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
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I mean chaotic flows. That is possible for chaotic maps.
What makes it possible to deduce or suppose the existence of chaos are the positivé exponents of lyaponuv and not the eigenvalues ​​(multipliers) associated with a fixed point or a cycle.
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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.
we can see this:
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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 do not think such a general question can have a definite answer. A particular question to classify the initial data leading to closed periodic trajectories (without collisions) for 3 body case makes sense and is of importance. I guess the most appropriate person to discuss all that is R Montgomery from UC Santa Cruz.
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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.
I understand that you want to solve a system of differential equations, but the details you provide is a bit confusing.
A second order differential equation is one that contains the second derivative of the unknown function and maybe, but not necessarily, the first derivative. These equations are linear if the unknown function and its derivatives are raised to the first power only; otherwise they are non-linear. Usually, by quadratic differential equation, we mean one that contains the square of the unknown function. They are different things.
The example you provide seems to be a first order differential equation, but could you write it more clearly?
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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?
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Hello sirs, as for Matlab code for fractional differential equations, I would recommend FOTF toolbox, FOMCON toolbox and Prof Garrappa's Matlab implementation.
There are also some open source softwares focus on fractional differential equations I am working on:
Any suggestion and advice are welcomed😀
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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
check out this paper using a Laplace transformation for solving nonlinear nonhomogenous partial equations
the solutions are not trivial which is different if your coefficients and the pressure P are constant
Hopefully it helps
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Which is the best software for finding Optimal Control for System of Fractional Order Differential Equations?
Thank You for your response sir, surely i read those ref papers you mentioned, i will learn and use FOMCON toolbox. @ Kibru Teka Nida
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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.
Dear Alireza Akbari looks like your equation is a nonlinear PDE, there are tables for those:
However I could not find yours, but don't worry, I tell you a trick we use in MHD.
1. You linearized it, i.e., you solve the PDE as a function of ei(k.r - omega t)
2. You get a complex polinom, but I don't see any parameters in your equation.
3. Anyway you can try an algebraic manipulator such as math or maple and find the roots. However, I find it strange that there is not a parameter, you need it to scan the complex solution.
Best Regards.
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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?
PDE's can be solved with the software Mathematica. They have a free version online.
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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
This is a good question.
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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
Many of these answers do not fit with my experience. It is certainly true that older authors often know how to write well and to choose good topics to research. Of course, some don't. There will certainly be cases of injustice in terms of the acceptability of a paper for publication, but this happens both ways: some papers that were rejected should have been published and some that were published ought not to have been. This happens in many journals. In the long run, though, if someone is always finding it difficult to publish, then perhaps they do need to rethink what they are doing.
It is very unlikely that a Chief Editor will know everything about the topic of his/her journal, which is why Assistant Editors are often appointed and papers are assigned to them which are more closely related to the AE's fields of expertise. Even that process is imperfect, though, and I find that it takes much longer to find appropriate reviewers in some cases and very great care has to be taken.
Three experiences of mine follow. They may be of some value.
1. In the last few years as an editor I have had to recommend that a paper by a very eminent scientist shouldn't be published. I admit that this caused me a lot of anguish, but the reviews were clear and there was no option. One has to be objective and judge the quality of the work, not the length of the CV, the accolades and prizes etc.
2. This one is an example of someone who did not learn from their mistakes or from the objective advice given by the reviewer (me). It involved a paper which I had to review five times, twice for one journal, twice for a second and finally once for a third. Somehow three different editors chose me as reviewer. The sadness here was that, once corrected, this paper would have been good enough to publish. I gave essentially the same review each time, and while minor cosmetic issues were addressed the central error remained unchanged and ignored by the authors. This cynical attitude on the part of those authors, where they continue to try yet more journals in the hope that sufficiently poor reviewers will eventually be found, ought to be deprecated. Strong words, yes, but standards do need to be maintained.
3. Many years ago one rejection I had of my own single-author paper was because a reviewer had said, "This is known as a difficult problem and I do not believe that the author has solved it". I complained to the editor and asked him to ask the reviewer to tell me where my error was. Within a week the paper had been accepted with no comment from the reviewer and with no revisions required! Perhaps this was a lazy reviewer> I have no idea. Perhaps there was a systemic problem with the journal for it no longer exists.
In my experience, then, there are some lazy reviewers and some who clearly do not understand the work that they have been asked to review. Authors often notice these. Strangely, there are some reviewers who provide what I would regard as very minor comments but who recommend Major Revisions. As a general principle, long reviews are often about as objective as one might get, and will signify that the reviewer knows what he/she is talking about. So do attend to those comments carefully. Short reviews (even those which are essentially "This is brilliant, the best thing ever, it must be published") can be unreliable.
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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?
A package for automating this can be found in Julia called NeuralPDE.jl.
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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!
The following steps are helpful for finding the dynamics of the system:
1. Check the stability of equilibrium by the Jacobian method.
2. Find the parametric conditions of stability/unstability.
3. Check what type of Hopf bifurcation (subcritical/supercritical) is there.
4. If the Hopf bifurcation is supercritical, then there is a stable limit cycle.
5. You can also solve the system numerically in Matlab/Mathematica and plot the limit cycle.
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For example, in terms of ordinary differential equation we have ordinary derivative in boundary condition.
.
Dear Dr. Sabermahani
Please, see the following paper in which there are boundary value conditions with fractional derivatives:
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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
These two expressions are the same. Maybe your mean is difference equations and differential equations.
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He is the author of one of the most important works of of the early twenty century in the qualitative theory.
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The original differential Equation associated with BVP or IVPs is transformed into an equivalent integral equation to solve.
At least two (related) reasons come to mind:
Integral equations involve bounded linear integral operators (or nonlinear integral operators that are at least continuous), whereas differential equations involve unbounded (discontinuous) differential operators. As a consequence, integral operators have better analytical behavior, making them more suitable for theoretical (e.g. fixed-point theorems) and for numerical (e.g. discretization) treatment.
Typically, solutions of the differential problem are also solutions of the integral problem, but the converse is not necessarily true. This is already the case for ODE initial value problems $\dot{x}(t) = f(t, x(t))$ where we allow discontinuities of $f$ in the first variable, and it becomes even more important in the context of PDEs. So, if the modeling context suggests that solutions with non-differentiable time dependence are relevant, then the integral formulation is the relevant formulation.
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i have two nonlinear coupled differential equations and want to know if they have unique or infinite solutions.
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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?
Thank you
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Under what conditions the solution of nonlinear ordinary differential equations belong to the space of continouse functions? Need opinions
The solution of ODE is an Integral function, I think it always belongs to the space of continouse functions.
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Is there a polynomial Hamiltonian vector field with a finite number of periodic orbits?
Thank you
Biswanath Rath The Hamiltonian vector field associated to H(x,y,z,w) is simply the following vector field:
x'=\partial H/\partial z
y'=\partial H/\partial w
z'=-\partial H/\partial x
w'=-\partial H/\partial y
So our question is the following: Is there a polynomial H such that the above vector field possess (exactly) k periodic orbit for a finite number k different zero?
The motivation for this question: In dimension 2 (rather than dim. 4) this situation can not occured(we have a bound of periodic orbits, if there is any)
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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?
According to my experience tow books are good
Partial Differential Equations I: Basic Theory
Springer-Verlag New YorkMichael E. TaylorYear:2011
and
Schaum’s Outline of Differential Equations
McGraw-Hill EducationRichard Bronson, Gabriel B. CostaYear:2014
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Would prefer a book for learners.
see
Anh C.T., Hung P.Q., Ke T.D., Phong T.T.: Global attractor for a semilinear parabolic equation involving Grushin operatot. Electron. J. Differ. Equ. 32, 1–11 (2008)
D’Ambrosio L.: Hardy inequalities related to Grushin type operators. Proc. Am. Math. Soc. 132, 725–734 (2004)
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for the approximate solution of different
types of differential equations"
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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.
Fixed point theory solved many nonlinear problems. In the future, it will play more crucial role in many real world problems.
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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.
Thanks and Regards
N Das
Dear Prof. Nityananda Das, in addition to all the interesting answers to your thread, most cases that have an analytical solution of the Laplace equation are in the following theoretical book and in its complementary book with the exercises:
• Equations of Mathematical Physics (Dover Books on Physics), 2011 by A. Tikhonov and A. Samarskii.
• A Collection of Problems in Mathematical Physics (Dover Books on Physics), 2011 by B. Budak, A. Samarskii, and A. Tikhonov
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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?
Filippo Maria Denaro Grazie per la tua risposta, dovrò cercare un po' circa la equazione di onda con coefficienti variabili. La verità è che io lavoro in calcolo stocastico da una prospettiva matematica, questa equazione è venuta fuori e volevo capire se avesse una qualche rilevanza dalla modelistica.
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I think the easiest and best way is using operational methods , see papers of professor G Dattoli
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find to Differential Equations with the same method
Your question needs clarification, so that we can share with you as required
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Is it useful to transform problems between difference equations and differential equations ?
Dear;
If you want analytic solution then difference equation can give you idea as to what type of solutions are there.
Regards
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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,
I don't know about analytic solutions but this is the equation for a harmonic oscillator in a constant force field with nonlinear damping. It is a like a mass suspended by a nonlinear spring in a gravitational field. There is a single equilibrium point at x=V and x'=0 whose stability depends on the value of a. Solutions will either be unbounded (go to infinity), which is probably unphysical for your electrical circuit, or they will decay to the equilibrium point, following the usual exponential law for a linearly damped oscillator as they approach it. Only the case a=0 has (neutrally) stable oscillations whose amplitude depend on the initial conditions just as you would expect for a harmonic oscillator without damping. There are no limit cycles and no chaos.
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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.
I hope the following attached textbook is useful.
You can find more textbooks online.
Best regards
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Studies should be recent and involving O.D.E and/or P.D.E.
Mathematical Modeling of Ecological systems and Epidemic systems.
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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?
Use Mathematica-modul NonlinearModelFit.
For example, for the data set: data= {{0., 61.375}, {0.25, 366.5}, {0.5, 305.375}, {0.75, 211.375},
{1., 165.125}, {1.5, 128.25}, {2., 121.875}, {4., 84.375}}
a simple Mathematica-program [Glucose Tolerance Test]:
model[al_?NumberQ, om2_?NumberQ, KK_?NumberQ, mu_?NumberQ,
nu_?NumberQ] :=
Module[{y, x},
First[y /.
NDSolve[{y''[x] + 2 al y'[x] + om2 y[x] == KK, y[0] == mu,
y'[0] == nu}, y, {x, 0, 10}]]]
Clear[x, y]
nlm = NonlinearModelFit[SDP0,
model[al, om2, KK, mu, nu][x], {{al, 5}, om2,
KK, {mu, 100}, {nu, 4000}}, x]
nlm[[1, 2]]
gives:
{al -> 4.97808, om2 -> 16.8225, KK -> 1650.52, mu -> 61.2647,
nu -> 3811.59}
Parameter Identification Problem for a set of nonlinear ordinary differential equations with some unknown parameters can be solved similarly.
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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...)
Hello
I agree with Maged about using a digital pen (stylus)....Regards
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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.
The logic is that we need to find two independent solutions. If they differ by an integer, one can be obtained from others by setting few zero coefficients on series. You can find the proof of it from Kreyszig Advanced Engineering Mathematics book (Appendix 4)
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Odes
For solving linear and nonlinear ODEs subject to boundary conditions, I recommend that you use the Duan-Rach modified Adomian decomposition method as discussed in:
1. J.-S. Duan, R. Rach, A new modification of the Adomian decomposition method for solving boundary value problems for higher order differential equations, Applied Mathematics and Computation, Vol. 218, No. 8, 2011, Pp. 4090-4118.
2. J.-S. Duan, R. Rach, A.-M. Wazwaz, Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems, International Journal of Non-Linear Mechanics, Vol. 49, 2013, Pp. 159-169.
3. J.-S. Duan, R. Rach, A.-M. Wazwaz, Temuer Chaolu, Z. Wang, A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions, Applied Mathematical Modelling, Vol. 37, Nos. 20/21, 2013, Pp. 8687-8708.
4. A.-M. Wazwaz, R. Rach, J.-S. Duan, Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions, Applied Mathematics and Computation, Volume 219, Issue 10, 2013 Pp. 5004-5019.
5. Jun-Sheng Duan, Zhong Wang, Shou-Zhong Fu, Temuer Chaolu, Parametrized temperature distribution and efficiency of convective straight fins with temperature-dependent thermal conductivity by a new modified decomposition method, International Journal of Heat and Mass Transfer, Vol. 59, April 2013, Pp. 137-143.
6. J.-S. Duan, R. Rach, A.-M. Wazwaz, A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method, Open Engineering, Vol. 5, No. 1, 2014, Pp. 59-74.
7. R. Rach, J.-S. Duan, A.-M. Wazwaz, Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method, Journal of Mathematical Chemistry, Volume 52, No.1, 2014, Pp. 255-267.
8. R. Rach, A.-M. Wazwaz, J.-S. Duan, A reliable analysis of oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, International Journal of Biomathematics, Vol. 07, No. 02, March 2014, 1450020 (2014), [12 pages].
9. J.-S. Duan, R. Rach, A.-M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method, Journal of Mathematical Chemistry, Vol. 53, No. 4, 2015, Pp. 1054-1067.
10. R. Rach, J.-S. Duan, A.-M. Wazwaz, On the solution of non-isothermal reaction-diffusion model equations in a spherical catalyst by the modified Adomian method, Chemical Engineering Communications, Vol. 202, No. 8, 2015, Pp. 1081-1088.
11. R. Rach, J.-S. Duan, A.-M. Wazwaz, Solution of higher-order, multipoint, nonlinear boundary value problems with high-order Robin-type boundary conditions by the Adomian decomposition method, Applied Mathematics and Information Sciences, Vol. 10, No. 4, 2016, Pp. 1231-1242.
https://doi:10.18576/amis/100403
12. Simulation of large deflections of a flexible cantilever beam fabricated from functionally graded materials by the Adomian decomposition method
R. Rach, J.-S. Duan, A.-M. Wazwaz
International Journal of Dynamical Systems and Differential Equations, Article In Press, 2019,
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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
You are most welcome. Some additional suggestions are as follows.
Conversion of the Mathematica script to a program written in C is discussed in the following links
and
Concerning the concept of an inverse function in Mathematica please consult
Before converting the solution to C it is advisable to investigate the functions playing with the formulae using the traditional method (pen and paper) The numerical solutions are also useful as the additional method of verification.
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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
Dear Norazrizal
I did not find any inverse Laplace transform in the link you sent me
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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?
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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
Dear Yves, I suggest you to see links and attached files on topic.
Dynamic Optimization in JModelica.org - Semantic Scholar
An Introduction to Trajectory Optimization: How to Do Your Own Direct ...
Introduction to Nonlinear Model Predictive Control and Moving ...
Hybrid Optimal Theory and Predictive Control for Power Management ...
Lifted collocation integrators for direct optimal control in ACADO toolkit
Real-Time Optimization
Fast Numerical Methods for Mixed-Integer Nonlinear Model-Predictive ...
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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?
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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!
entropy solution, normalized solution and weak solution for a nonlinear parabolic equation in a paper by clicking the following link:
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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.
You may search within
these references
Good Luck
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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
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
is the journal support the bifurcation theory papers
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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 :
I stopped working in this field nearly 20 years ago. But even than there existed several methods that outperformed RK4 by lengths, the comparison is done on the estimated accuracy obtained by the same number of calls to the ODE's right hand side function. Have a look in the following books
Hairer, E. and Nørsett, S. P. and Wanner, G.: Solving Ordinary Differential Equations, Part I - Nonstiff Problems, Springer Verlag, 1987.
Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations, Part II - Stiff and Differential-Algebraic Problems, Springer Verlag, 1991.
There the methods (for non-stiff ODE) of Dormand/Prince are highly recommended (I used one in my time). Somewhere in the middle of the first book is a twosided graphics where the orbits of a specific problem are shown obtained by different methods (and the same maximum number of rhs calls). The orbits of the chosen ODE are known from theory to return to the starting point. The impressive fact of the graphics is to show how good this feature is reproduced by the methods applied: the orbit of 1-step Euler metod leaves the pages and does not return, the orbit of the RK4 method leaves a large gap between start and end points, the orbit of the DP method without stepsize control leaves a small gap, and the orbit of DP with stepsize control leaves no visible gap; even more, the last method needed much less rhs calls!
This was state of the art 20 years ago. Maybe that meanwhile better methods have been developed, maybe the initially discussed PAM is one. Anyway, I recommend that you replace the RK4 method in your code at least by the DP4/5 method. If you use Mathlab then apply ode45 as ODE solver, if your code is in Fortran or C then search the internet for DOPRI5.
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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?
HAM and HPM coupled with the integral transformations like Laplace, Fourier, etc. Especially if you apply them to linear/nonlinear problems by considering the fractional operators without a singular kernel, I hope you will obtain great results.
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For example there exists methods to find expansions of Gauss Hypergeometric Function( Binomial Sums, Differential Equation method etc.)
When we are talking about expansion the one of the question is in terms of what function we want to get result. There are numerical approach:
Z.W. Huang, J. Liu, Comput. Phys. Commun.184(2013) 1973 ;
D. Greynat, J. Sesma, Comput. Phys. Commun.185(2014) 472 ;
There is universal for values of parameters approach:
Ancarani and G. Gasaneo:
. U. Ancarani and G. Gasaneo,Derivatives of any order of the Gaussian hypergeometricfunction2F1(a,b,c;z)with respect to the parametersa,bandc, Journal of Physics A:Mathematical and Theoretical42, 395208 (2009).
L. U. Ancarani and G. Gasaneo,Derivatives of any order of the hypergeometric functionpFq(a1,...,ap;b1,...,bq;z)with respect to the parametersaiandbi, Journal of Physics A:Mathematical and Theoretical43, 085210 (2010).
L. U. Ancarani, J. A. D. Punta, and G. Gasaneo,
Derivatives of Horn hypergeomet-ric functions with respect to their parameters, J. Math. Phys.58, 073504 (2017)
V. Sahai and A. Verma,Derivatives of Appell functions with respect to respect to parameters,Journal of Inequalities and Special Functions6, 1 (2015)
It is unversal, in application to expansion around any small parameter, but produces new extented set of Horn-type hypergeometric functions. Another limitation is that this method is not applicable to some multiple hypergeometric funcions (when one of summation include negative values).
There is my approach, based on the iterative solution of Pfaff system of differential equations:
This technique is applicable to the expansion of hypergeometric functions of a few variable in terms of multiple polylogarithms (relevant for physik), but it is not algoritmically closed since at the some moment problem of expansion is reduce to problem of finding transformation which convert the given set of algebraic functions to set of rational functions.
Nevertheless, I have successfully applied this technique to a few function where expansion was not able to construct using another technique.
Unfortunately, the further extension and application of this project has not get financial support from external sources
and at the present moment it is frozen.
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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)?
see my theory
In which I discuss the possibility of rapid movement given specific situations arise in my theory... You speak of some in your question...
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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?
Hi Michael Short Thank you a lot for your answer. This is exactly what I was looking for and you understood my question perfectly. I will try to implement this soon. Again, thanks for your time investment and the detailed answer.
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What is the general solution of x^{(k)}+(\cos t + \sin\sqrt{2}t ) x = 0 ordinary differential equation for k>=2?
((d^{k}x)/(dt^{k}))+(cos t+sin√2t)x = 0.
Solution: Let k = 3
x₁ = x(t)
x₂ = dx(t)/dt
x₃ = d²x(t)/dt²
This will transform the ODE into the homogeneous non- autonomous system:
dx₁/dt = x₂
dx₂ /dt = x₃
dx₃/dt = - (cos t+sin√2t)x₁
Now, with given initial conditions, we write the system in matrix form
and find a general solution using the usual method.
(See the attached article).
Similarly, you can choose any positive integer value for k > 2.
Best regards
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It's very similar to modified Bessel function. But it's second coefficient is 2 rather than 1.
r^2dX^2(r)/dr^2+2rdX(r)/dr-(r^2+p^2)X(r)=0 , p is a known quantity.
Yes, I solved it in MATHEMATICA, see the attached.
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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
Search by the elements of real analysis by Bartle and Sherbert.
(For its first edition )
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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
First of all, the lambda/4, should the minimum distance of Floquet port from the FSS to ensure that higher order modes are sufficiently attenuated ( more than 50dB) before hitting the FSS surface. So we could eliminate those modes.
Therefore, the Lambda value should be computed with the lowest frequency of the frequency band. This remains true for dual band systems.
You can also use mode calculator in HFSS to check the effects.
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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 !
To understand ODEs or PDEs, classes in Engineering and Science are essential. See my Casebook titled "Engineering Design Optimization using Calculus Level Methods- A Casebook Approach", http://fortrancalculus.info/textbooks/ . Their are some 20+ Engineering and Science problems showing their math problems and Optimal solution. Plus a little write-up / intro to each problem.
Try the FortranCalculus (FC) language that got its start by NASA's Apollo Space program that got us to the moon!
FC-Compiler™ application is a (free) Calculus-level Compiler that simplifies Tweaking parameters in ones math model. FC solves Algebraic through Ordinary Differential Equations Equations; Laplace transforms; etc. FC is based on Automatic Differentiation that simplifies computer code to an absolute minimum; i.e., a mathematical model, constraints, and the objective (function) definition.
Most math Modeling, Simulations, & Optimizations can be easily solved with a Calculus-level language/compiler. A half-hour per problem should be enough time to code and execute most math problems. Solves Algebraic Equations through Ordinary Differential Equations, for more visit http://fortrancalculus.info/apps/fc-compiler.html . Equations may be implicit, non-linear, Boundary Value Problems (BVP), Initial Value Problems (IVP), etc. Download, install, & execute some demo problems. Simple and its Freeware! (PDE solvers are in the works!)
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I'm researching about a Timoshenko Beam with a Transverse vibration. The displacements are very small.
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?
The total acceleration of the moving mass is given by
\frac{d^2}{dt^2} = \frac{\partial^2 w}{\partial t^2} + 2 v \frac{\partial^2 w}{\partial x \partial t} + \frac{\partial^2 w}{\partial x^2} v^2 + \frac{\partial w}{\partial x} a
where v denotes the velocity of the moving mass and a the acceleration of the moving mass. For a constant velocity, the last term vanishes.
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Let f be the pdf of a n dimensional N(0,C) distribution i.e up to a multiplicative constant, f(x)=exp(−0.5 xC−1x).
Which vector fields F are so that div(F)=f ?
According to Helmholtz decomposition in 3 dimensions, you can allways write F= -grad (u)+curl (A), where u is a scalar field and A is a vector field. So, operating with the divergence in both sides you get lap (u)=-f, where "lap" stands for Laplacian. Then you use boundary conditions on F to select those for u and A. For instance, in the relevant case of full 3-dimensional space demanding that F and all its derivatives vanish at infinity, one solution to your problem is simply F=-grad (u), where u is the convolution of your Gaussian with 1/|r|^2. Additional solutions can be found by adding to the above solution the curl of any vector field A so that curl (A) vanishes at infinity (i'm not sure whether such a vector field A exists, if we additionally demand it to be smooth).
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Could there be nonlinear ODEs that could not even be solved numerically?
I am working on a third-order nonlinear problem which is giving accurate results for a given set of initial conditions (ICs) but for other ICs the matlab ODE solver (checked 4-5 algorithms) do not converge.
If your equation/system of equations is prone to stiffness or strongly nonlinear the ode 45 solver may not be adapted for some ICs because it uses a fully explicit numerical scheme. This page can help you to chose better adapted solvers: https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html
If none of these solvers works then you may need to write yourself a numerical scheme adapted to your case, probably an implicit or semi-implicit scheme (more stable than explicit schemes) since matlab solvers don't work.
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As for example, light beam attenuation is described by the differential equation
dS/dx = -S
which solution is S~e(-x).
But what physical processes could be described by the differential equation:
dS/dt = -t*S or dS/dx = -x*S
which solution is S~e(-t^2) or S~e(-x^2), with t as time and x as distance.
Do you have ideas?
Thank you very much in advance,
Algis
In continuation, what is internal expectation from dynamical insights after your current equation of concern?
What is its physical model context?
The moderation stems from the new appearance of "position coordinate" on the RHS of your equation.
Thanks to all having participated in this discussion.
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I am solving a problem from fluid dynamics; in particular tightly coupled nonlinear ordinary differential equations. The following is a scaled-down version of my actual problem. I have solved system of coupled odes many times in the past but this case is different since double derivatives of one variable depends on the double derivative of another variable. How do I implement it in ode45? I need 3 x 2 = 6 plots of x, x-dot and x-ddot versus time for t, 0 to 2. All required initial conditions have zero values at t = 0
How do I store the updated value of the double derivatives as the ode45 code runs? The way ode45 works, I get x and x-dot as output but not the double derivatives. Any help will be highly appreciated.
Hi Vikash.
Congratulations on getting the code to work. Sounds as if it was quite tricky!
Re the very slight differences, they could possibly be due to the number of significant figures used. Or it could be that the maximal Lyapunov exponent could occasionally be positive - this could be tested by varying very slightly the initial conditions which you used.
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The function assumes a direct and reverse law. What do we know about the inverse function? Never mind. This is just the shadow of the direct function. Why don't we use the inverse function, as well as direct? ------------- I propose the concept of an unrelated function as extended concept of reverse function. ------------ There is a sum of intervals, on each of which the function is reversible (strictly monotonic) -nondegenerate function. ---------- For any sum of intervals, there is an interval where the function is an irreversible-degenerate function.
@ Vasiliy Knyshev ,
Kindly state and explain clearly the Newton's Second Law of the third order.
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In nonlinear systems, we know several bifurcations (i.e. Saddle-node, Pitchfork, Transcritical, and Hopf). The question is: does there exist a specific bifurcation that merges two "stable" limit cycles to one "stable" (and probably with larger amplitude) limit cycle?
Try x'' = ax - x^3 + (1 - x'^2)x' in the vicinity of a = 1.87525 (the Rayleigh-Duffing two-well oscillator).
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Explicit numerical schemes for fractional partial differential equations.
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Is there any way of producing periodic Gaussian pulses such that the bell shape repeats itself with a defined period. Does there exist any solution for some differential equation which resembles the aforementioned function?
you can use the following function and change the parameters (a, b and c) to fit your data: y=exp(-((a*sin(x)-b)/c).^2)
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Is there anybody with an insightful note or explanation on how to establish the stability of the solution of ODE (bvp) using Poincare-Lyapunov Theorem?
Reduction of the bvp to initial values problem and generation of the Lyapunov function do not incorporate the boundary conditions. We believe these are not acceptable.
Do you know how to include the Neumann boundary condition into Jacobian matrix or how to generate Lyapunov function suitable to establish the stability of the solution of ODE bvp; see the attached problem.
The notion of stability for BVP is different than the notion of stability for IVP. Hence the standard linearisation theorem cannot be directly applied.
The main question that you want to address is: what does it mean for a solution of a BVP to be stable (with respect to what perturbations). So, you need a definition.
If the problem comes from a variational formulation based on the energy, then looking at the positive definiteness of the second variation is the standard way to define and prove stability. Intuitively, the question is to see if nearby solutions around a given solution have lower energy. You can see examples in my paper with Thomas Lessinnes that deals with Neuman conditions. In particular we give a full computation for a simple 2nd order system for which we establish stability. Maybe it would be helpful if this is what you have in mind. Otherwise, you will find references in there.
The reference is
T. Lessinnes and A. Goriely, 2017 Geometric conditions for the positive definiteness of the second variation in one-dimensional problems. Nonlinearity 30, 2023..
And I believe that it can be found on research gate (or ask for it).
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I have a problem of understanding the Definitions of the Fractional-Order Calculus which make the foundation of the fractional order controllers.
How I can study and understand this theory in a simpler manner?
Are there any references simplify the understanding of this theory with numerical examples and figures?
Note: I'm an engineer, and I studied Calculus, differential equations, control theory, and some other math courses.
I would like to refer the following books for your study:
• Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering. V. Academic Press. ISBN 0-12-525550-0.
• Miller, Kenneth S.; Ross, Bertram, eds. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. ISBN 0-471-58884-9.
• Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN 2-88124-864-0.
• Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag Telos. ISBN 3-211-82913-X.
• Podlubny, Igor (1998). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. 198. Academic Press. ISBN 0-12-558840-2.
• West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003). Physics of Fractal Operators. Springer Verlag. ISBN 0-387-95554-2.
• Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 0-444-51832-0.
• Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer.
• Daftardar-gejji, Varsha (2013). Fractional Calculus: Theory and Applications. Narosa Publishing House.
• Herrmann, R. (2014). Fractional Calculus - An Introduction for Physicists. Singapore: World Scientific.
• Li, C P; Zeng, F H (2015). Numerical Methods for Fractional Calcuus. USA: CRC Press.
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