Boundary Value Problem - Science topic

A small and direct answer is 'No'.

It depends on several parameters. The PDE is linear or nonlinear? The boundary conditions are Dirichlet, Neumann or mixed type? The dimension of the problem? Is it time-depends or no?....

If such algorithm exists, it will be a revolution in the field of numerical methods. 'if'

Best regards

Murtadha Shukur "No, radiation from accelerated force field sources isn't simply the field adapting to a new velocity. It's a more complex phenomenon."

I never claimed that adapting the static force field around an accelerated field source to the new velocity is simple. The exact claim is that this field adaptation is subject of an initial / boundary value problem.

The point then is that initial / boundary value problems always have an aperiodic part in their solution.

This aperiodic part is not considered in the standard physics approach to the problem of radiation exerted by accelerated charges or masses.

Dear friend Rituparna Bhattacharyya

Now, let's dive into solving that fuzzy boundary value problem using the Fuzzy Laplace Transform method. The Fuzzy Laplace Transform is an extension of the classical Laplace Transform, incorporating the concept of fuzziness.

Here's a general guide on how you Rituparna Bhattacharyya might approach solving a fuzzy boundary value problem:

### Steps:

1. **Formulate the Problem:**

- Define the fuzzy boundary value problem clearly.

- Specify the fuzzy boundary conditions and any other relevant parameters.

2. **Fuzzify the Problem:**

- Represent the fuzzy quantities using appropriate fuzzy sets and linguistic variables.

- Convert fuzzy linguistic descriptions into fuzzy numbers or fuzzy functions.

3. **Fuzzy Laplace Transform:**

- Apply the Fuzzy Laplace Transform to convert the fuzzy boundary value problem into a fuzzy algebraic system.

4. **Solve the Fuzzy System:**

- Depending on the complexity of your fuzzy system, you Rituparna Bhattacharyya may need to use numerical methods or analytical techniques specific to fuzzy algebra.

5. **Defuzzify the Solution:**

- Convert the fuzzy solution back into a crisp solution using a defuzzification method. Common methods include the centroid, mean of maximum, etc.

6. **Verify and Interpret:**

- Verify the obtained solution and interpret the results in the context of your original problem.

### Tips:

- **Literature Review:**

- Since you Rituparna Bhattacharyya mentioned there are papers available on solving initial value problems using Fuzzy Laplace Transform, consider looking for similar approaches in the literature. They might provide insights into adapting the method for boundary value problems.

- **Software Tools:**

- Explore if there are software tools or programming environments that support fuzzy computations. Some languages like MATLAB or Python with appropriate libraries might be helpful.

- **Consult Experts:**

- If the problem is particularly complex or if you encounter challenges, consider consulting experts in fuzzy logic or numerical methods.

Remember, this is a general guideline, and the specifics might depend on the details of your particular problem. It's also worth noting that fuzzy methods can be computationally intensive, so choosing appropriate numerical methods is crucial. Good luck with your fuzzy boundary value problem!

Thank you, dear Anwar, for the answer. Indeed, when I searched by the title "A novel efficient method for nonlinear boundary value problems" I saw the paper with the names of two other authors. I think the problem is in this. My name is not included with the paper.

implementation details are missing anyway, for analytical derivatives, use of flow thrust trajectory optimization problem via indirect optimal control is applicable Finite difference derivatives are not accurate enough to achieve convergence in some perspectives, this can be done through transition matrix using instance of chain rule to derive required analytic partial derivatives.

Good luck.

I have also applied the principle of superposition to combine multiple analytical solutions (any one of which satisfies the governing partial differential equation) by solving a minimization problem of the residuals between the approximated and desired values. For example, a bunch of exponentially decaying sources in a field (which could be almost anything), adjusting the position and strength of each in order to best match some expectation. I used this to figure out how much of a contaminant was dumped and based on how far it had spread, when and where it was dumped. A guy who was paid to transport 8000 gallons of TCE to a disposal site parked the truck in the vacant lot next to his mistress' house and opened the valve.

Dear Ms kfelati Thank you very much for your useful answer

How do we solve the fractional-order initial boundary value problems in Mape?

Answer or code may be sent to abiodunowoyemi9@gmail.com, please

As the problem is fuzzy boundary value problem, you may get different existence conditions for the solutions depending upon the fuzzy variables of the equation which is pre- defined by you. Also you shall get infinitely many solutions in fuzzy sense as the solutions are parametric fuzzy solutions, the parameter belongs to [0,1].According the application, we shall choose the value of parameter.

Dear Mark Stephen Luloff

A conformal map is a function in mathematics that preserves angles but not necessarily lengths locally. In more technical terms, let and be open subsets of At a location, a function is said to be conformal (or angle-preserving) if it retains angles between directed curves while also retaining orientation.

the conformal map, A transformation of one graph into another in which the angle of intersection of any two lines or curves remains constant. The most prominent example is the Mercator map, which is a two-dimensional depiction of the earth's surface that includes compass directions.

In contrast, if the Cauchy-Riemann equations are met and the derivative at the point is not zero, one may demonstrate that there is an a > 0 and such that the above is true. As a result, the map retains angles. As a result, a map is a conformal map if and only if it is a one-to-one, onto analytic function of D to D.

In addition, have a look to these links also:

1. https://www.researchgate.net/publication/228448714_Conformal_mapping-based_3D_face_recognition

2. https://math.stackexchange.com/questions/187954/conformal-map-in-3d?rq=1

3. https://gisgeography.com/map-projections/

Kind Regards

I would prefer Galerkin or Petrov-Galerkin Method. And if you want to get more specific try out point collocation method.

I suggest for you the following papers Dr Sina Etemad

[springeropen.com] New applications of the variational iteration method-from differential equations to q-fractional difference equations

You can also read the following paper

On solutions of variable-order fractional differential equations

DOI:10.11121/ijocta.01.2017.00368

One advantage of using Green's function is that it reduces the dimension of the problem by one.

For example, Legendre and Associate Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom and in the determination of potential functions in the spherically symmetric geometry etc

Examples on practical applications in fractional boundary value problems;

Fractional differential equations /models are applied in different fields of studies: mathematics, physics, chemistry, biology, medicine,…

They are used to describe different physical process such as dynamics of blood flow, wave motions, elasticity,…

For more information, read attached files!

You can see the following links:

You can do it yourself in Fortran or Basic. Two procedures are enough: minimizing a function e.g. Marquardt algorithm and integrating differential equations e.g. RKF algorithm. The right starting point, and you're out of trouble.

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.

Hi, I think in your case it is easier to implement your problem yourself in MATLAB for example by using the fmincon optimizer. As a matter of fact, your dynamics equations can be integrated analytically. Best

Dear sirs ,

Yours noted with thanks but I can't provide you the answer right now.

let me analyze deeply the question in order to be able to answer.

Best regards,

Bunga Paulo Teka

Hi, you can first rewrite your equation as a system of two first-order equations:

u'(x)=y(x)

y'(x)=u(x)+(4x^2+6)exp(x^2)

u(-1)=0 u(1)=0

Then you can define the shooting function whose parameter is z=y(-1) :

S: z --> S(z)=u(1)

To compute numerically S you simply have to call ode45.

Finally, using fsolve you can search for a zero of S, i.e. z such that S(z)=0.

Only some short MATLAB modules are required.

Best

Thank you so much Prof. Dudley J Benton for your sharing knowledge.

Richard Epenoy I am studying the debonding problem of an adhesive structure (such as two laminated beams). The whole length is L, and in the range 0~L/3, the two beams are debonding; in the range L/3~L, they are bonded well.

The first equation represent the adhesion stress of the interface, so in the subregion one, the first equation equal to zero.

Your comment is correct, in the sense that for a first order initial value problem, under some regularity assumptions the initial value problem (also known as Cauchy problem) has a unique solution, so that in general the boundary value problem would have NO solutions. However, IF you introduce a free parameter r,

you may try to find the value of that parameter for which you the solution of the initial value problem actually solves the boundary value problem. However, existence of the solution of this modified is probably impossible to prove in general (at least, I think it would be possible to come up with simple counterexamples). It could then be solved in a least square sense, as inverse problem in which you are trying to fit a model (that is, the first order ODE) to your boundary data.

see

Using the shooting method to solve boundary-value problems involving nonlinear coupled-wave equations

Y. H. Ja

Fractional differential equations can be presented with

differential operators of both Caputo and Riemann-Liouville. Here comes the first practical advantage of working with practical problems using the Caputo definition. When using the RiemannLiouville definition, it is necessary to know the fractional derivative of function as an initial or boundary condition, which is always very difficult. On the other hand, Caputo's allows only the value of the function itself, or the value of integer derivatives, which can be measured and easily interpreted.

It is not the purpose of this paper to show how the solution is

analysis of a fractional differential equation, which can be seen in (PODLUBNY, 1999). However, it is worth noting that a widely used technique is that of Laplace transform, already existing much development in the area. The Laplace transform of the Caputo fractional derivative is described in this article: https://proceedings.sbmac.org.br/sbmac/article/view/2315/2331

Looks like some sort of convective boundary layer problem from a cylinder.

These equations and boundary conditions form a typical voundary value problem. The equations only look complicated because of a large number of constants due to the nanofluid presence. If you are able to solve the Blasius boundary layer equations using a shooting method, or, better, the boundary layer equations for convection from a vertical surface - these date back to the 1950s, then this is just more of the same with no extra difficulties.

Re the boundary condition at infinity, a sufficiently large eta_max will do, but one will have to increase that gradually in order to check that your chosen eta_max is large enough. But I have a feeling that the solutions for theta and f' may well decay algebraically (a negative power of eta) rather than exponentially, so the "compactified" rescaling may eventually need to be done. But I would try without it first in order to make sure that your code runs and gives physically realistic looking solutions.

FC reminds me of the Continuous Systems Modeling Program that was around decades ago. Prof. Speckhart taught at my almamater (UTK) and wrote the book on CSMP (still available at Amazon). Whatever Maple can't solve analytically, I can knock out quickly with C. I've seen several people on RG looking for a freebie FORTRAN compiler.

Thank you for telling me the above information.

These links may help you

Demetris,

Thank you for your comment. It reminds me of a story by Isaac Asimov, where the word 'computer', originally meant for humans, became used only by machines, and so no one would even know how to add two numbers themselves, without a computer, until... they rediscovered arithmetic.

So, it seems important, and I will deal with it in the second report, but it includes a misconception. Mathematics is not computing, and more can be learned by understanding how things work, such as scaling laws, instead of blindly solving a differential equation. I see that several times.

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).

Issam> Example: Consider the initial value problem:

I think, that the solution is unique! Additionally, the following objections appear:

1^o the examples y_s ( s>0) are not defined for 0< t \le s, unless there should be: y_s (t)=0 for 0\le t \le s, instead of y_s (0)=0 for t<s.

2^o Even after the correction of 1^o , y_s is not a solutions: the RHS is y_s'(t) = 0 for t\le s, whereas, the RHS equals 2 \sqrt{t} \ne 0, unless t=0.

3^o Probably, the ODE was dy/dt = 2√y, y(0) = 0. But then the "direct integration" gives \sqrt{y(t)} = t + C, which together with the initial condition implies y(t) = t². And indeed, Issam is right that this is not the unique solution of the improved ODE. And then also the additional functions satisfy the initial condition proving nonuniqueness of the new problem.

4^o There is a suggestion, that if the uniqueness theorem's conditiona are not satisfied, then the solution s not unique. Probably this is only an error of formulation. Lack of satisfying assumptions can only serve as an explanation of presence of nonuniqueness, BUT it does not imply nonuniqueness (or there is a theorem which says, that the differential equation possesses exacly one solution satisfying the given initial condition if and only if . . . .. Frankly, without additional introductory assumptions such a theorem is not known to me.

Best wishes to all followers in studying this interesting question,

Joachim

Yes in classical Cubic/Quintic B Spline collocation method, it is easier to deal with Neumann's BC than Dirichlet's BC. From Dirichlet's BC we get derivative condition and use it at the boundary. For more detail, you can see our papers with Geeta Arora. Due to this reasons we have modified Cubic B Spline and Quintic B Spline on the boundary so that Dirichlet's BC can be applied with ease.

Joachim> I have got into some blackout state,

I assumed you were just temporarily misled by your "constant restoring force" (V = |y|) analysis :-)

Actually, the problem of how the return time varies with (kinetic) energy is quite interesting and complicated, and varied. Which is why I hid behind the intuition phrase. It is easy to construct potentials where almost any behaviour can occur. So, I just believe that the specific differential equation under discussion leads to the behaviour I conjectured; other cases will very likely be different.

The equations analysed in the AMS paper can be interpreted as one-dimensional motion of a point particle influenced by a time and position dependent force. I am rather unimpressed by the results in that paper; they look like mechanical trivialities dressed in abstract mathematical jargon, decorated with complicating concepts and notation. (I think am allowed to say that, since I graduated is from a department of mathematics :-D)

Thank you Farid and Smrity but I think You have misunderstand my question. I mean when solving boundary value problems like an optical fiber wave guide for example, the z component of the electric field in TM mode inside the core has the form:

Ez = A*Besselj(h*r)*exp(i*n*phi)*exp(-i*betta*z)

where the Hankel function is not used since it is not regular at the origin. But the Phi component of the electric field using Maxwell equations is:

Ephi = i/h^2*(-betta/r*d(Ez)/d(phi)+...)

where it can be seen that Ephi is singular at the origin (r=0). So I think our 1st justification to not using Hankel function is some how negated.

Thanks.

tau is the integration variable...that means that for computing h at a time t, you have to compute an integral function. Once a value t is fixed, the integral is definite and you have to integrate in dtau (the dummy variable can be denoted as you want, for example t')

I have all papers of Mr. Wu but still unable to get the results. Thanks for your reply. 

 Basically, BI-RME use FEM technique. This paper may help you...

Shape analysis with the ‘Boundary Integral–Resonant Mode Expansion’ method by P. Gambaa , L. Lombardib, Image and Vision Computing 17 (1999) 357–364

.

If you want to use analytical formula, you can use the solution of waveguide. You can consult this book. Advanced Engineering Electromagnetics_C A Balanis_1989

.

As you are working on SIW, please consult Prof. K. Wu's paper or Microstrip Lines and Slotlines_3e_Ramesh Garg_2013

I never seen BI-RME using analytical technique. Every body do prefer to use FEM. Please go on this particular topic.

Sorry, but I think you asked for solution manual? What Tankam has uploaded here is the book not solution manual!?

The issue you are dealing with is how to construct what are called "Artificial Boundary Conditions" (ABC) that are used to replace the condition at infinity in numerical computations.  There is a very readable review paper by Tsynkov that you should look at -see attachment. You can also look at one of my papers " Downstream development of a two dimensional viscocapillary flow ", which you can download from RG

From the attachment it is clear that you have a couple of first order ODE with initial conditions 𝑇𝑎𝑖𝑛= 𝑇𝑎(0) and 𝑇𝑝𝑖𝑛 = 𝑇𝑝(0) . Solve them Runge Kutta method and find the solutions to obtain T_a0 and T_pout and use them in ∆𝑻 

There are various ways to solve nonlinear BVP numerically. If the BVP problem is  an ODE one can use a collocation method, Finite Difference method, Finite Element method and a shooting method. All all cases you will end up with a system of nonlinear algebraic equations that have to be solved. And Newton's method  at this point is normally your best choice, especially if you have a reasonable guess for the solution.

Here are some links where you can download some examples (using Mathematica  as the  software):

Spectral Collocation Method:

Shooting Method:

Finte Difference Method:

Thanks Ayad for the advice.

However, I got a problem with homogeneous boundary conditions. The paper is hereby attached hopefully will add to your academic knowledge.

You cannot define both Dirichlet and Neumann conditions on Gamma_0.. Indeed, the Neumann value of your field on Gamma_0 is uniquely determined by the Neumann conditions on Sigma_pm and the condition u=0 on Gamma_0.

You may start with the highest order derivation for approximation

Thank you for your answer. In ABAQUS, I only dealt with the local problem in its user subroutine. The local Jacobean was also given in the subroutine. The global iteration is Full-Newton method in this software. Using one element mesh, I can obtain a smooth stress-strain curve. However, if this one element was divided into many elements, the solution diverged immediately after the peak. Please have a check on the figures.

More severe case is the footing case. The elements suffer very low initial stress at the beginning of the simulation. Local non-convergence happened in all the elements, so the solution diverged immediately after start.

Hello Muhammad Usman

I believe this problem can be solved analytically using the method of separation of variables. It can be shown that one set of the possible solutions are the infinite series:

V(x, t) = p(t) + Σ [a1*cos(2β²t) + a2*sin(2β²t)]*[b1*Cosh(βx)cos(βx)+ b2*Cosh(βx)sin(βx) + b3*Sinh(βx)cos(βx) + b4*Sinh(βx)sin(β x)]

w(x, t) = p(t) - (1/2)* Σ (1/ β²)* [a1*cos(2β²t) + a2*sin(2β²t)]*[ -b1*Sinh(βx)sin(βx)+ b2*Sinh(βx)cos(βx) - b3*Cosh(βx)sin(βx) + b4*Cosh(βx)cos(βx)]

Where d²p/dt² = F(t) and the series runs from β = certain value to ∞ and the constants a1, a2, b1.....b4 are functions of. β

Four boundary conditions and need to applied to determine the constants b1, b2, b3, b4. When the boundary conditions are applied an eigen value equation will be obtained for determining the admissible values of β. The coefficients a1 and a2 can be determined as Fourier coefficients when the two initial conditions are applied. When some boundary values are non-zero the above solutions need to be suitably modified by adding linear function of x..Note that another solution set is also possible in which the time functions are replaced by exp(-2β²t)

A paper on the method: http://www.orcca.on.ca/TechReports/TechReports/2001/TR-01-02.pdf

If you are interested in finding numerical solutions then apply shooting method to original system (1). I understand they are a coupled equations of second order. Assume p(0) = alpha1 and q(0) = beta1 and try to find solution at t=2 by Runge-Kutta method. Now you take another set of values p(0) = alpha2 and q(0) = beta2 and find solution for t=2. Interpolate so that you get values of alpha and beta where p(2) = 0 and q(2) =0. Find final solutions by Runge- Kutta method by these alpha beta.

Did you test it with diffefrent grid sizes?

Hi,

Start studying below link...

http://shellbuckling.com/imperfectionSensitivity.php 

All the best

You can use  quasilinearization method for second order four- point boundary value problems.

Please see the attached research paper

Please check it out in the book below:

Orthogonal Stagnation Point Flow (Hiemenz Flow):

y''' + yy'' - (y')^2 + 1 = 0 subject to y(0) = 0, y'(0) = 0, y'(infinity) = 0

For the above, check these websites:

1) http://www.leb.eei.uni-erlangen.de/winterakademie/2007/report/content/course01/pdf/0114.pdf

2) https://books.google.de/books?id=fwj1hKnLrcUC&pg=PA657&lpg=PA657&dq=Boundary+Value+Problems+for+Third-Order+Nonlinear+Ordinary+Differential+Equations+in+Fluid+Mechanics&source=bl&ots=5DcLKfXxmj&sig=TmV1j4_cSgn6mUtCpMWQB68wJ4M&hl=en&sa=X&ei=GAsdVZ7_FojVapeRgeAK&ved=0CDoQ6AEwBTgK#v=onepage&q=Boundary%20Value%20Problems%20for%20Third-Order%20Nonlinear%20Ordinary%20Differential%20Equations%20in%20Fluid%20Mechanics&f=false

OK, sir Can you suggest me name of books  for spline method?

Hmm - I'd say it's difficult. As far as I know, quite some models are tuned for channel flows w.r.t. parametrisation; so, going for a complex geometry such as porous media is probably not so simple. But quite much work already exists, you may e.g. check out for a start:

- Kalarakis et al; Mesoscopic Simulation of Rarefied Flow in Narrow Channels and Porous Media, Transp Porous Med, 2012

- Tang et al; Three-dimensional lattice Boltzmann model for gaseous flow in rectangular microducts and microscale porous media, J Appl Phys, 2005

- Tang et al; Gas slippage effect on microscale porous flow using the lattice Boltzmann method, Phys Rev E, 2005

First of all, you need a dynamic model of your industrial robot (including the maximum values of the actuators torques) since the cycle time depends on the available torque on each joint.

Then, there are several methods to solve the minimum time problem. You can look for example on  the early works of Dubowsky and  Shiller, almost 30 years ago...

I would suggest the Painleve analysis of the equation. It may happen the resonances depend on the parameter q. If they do then you can obtain an information about essential singularities. If the resonances are all integers independent of q then you are dealing with a possibly integrable equation for all q.

Thank you sir.

I agree that the choice on Matlab is bvp4c or bvp5c. However, if your boundary value problem is in a high order form (i.e. the states are in cascade), scilab has the bvode based on Ascher et al fortran routines and that we have successfully used in the following references. 

More precisely, if your are solving an optimal control problem and if your dynamics are in normal form, the the adjoint states have a dual normal form and the boundary value problem is in n-th order form. In this case, bvode is faster and more accurate.

Inversion in indirect optimal control: constrained and unconstrained cases

Inversion in indirect optimal control of multivariable systems

I highly recommend Asher's book

François

Numerical methods like finitr differece Crenck Nicolsion schem and finit element method in some cases applicable.

If the cavity is spherical, then you can use the respective Green function. I.e., you only need the solution for a point charge somewhere in the cavity, and then sum or integrate over the charges in your cavity.

If it's not spherical, then you can use Comsol Multiphysics. I think they offer a 1 month trail version; this will be enough to try it. It is also possible with Mathematica/Maple/MatLab, but it will be harder.

Right, if the equation is written as y''=g(x,y), then g is singular at y=0 but it is regular in the independent variable x. In a sense, this makes the problem easier, but it is necessary a careful checking. I can recommend the monography

I. Rachůnková, S. Staněk, M. Tvrdý: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations. Hindawi Publishing Corporation, New York, USA, 2009, 268 pages.

for this type of problems. But typically the mathematical methods we use are robust, that is, usually we get an open interval of available values for the parameters...