terry-jones-barbarians-2496173287.html#:~:text=The%20Celts%20developed%20solar%20calendars,elderly%2C%20the%20infirm%20and%20children. Accessed 28 Aug. 2023.

Dahmer, Adam . "Revisiting the achievements of the Ancient Celts : evidence that the Celtic civilization surpassed contemporary European civilizations in its technical sophistication and social complexity, and continues to influence later cultures.." ir.library.louisville.edu. http://doi.org/10.18297/honors/11. Accessed 28 Aug. 2023.

Dahmer, Adam, "Revisiting the achievements of the Ancient Celts : evidence that the Celtic civilization

surpassed contemporary European civilizations in its technical sophistication and social complexity, and

continues to influence later cultures." (2013). College of Arts & Sciences Senior Honors Theses. Paper 11.

ENCYCLOPEDIC ENTRY. "Europe: Resources ." education.nationalgeographic.org . education.nationalgeographic.org/resource/europe-resources/. Accessed 5 Sep. 2023.

Ohnemus , Alexander . "Interplanetary Reparations CRT Welfare State Proposal." ResearchGate.net . www.researchgate.net/publication/372948149_Interplanetary_Reparations_CRT_Welfare_State_Proposal. Accessed 19 Aug. 2023.

C. CORDUNEANU, Sopra I problemi ai limiti per alcuni sistemi di equazioni differenziali non

lineari, Rend. Acad. Napoli 4 (1958), 98-106.

Cartoon Cosmological Physics: South Park takes place in another universe so it can be absurd.

Differential Equations:

(F)' = A

F: Fiction

A: Absurdity

The show's absurdity is a derivative of being fictional.

What are your thoughts?

Dear colleagues

To find coexisting attractors in a chaotic system, I use the continuation diagram. Here in each iteration, the initial conditions x(0) for the chaotic system are set as the final conditions x(t_final) from the previous simulation.

We do so as we increase the parameter under study (forward continuation diagram), and as we decrease the parameter (backward continuation diagram).

In a system I am studying though, I still know that coexisting attractors exist, and using both continuation diagrams, I still cannot depict all of them. The diagram cannot 'catch' them.

Is there an alternative, or a solution to this?

Which is the best software to solve Fractional Order Differential Equations?

to solve Fuzzy Fractional Integro Differential Equations If anyone owns it, can you send it to me?

Dear all Mathematician,

Many Mathematician written in his/her research paper that fractional integral and differential Equations used in science and technology (write many fields), etc. But actually How we corelate it? can we give some exact practical example of it?

Are there some methods to handle stochastic partial differential equations with an integral term as drift coefficient? One method is semigroup theory but are there other methods to find solution or show the existence of solution. Any references are also welcome.

It seems to me that "determinism" is not a rigorously defined concept. It obviously involves

the order-structure of time T(what determines "before" and "after") as well as the possibility of capturing the instantaneous state of the universe at a given time t in T by an element in a certain phase-space Q.

Our notion of "determinism" will greatly depend on the order-structure of T as well as Q (for instance, its cardinality: is an infinite amount of information required to specify the state of the universe).

The popular concept of "determinism" corresponds to finite computational determinism. T is given the order structure of the natural numbers N and Q is finite. Then the state q(t) of the universe at time t can be computed via a recursive function F from the states q(t') at previous times for t' < t (more commonly the immediately preceeding state state is enough ?).

But suppose that F were not recursive but belonged to some other order of the arithmetical hierarchy (let us say Sigma^1) ? Could we still speak of "determinism" ? What if F were beyond the arithmetical hierarchy ?

What is the best way of extending our notion of "computability" to the case in which T has a dense linear order and/or in which Q has infinite cardinality ? How do we express the "determinism" paradigm of differential equations in a rigorous way ? What if the coeficients of analytic solutions are not computable ?

By "predetermination" I mean the idea that the entire evolution of the universe through time already "exists". Suppose that the law of evolution of the universe F were undefinable in first-order logic but that we had predeterminism. I call this "metaphysical predetermination".

What criteria or what experiment can we conceive of that could distinguish pure chance or free will

from metaphysical predetermination ?

I also note that for us conscious beings it seems arguable that finite computational determinism at least is false.

If we take a description of the solar system in terms of Newton's equations then the solutions are time-reversible.

But many phenomena in nature are observed to be non-reversible, "dissipative", hence not having time-reversible solutions. For instance, a glass falling off the table and breaking.

The big question is: can the second law of thermodynamics be

*deduced*from the fundamental differential equations of physics ?Or more generally are there differential equations whose solutions are mostly entropy-increasing ?

On the other hand can we find (a system of) differential equations whose solutions are generally entropy-decreasing ? Or in which entropy-decreasing phenomena occur in relatively frequent bursts ? Differential equations which would have solutions in which the pieces spontaneously assemble into the glass on the table ?

Contemporary physics is essentially incomplete (cf. the need for dark matter, dark energy, extra dimensions, etc.). Perhaps in the complete picture

*entropy is actually strictly conserved*. The entropy-increasing forces/fields are counterbalanced by (at present unknown) entropy-decreasing ones, in which entropy-decreasing phenomena occur in relatively frequent bursts.Then it is this entropy-decreasing aspect of nature that is the main cause of life, the cause of the relatively frequent bursts of increased self-organisation and complexity (which would then be further modulated (or "selected") by the constraints of the environment and the ecosystem).

Perhaps the "collapse of the wave-function" could be approached thermodynamically as well ?

Consider the class of real elementary functions defined on a real interval I. These

are real analytic functions. How can we characterise their power series ? That is, what

can we say about their coeficients, the structure of the series of their coeficients ?

For instance there are coeficients a(n) given by rational functions in n , or given by combinations of rational functions and factorials functions, computable coeficients, coeficients given by recurrence relations, etc.

It is easy to give an example of a real analytic function which is not elementary. Just solve the equation x'' - tx = 0 using power series. This equation is known not to have any non-trivial elementary solution, in fact it has no Liouville solution (indefinite integrals of elementary functions).

I am working on topology optimization for photonic devices. I need to apply a custom spatial filter on the designed geometry to make it fabricable with the CMOS process. I know there exist spatial filters to remove the pixel-by-pixel and small features from the geometry. However, I have not seen any custom analytical or numerical filters in the literature. Can anyone suggest a reference to help me through this?

Thanks,

I want to learn about Solving Differential Equation based on Barycentric Interpolation. I want to learn this method, if someone has hand notes it would be great to share with me. I need to learn that in 2 weeks. Thanks in advance.

Consider the Newtonian n-body problem. An initial condition must specify the initial positions and velocities for each of the n point masses. Thus the space of initial conditions has dimension 6n. I am interested in the subset G of initial conditions which yield solutions that:

1. Are global (defined for all t > t_0)

2. Do not have have collisions or any particle escaping into infinity

3. Are real analytic: at each t there is a neighbourhood U(t) such that each position component of each particle is given by a convergent power series in t.

Note that real analytic functions which are real analytic on the whole R need not be given globally by a convergent power series as in the complex analytic case (of entire functions).

For if we extend a real analytic function to the complex numbers, such as 1 /1 + x^2, then it may well have a pole. We call such real analytic functions piecewise-entire.

What can be said about G topologically ?

When are the coeficients of the convergent power-series computable (possibly different for each member of a countable cover of the reals) ?

Are there examples of solutions which satisfy 1 and 2 but not 3, i.e. are smooth but not real analytic ?

I mean chaotic flows. That is possible for chaotic maps.

I’m trying to fit some kind of causal model to continuous value data by solving differential equations probabilistically (machine learning).

Currently I’m solving complex-valued vector quadratic differential equation so there are more cross correlations between variables.

dx(t)/dt = diag(Ax(t)x(t)^h) + Bx(t) + c + f(t)

or just

dx(t)/dt = diag(Ax(t)x(t)^h) + Bx(t) + c

diag() takes diagonal of the square matrix.

But my diff. eq. math is rusty because I have studied differential equations 20 years ago. I solved the equation in 1-dimensional case but would need help for vector valued x(t).

Would someone point me to appropriate material?

EDIT: I did edit the question to be a bit more clear to read.

Fuzzy differential equation (FDE) is a new area in fuzzy analysis. It plays a vital role in modelling of real physical problem involving uncertainty parameters. There are many ways of interpreting the FDE. Which one is the best so far?

I am trying to solve the differential equation. I was able to solve it when the function P is constant and independent of r and z. But I am not able to solve it further when P is a function of r and z or function of r only (IMAGE 1).

Any general solution for IMAGE 2?

Kindly help me with this. Thanks

Which is the best software for finding Optimal Control for System of Fractional Order Differential Equations?

Hi

I wanna solve partial differential equation in terms of x and t (spatial and time), As I know one of the most useful way for solving pde is variable separation. well explained examples about mentioned way are wave equation, heat equation, diffusion....

wave equation is Utt=C^2 .Uxx

in other word; derivatives of displacement to time, equals to derivatives of displacement to spatial multiplied by constant or vice versa.

however my equation is not like that and derivatives are multiplied to each other.for example : Uxx=(1+Ux)*Utt

Im wondering how to solve this equation.

I will be thankful to hear any idea.

I want to solve analytically a coupled 2nd order space-time problem, originated from an optimal control problem. One of the problems is forward, another is backward in time. For example, (i) $y_t-y_{xx}=u, y(x,0)=0, y(0,t)=0, y(1,t)=g(t)$ (ii) $-p_t-p_{xx}=y, p(x,T)=0,p(0,t)=0, p(1,t)=h(t)$ with the coupling condition $p(x,t)+c*u(x,t)=0$ in $(0,1)\times (0,T)$. I have tried separation of variables, but it is getting complicated, any suggestions?

Dear all,

I have to model three point beam bending test on asphalt concrete beam with geosynthetic interface layer at 2/3rd depth from the top(using abaqus 6.12-3). I have modeled three parts separately i.e.. asphalt, geosynthetic material and cohesive elements(to define interface). When I give the crack region for XFEM crack, there is an error showing that crack region should contain only one instance. Can someone suggest me how to move further in this problem.. Suggestion would be really helpful. Thanks in advance.

Regards

Prashanthi

Are journals accepting the papers based on contents or Based on the Profile of authors ? The same kind of paper gets reject from a journal as out of scope but the same kind of papers are published

I want to work on solving differential equation using artificial neural network. I saw some paper is working on closed form solution. But will that be a good idea? But in this way it is not possible to deal with real data which may be discrete . Is there any paper which works to solve differential equation using ANN using totally numerical way?

The Van der Pol oscillator can be give in state model form as follows:

dx/dt = y

dy/dt = mu (1 - x^2) y - x,

where mu is a scalar parameter.

When mu = 0, the Van der Pol oscillator has simple harmonic motion. Its behavior is well-known.

When mu > 0, the Van der Pol oscillator has a stable limit cycle (with Hopf bifurcation).

While we can show the existence of a stable limit cycle with a MATLAB / SCILAB plot with some initial conditions and some positive value for mu like mu = 0.1 or 0.5 (for simulation), I like to know if there is a smart analytical proof (without any simulation) showing the existence of a limit cycle.

Specifically I like to know - is there any energy function V having time-derivative equal to zero along the trajectories of Van der Pol oscillator? Is there some smart calculation showing the existence of a stable limit cycle..

I am interested in knowing this - your help on my query is most welcome. Thanks!

For example, in terms of ordinary differential equation we have ordinary derivative in boundary condition.

.

Hi,

I am doing research on differential equations but came around the term dynamic system and dynamical system. Some papers says dynamic systems and some other says dynamical systems. I can't figure what the exact technical difference between these two terms or are they just two different words used with the same meaning. Thanks in Advance

He is the author of one of the most important works of of the early twenty century in the qualitative theory.

The original differential Equation associated with BVP or IVPs is transformed into an equivalent integral equation to solve.

i have two nonlinear coupled differential equations and want to know if they have unique or infinite solutions.

Let $X$ be a hamiltonian vector field on the plane. Then either has no closed orbit or it has infinite number of closed orbit. Now what can be said about higher dimensional hamiltonian vector fields? Is there a hamiltonian vector field on R^4 which has a finite number of periodic orbits?

Please see the following corresponding MO link:

Thank you

Under what conditions the solution of nonlinear ordinary differential equations belong to the space of continouse functions? Need opinions

Is there a polynomial Hamiltonian vector field with a finite number of periodic orbits?

Please see this MO question:

Thank you

Differential equations are known to be potential tools for modeling and predicting recurrent phenomena in time and space, whether simple or more elaborate models. In this sense, how to approach teaching in this area in a more didactic and methodological way?

As equações diferenciais são conhecidas por serem ferramentas potenciais para modelar e prever fenômenos recorrentes no tempo e no espaço, sejam modelos simples ou mais elaborados. Nesse sentido, como abordar o ensino nessa área de forma mais didática e metodológica?

As we know, the vastness of the subject is realized by the variety of interdisciplinary subjects that belong to several mathematical domains such as classical analysis, differential and integral equations, functional analysis, operator theory, topology and algebraic topology; as well as other subjects such as Economics, Commerce etc.

I need to solve the Laplace Differential Equation Del²Φ = 0 and find the electric potential at different points due to a charged plate with uniform surface charge density λ/ m².

Dear colleagues, can you please help me with your valuable suggestions?

Thanks and Regards

N Das

First of all I should say that I work in probability and my knowledge about PDEs is quite small, so this question could make little sense, please let me know if something is not well stated.

While dealing with approximations methods for SDEs a I've noticed a particular connection between an SDE and a deterministic PDE of the form:

u_t+σ(t)⋅ u_x+x⋅u=b(t,x,u)

where b is a Lipschitz continuous function in u.

I've tried searching online for application of this kind of equation but unfortunately I wasn't able to find anything concrete. In the book by Moussiaux, Zaitsev and Polyanin "Handbook of first order PDEs" they discuss methods for solving this kind of equations but they don't provide examples of applications.

I suspect this could be connected somehow to the transport equations, but I am not entirely sure. Do you know of some references for applications of this particular equation?

Thanks in advance for your replies.

find to Differential Equations with the same method

UPDATE: The values of the variables that I am currently concerned with are:

a~65

V~3.887

While trying to solve a circuit equation, I stumbled onto a type of Lienard Equation. But, I am unable to solve this analytically.

x'' + a(x-1)x' + x = V-------------------------(1)

where dash(') represent differentiation w.r.t time(t).

The following substitution y =x-V and w(y) = y', it gets converted into first order equation

w*w' + a(y+V-1)w + y = 0; ---------------------- (2)

here dash(') represent differentiation w.r.t y.

if I substitute z = (int)(-a*(y+V-1), (int) represent integration. The equation gets converted into Abel equation of second kind.

w*w' - w = f(z). -------------------- (3) differentiation w.r.t z.

it get complicated and complicated.

I would like to solve the equation (1) with some other method or with the method that I had started. Kindly help in solving this,

Thank you for your time.

I'm trying to solve an important functional differential equation. But the problem is, it involves functional (variational) derivative, instead of partial ones. Can anyone suggest me a book where it is discussed? Any kind of material related to it may help.

Studies should be recent and involving O.D.E and/or P.D.E.

I have a set of nonlinear ordinary differential equations with some unknown parameters that I would like to determine from experimental data. Does anybody know of any good freely available software, or good reference books?

Which writing style do you prefer when doing the following tasks, using a Stylus or pencil and paper? why?

1- Writing text for a scientific paper.

2- Solving a differential equation that require a number of pages.

Your comparison is based on being (faster, easier, storage and backup, etc...)

What would be the logical mistake if indicial equation roots are actually different by an integer, but still mathematically one solves the equation following the route of non-integer difference, still evaluated at ordinary singular point? Suppose,the ODE is of second order. Would doing so might result in two linearly independent solutions whose one particular linear combination in a terminating series? I am following textbook on Ordinary and Partial differential equations by Dr. M.D. Raisinghania, but logic behind the method is not mentioned in the book.

Hello;

I have this 2 nonlinear 2nd order differential equations.

ode = (diff(y,t))^2 == (V^2)/(A-B*cos(y)); ( in matlab format) IC:y(0)=0

ode = (diff(g,t))^2 == (V-(t-X))^2/(A-B*cos(g)); ( in matlab format) IC: g(Tfin)=XFin

y(t)=?

g(t)=?

How do i solve these equations other than using numerical solutions? Matlab has no analytical solution for thEse.

I need to find X by equating y(X)=g(X). So I need to find y(t) and g(t) analytically to equate them. Then I would solve for X to find it symbolically, that was my plan.

If I choose runge kutta 4th order or any other numerical solution, i dont think I can find X symbolically but I am not sure. Any comment would be appreciated!

Caner

Dear Pr Mainardi,

I am a teacher researcher at Badji Mokhtar University Annaba,Algeria. I have a problem with the inverse Laplace transform of fractional differential equations.

In your article : Integral and Differential Equations of Fractional Order Rudolf GORENFLO and Francesco MAINARDI you used the Hankel path to find the inverse Laplace transform.

If you can help me find the inverse Laplace transform of fractional differential equations.

Best Regards,

Dr Benchettah

I have started a research on the modeling of multibody systems dynamics, and since dynamics mean to determine the Differential Equations of Motion, i am called to make the resolution of these OEM.

I need to know how to do so, so i started working on an easy example, but i find it difficult because i have no background on mathematics especially numerical integration.

My question is how to proceed to solve a DAE numerically using one of these methods ; Runge kutta, Adams - Bashford or Moulton, Central differences or Gaussian method?

Hi dear researchers,

I am currently implementing

**direct collocation NMPC**,However, I have a doubt on the control value for the first control u_0

that should be applied to the system once the control trajectory is found as collocation downsamples the system dynamics before solving the dynamic NLP problem.

will the control value u_0 be (u*(0)+u*(1))/2?

x_k+1 = f(x_k,u_0);

Should this be the case, will that not have a detrimental effect in the accuracy of the true control?

Kind regards

I have solved nonlinear FODE for an IVP, but in case of solving linear FODE, my question is can I solve the linear set of equations using the same algorithm?

Thank you in advance.

**A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard, hence we need an answer for the above question . Thank you very much!**

Greetings...

Is there a "general" solution to the telegrapher's equation {voltage or current on a transmission line}?

I mean by "general": without ignore of losses and without assuming sinusoidal excitation.

Kindly refer to some references on that topic.

Dear colleagues,

I am very pleased to announce the launch of the new issue of the

*Communications in Advanced Mathematical Sciences-CAMS.*The CAMS will continue to be an international journal mainly devoted to the publication of original studies and research papers in all areas of mathematical analysis and its numerous applications.

All Papers in Volume II, Issue I of CAMS:

**1) A Unified Family of Generalized**

*q*-Hermite Apostol Type Polynomials and its Applications**Authors:**Subuhi Khan , Tabinda Nahid

**2) Analytical and Solutions of Fourth Order Difference Equations**

**Authors:**Marwa M. Alzubaidi, Elsayed M. Elsayed

**3) A New Theorem on The Existence of Positive Solutions of Singular Initial-Value Problem for Second Order Differential Equations**

**Authors:**Afgan Aslanov

**4) Analysis of the Dynamical System**

**in a Special Time-Dependent Norm**

**Authors:**Ludwig Kohaupt

**5) An Agile Optimal Orthogonal Additive Randomized Response Model**

**Authors:**Tanveer A. Tarray , Housial P. Singh

**6) On Signomial Constrained Optimal Control Problems**

**Authors:**Savin Treanta

**7) Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green's Function**

**Authors:**Habib Djourdem, Slimane Benaicha, Noureddine Bouteraa

**8) Solution of Singular Integral Equations of the First Kind with Cauchy Kernel**

**Authors:**Subhabrata Mondal, B.N. Mandal

*For more details about this issue please visit*http://dergipark.gov.tr/cams/issue/44087

**:**I hope you will enjoy this new issue of CAMS and consider submitting your future work to this promising academic venue.

With my best regards,

Emrah Evren Kara

Yes, there is a new method which is called Piecewise Analytic Method (PAM). It does more than Runge-Kutta.

1. PAM gives a general analytic formula that can be used in differentiation and integration.

2. PAM can solve highly non-linear differential equation.

3. The accuracy and error can be controlled according to our needs very easily.

4. PAM can solve problems which other famous techniques can’t solve.

5. In some cases, PAM gives the exact solution.

6. ....

You can see :

Also, You can write your comments and follow the update of PAM in the discussion

There are various methods that have been used in solving the fractional differential equations, but I am wondering what are the most powerful and efficient ones that can be applied effectively in solving the fractional differential equations?

For example there exists methods to find expansions of Gauss Hypergeometric Function( Binomial Sums, Differential Equation method etc.)

When plotting a bifurcation diagram in nonlinear dynamics, the axis x displays a given phase parameter. Are there examples in which the phase parameter stands for time passing (for example, from the value T0 to the value T200 seconds, or months, or years)?

Thanks!

To make an example, I was thinking to something like the one in the Figure below, concerning the phase transitions among liquids, solids and gases: if you leave, e.g,., that the temperature raises of one degree every second, can we say that the axis x displays time (apart temperature values)?

Hello,

I am using a Kalman-Filter for a System with a watertank . Theres a known voltage input signal

**U_pump,**that is used to control a waterpump. The waterpumps flowrate is proportional to the voltage U_Pump, so the relationship between U_Pump and the flowrate Q is**Q = K * U_Pump.**As you can see, this is just a very easy model and probably not very accurate. So I have a Sensor, and want to decrease the uncertainty in this model by using the sensor to update the flowrate Q.

Therefore I have set Q as a state.

So the Differential Equation for the Flowrate Q is.

Q_dot = 0* Q + 0*U_Pump;

with A = 0; B = 0;

Because Q_dot is just the Rate at which Q changes over time, I cant put the K * U_Pump in there.

This means, in the discrete form:

Q_k = Q_k-1, which is a static model for the Flowrate Q, although I have the model Q = K*U_Pump

So my predicted Measurement of Q with the Kalman-Filter is just

Q_k_predicted = 1 * Q_k-1; which is not correct when I am adjusting the Voltage U at this timestep, therefore I can only use this for a static Voltage.

So is there a way to update U_Pump with the sensor value, or is there non?

What is the general solution of x^{(k)}+(\cos t + \sin\sqrt{2}t ) x = 0 ordinary differential equation for k>=2?

It's very similar to modified Bessel function. But it's second coefficient is 2 rather than 1.

r^2dX^2(r)/dr^2+

**2**rdX(r)/dr-(r^2+p^2)X(r)=0 , p is a known quantity.Dear All,

I am hoping that someone of you have the

**First Edition**of this book (pdf)**Introduction to Real Analysis by Bartle and Sherbert**

The other editions are already available online. I need the First Edition only.

It would be a great help to me!

Thank you so much in advance.

Sarah

In the process of simulating a periodic HIS, a Floquet-Port needs to be assigned at an appropriate height over the surface. How much is that height supposed to be? and how can I be so sure of the results if any small change in this height dramatically changes the results??

Also, do I need to use PML?

P.S.: I am using HFSS

What's essential when it comes to learning, understanding and teaching differential equations?

What concepts are important to cover? And, how to cover them?

What does research tells us about all this?

Your suggestions/references to literature are very welcome !

I'm researching about a Timoshenko Beam with a Transverse vibration. The displacements are very small.

Equation of moving load is:

Fm=-(N+m*d^2v/dt^2) δ(x-ut)

N=mg+f , mg is load weight and f is a constant force , V is the transverse displacement

When velocity is constant, with the change of a variable the equation can be solved, but when the velocity is variable, how can I solve the equation of beam?