Integral Equations - Science topic

Advanced Applications of Differential and Integral Equations and Related Theorems in Engineering Sciences

A question that has been puzzling me for some time,

There are physics questions that have not been solved yet and some institutions have set prizes for those who want to put their name in the column of the stars of physics history with a group of scientists, some of these equations have not been solved for nearly four continuous centuries

Do you think that artificial intelligence is capable of solving these equations such as the Hodge conjecture or the Yang-Mills theory and the Navier-Stokes equations and others

Can artificial intelligence reach a solution to those equations that have puzzled stars of scientists such as Isaac Newton or Einstein and Riemann ....

Differentiating and integrating

I'm trying to solve the integral shown in the picture.

I'm using python libraries to plot the integrand (numpy and matplotlib.pyplot), as well as scipy.integrate library to solve the integral.

However, I'd like to see other suggestions or tips to solve this problem.

Any comment will be well appreciated.

Thanks, Pablo

The numerical solution of the singular integral equations has been developed by Erdogan and many other researchers (e.g., Erdogan, F., Gupta, G.D., 1972. On the numerical solution of singular integral equations. Q. Appl. Math. 29, 525–534. DOI: ), where the Gauss-Chebyshev quadrature method was proposed. However, if the kernel function of the integral is already regular, then may I ask what is the proper numerical method to deal with this issue?

A necessary condition to utilize Erdogan's method is that the coefficients a_ij (please refer to the picture in the file) of the singular term are non-singular, which is not the case I would want to deal with, since those coefficients are already zero in the regular integral equations.

I would be very grateful if you can give me some guidence! Thank you very much!

The following formula for calculating the average reflectance within a wavelength range appears in many literatures.

But R(n) and E(λ) are curves obtained through testing, and there is no clear equation (if there is one, please tell me and explain the source, thank you very much).

I want to know how to calculate the integral equation in the numerator. The separate integral results of E and R can be easily obtained using origin.

How can we numerically calculate integration like this in the picture, in which the denominator of the function becomes zero? any help will be appreciated.

This kind of integral usually appears in graphene and semi-metals. The denominator becomes zero at several points in the range of calculation, where Ω=2*ε.

Under dominant mode I loading, planar cracks have been observed to move from zero velocity v= 0; for a certain value v= v₁, these turn into non-planar crack configurations. An explanation is offered below.

This subject is important because evidence of conoidal rough cracks is observed experimentally on various macrographs of broken specimens, under fatigue for instance. Our recent works (see below in answers) provides associated physical quantities.

in integration

Hello everybody!

I need some computational solutions of Integral Equations for my studies can you recommend to me ?

To the best of my knowledge, the first rigorous approach was presented here

and all studies followed that approach.

But I wonder what about the approach to follow in case of an integral equation that is discretized?

Have you ever seen some study wherein the modified integral equation is illustrated?

Higher-order basis function (HOBF) settings are ignored when are used in conjunction with planar multilayer substrates in FEKO. Does anyone know how can i activate a similar setting?

Thanks in advance

Hello

I would be appreciated if you could guide me on how to integrate the product of an arbitrary function and Dirac delta function as attached?

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

Hello, i am working on a project with LS PLCs that using IEC language. i need to calculate integral for any curves. like SIN,COS, X² or ...

do you have an idea what should i do?

thanks

For a function, usually sign of second derivative (and if it is zero, even/odd index of higher order derivative whose numerical value is zero) is enough to detect whether the extreme point is maximum/minimum/saddle point, if first derivative is zero. For a functional (NOT A FUNCTION), Euler-Lagrange equation plays the role of first "derivative" of Functional. However, it the RHS of Euler-Lagrange equation is set to zero and the resultant differential equation is solved, then how to find whether this function (as solution to differential equation) corresponds to minimum, maximum or saddle "point" of functional?

Unfortunately, the nature of extremum of a functional is usually declared to be "beyond the scope" of most preliminary/introductory functional analysis resources (I have not checked all). How difficult is that mathematics and what are the prerequisites to understand the mathematics involved in finding the nature of functional extremum?

Please note my knowledge on variational calculus, integral equations and transformations as well as group theory and advanced differential geometry is rudimentary.

I realized that students do not understand integral conceptually.

solved

All codes of above-mentioned programs for

1) Differential Equation DE

2) Fractional Differential Equation FDE

3) Integral Equation IE

4) Fractional Integral Equation FIE

5) Integro-Differential Equation IDE

6) Fractional Integro-Differential Equation FIDE

Please, if any one can help me to reach

I want to extremize a functional by using standard Euler-Lagrange method, but the constraint contains unknown function inside the integral that is to be equated with a constant. I have no experience on solving integral equations.

The definite integral of the first derivative of a function is trivial. What if the argument is a non integer power of the first derivative of a function? any reduction formula? Thanks

Please find attached a problem taken from book "Linear and Non linear Integral Equations" by Wazwaz. The given system of Volterra integral equations can be easily solved using Adomian Decomposition method, Variational Iteration method etc. I also tried solving problem using Laplace transform and i am unable to get the results. I even tried to represent transformed form as a series and then tried taking Inverse Laplace transform to get the solution but it was not fruitful. The given system is linear and Laplace transform is linear operator but i don't understand why result is not achievable. If someone can provide solution using Laplace transform then i shall be grateful.

Exact solution is u(x)=1+x²

and v(x)=1+x-x³

Saad Sharjeel

we need to solve this integral as a analytical result, is this possible ?

Please I want to work on this research topic. Any ideal on how to use any of the methods in integral equations to develop a numerical scheme for the above topic. Thanks

Hi all,

I am working on the EFIE and MFIE formulations recently. However, I was confused about the integral below. It seems to be zero if there is no overlap between the basis function and the weight function.

$$

\int_{T_m^ \pm } {{{\bf{f}}_m}({\bf{r}}){{\bf{f}}_n}({\bf{r}}')dS} \\

= \int_{T_m^ \pm } {\frac{{{l_m}}}{{2A_m^ \pm }}\rho _m^ \pm ({\bf{r}})\frac{{{l_n}}}{{2A_n^ \pm }}\rho _n^ \pm ({\bf{r}}')} dS

$$

If anyone know this integral please kindly help me. It would be very great to talk about the possible ways or just give me some hints to the results.

I appreciate all your words or comments on this question.

Thank you in advance!

Best,

Bin

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:

Authors: Subuhi Khan , Tabinda Nahid

Authors: Marwa M. Alzubaidi, Elsayed M. Elsayed

Authors: Afgan Aslanov

4) Analysis of the Dynamical System in a Special Time-Dependent Norm

Authors: Ludwig Kohaupt

Authors: Tanveer A. Tarray , Housial P. Singh

Authors: Savin Treanta

Authors: Habib Djourdem, Slimane Benaicha, Noureddine Bouteraa

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

Dear All,

I am hoping that someone of you have the First Edition of this book (pdf)

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

Hi,

I need your opinion for finite integration method, I was reading the original paper:

Finite integration method for partial differential equations by

P.H. Wen, Y.C. Hon , M. Li , T. Korakianitis

I started to solve simple ode: y'(x)-y(x)=0;y(0)=1, what they did integrate both side we get---> y(x)=1+int(0-->x) y(t)dt...right? This is a Volterra integral equation, then they used numerical integration say Trap or Simp. what is new? what is different with numerical solution for VIE? one can even use ADM, PIM...etc to solve the obtain integral equation? what is the novelty?

May be I am wrong, I just want to know exactly what is new in the finite integration method

Best regards,

Majeed

Good evening, i am studing a crack under a mode I (opening) loading located in a structure with non-homogeneous bondes, this problem can be reduced to a singular integral equation with a simple Cauchy type singular part and another complex-valued part, the obtained form is shown in the fingure below, i need to now what is the normaly used numerical procedure to solve this kind of SIEs that involve a complex Kernel part. thanks

Up to my knowledge, I find that the simulation with regard to the simulate of two-dimensional mixed Volterra-Fredholm equatio is confined to be in a closed subset of R^n and there are several wonderful methods. However, I happen to get a integral equation in the following form (please see the attached picture). The only difference with the existing ones is that the integration with respect to x is from 0 to infinity and the boundary condition available is f(x,T)=h(x), with h() given, which is also quite weird. I desire to do a rough simulation in the last part to make a brief illustration but I get no idea and have no inspirations from the literature. Also, since I get little knowledge and experience in simulation of solutions to equations like that, I find it's hard for me to think out a practical method to do the simulation. I'm asking if I can get some useful inspirations from you, who may be experienced in numerical simulation of integral equations. Thanks a lot for your generouse help, your attention and your precious time.

It seems I am not so familiar with definite integral being changed to differentials.Can anyone justify me how the T1,P1 values and the limits are being changed to simply T and P as given in the attachment.

I am working on the numerical solutions of the integral equations with noisy data. To get a suitable feedback in this area I need some papers about its theory and existing numerical methods in the literature. 

Proving existence results for some initial and boundary value problem, we usually find a corresponding integral equation first and then use some fixed point theorem to prove the existence of solution of differential equation. Why finding corresponding integral equation is not enough and it is important to use fixed point theorems to prove existence of solutions? some counter example will be helpful.

Dear friends, 

                    Many mathematicians in this forum have studied different aspects of Differential equations (including fractional order)  under integral type of boundary conditions. I search google with  words "integral type boundary conditions."  I see many papers and blogs in most of which the following topics are discussed , solution approximation, existence and uniqueness,  positive solutions, etc.

But I do not find a single page which demonstrate the application of differential equations with integral type boundary condition is used to model any useful physical phenomena. It does not means that such problems are useless. 

If some one in this forum knows about a solid application of such problems. Please share it with us.

Hi,

How I want to find a numerical solution for integral equation between number to teta?

example:

int[f(x),4,x]

f(x)= (rho*g*(H/2)*cosh(k*h+r*sin(x))/cosh((k*h)*cos(k*r*cos(x)-sigmma*tt/T))

Hello, Every researcher, I have one question to you. I solve one dynamical integral equation but I see different behavior between to exact solution and approximation solution. Is it true? I don't know this results is true or not. Many thanks in advance Elham

I solved the following equation and got the amount of q":

q"=inv(M)*(F+G-C)

How do I get the integral equation and get the amount of q' and q???(In MATLAB)

By adding to the title and its anisotropy of this project the non-local or (singular) integral operators = my response to an ask of feedback to this paper, see below (paper put in reference to my own project)= (+ -via google translation, French text below) it is quite interesting = the anisotropy is here xi-cartesian constant (and the study comes out of the 1d to speak of the true multidimensional) and one can imagine more without difficulties The localized varying declinations of this anisotropy with Cartesian bases and ellipsoids/paraboloids and weights and functions, and your localized exponents ai, varying in x all of them in their own way, in centers, angles and Rotations, values etc, for your operators and various weights and functions on which your operators apply, variabilities more or less slow, fast, (ir /) regular, adapted, local, nice or not etc. It is quite speaking when the function is the derivative or the gradient of another or with studies in spaces of besov, sobolev or all functional spaces Es,p,q, with s not zero, s being the index (Integral) of derivation (instead of the spaces with s = 0, Lp, Lq etc) with s, p and p variables in x = we approach geometric anisotropy, foliations and associated irregularities which can eg to account for very natural situations in mathematical physics such as vortex patches and many others. (+-via google traduction, texte francais plus bas) c'est tout a fait interessant = l'anisotropie est ici xi-cartesienne constante (et l'etude sort du 1d pour parler du vrai multidimensionnel) et on peut imaginer de plus sans trop de difficultés, les declinaisons variables localisées de cette anisotropie avec des bases cartesiennes et des ellipsoïdes/paraboloides et des poids et fonctions et vos exposents 'ai', tous localisés, variant en x tous chacun d'eux a sa facon, en centres, angles et en rotations, valeurs etc, pour vos operateurs et poids et fonctions diverses sur lesquels s'appliquent vos operateurs, variabilités plus ou moins lentes, rapides, (ir/)regulieres, adaptées, locales, gentilles ou pas etc. c'est assez parlant quand la fonction est la derivée ou le gradient d'une autre ou avec des etudes dans des espaces de besov, de sobolev ou tous espaces fonctionnels Es,p,q, avec s pas nul, s etant l'indice (integral) de derivation (au lieu des espaces a s=0, Lp, Lq etc) avec donc s, p et p variables en x = on s'approche de l'anisotropie geometrique, les feuilletages et foliations et des irregularités associées qui peuvent eg bien rendre compte de situations tres naturelles en physique mathematique comme les vortex patches et beaucoup d'autres.

(paper and project "Weighted Anisotropic Morrey Spaces Estimates for Anisotropic Maximal Operators" and "weighted anisotropic Morrey spaces...." by Ferit Gürbüz= I pronounce my self on its/their subject and not on the novelty that its author brings to it). the rg-profile of the author has 2 (slightly different) papers with same title, but one of them has a "full text" with the title ending with: "...AND 0 -ORDER ANISOTROPIC PSEUDO-DIFFERENTIAL OPERATORS WITH SMOOTH SYMBOLS" with an additional math paragraph.

Hi Dears, I have a problem in solving system of linear equations involving singular matrix. How we can fix the singularity of the matrix?

The attached file is useful.

Dear All,

Can I transform a linear fractional Volterra integro-differential equation into a fractional differential equation? If yes, then how?

Thank you very much in advance for your help.

Sarah

Dear All,

I have a question regarding the the existence and uniqueness of a system of linear Volterra integral equations.

Does the solution exist? Is it unique?

What are the conditions on the kernels for the existence and uniqueness of this system of equations?

I would appreciate your help greatly. Thank you.

Sarah

Dear all,

I have developed a new technique which solves linear integro-differential equations of fractional type. This includes Fredholm and Volterra equations.

I am looking for an application which can be modeled into such equation so I can apply my method. It can be any kind of application.

I also solve mixed system of equations e.g.1. A system of multiple Fredholm equations of different order of fractional derivatives (0, 1/2, 1, 3/2, etc..) or e.g. 2. A system of same or different kinds of Volterra equations. So, if there is an application to this kind of equation, it would be great!

I would appreciate your help. Please refer me to an article.

P.S. Only linear equations please.

Thank you very much in advance.

Sarah

Are there some researches about the next dynamical system: x'=Z(x) where Z is the classical Riemann zeta function?   The equation is deffined in R^2\{1} and    x'=||x-1||^2 Z(x)   is an equivalent formulation. Riemann zeta is pretty important and a system like this may be studied by someone, right?

What is the following integral in closed-form?

Hi All,

To get EOM of a specific type of vibrating shell structure, I constructed the Hamilton equation of the system with messy and complex integrand (so many derivatives and variables are included). Now I have to calculate the variation of the Hamilton equation (del(int((T+U-V),t=t0..t1)=0) to get the EOMs and BCs of the system.

Is there anybody to know how to take a variation of such bulky integral in Maple?

Is there anyone to have experience in finding the EOM of shell structure using variational calculus in Maple?

Any comment will be useful.

Regards,

If I have a general integral equation with a semi-bounded limits from zero to infinity and I want to apply this equation on some deterministic values. The integration in this case must be converted to a deterministic integral, but how?

for example, as shown in equation attached here, I want to find the integral for a specific range of frequencies.

Please help me to solve the integral:

$$\int_0^{-2w}\phi^{-k-\frac{1}{2}}\frac{{_2F_1(k-\frac{1}{2},k,k+\frac{1}{2};1-\frac{k-\frac{3}{2}}{\phi})}}{\sqrt{-\phi-2\omega}}$$ (see the attachment)

I have solved this in Mathematica.but I am not able to way a

general result even in the table of integrals. Please help me out. Thanks in advance

Dear All

Could you please help prove two integral inequalities in the picture uploaded to this message?

Best regards

Feng Qi (F. Qi)

I have a question for how create the initial conditions for differential equation or integro- differential equation ?

Is there any specific conditions to be identified?

In the course of solving a problem, I am stuck with solving a problem which is a combination of hypergeometric function and two algebraic function. Here the integrals runs from 0 to a value 'u'. I have gone through almost all books available online and those in the library. Could you please help me out by suggest a book

I have attached the file which contains the integral to be  solved. Please note that, in the expression B,phi are constants. Please help me out

Why series solution method is not applicable to Fredholm integral equation?

It is mainly used for solving Volterra integral equations.

mathematics

optimization

least squares

numerical analysis

I want to draw upper and lower rectangular for sin(x)dx over [-π,π] dividing the partition into 6 equal intervals. 

I only want the drawing and I can solve it.

Best Regards,

I am working in theoretical plasma physics. In the derivation of nonlinear  dispersion relation for plasma the author integrated the equation along unperturbed orbits. The result was that he just multiplied with a factor of exp(ik.vt).I would like to know the concept behind this

i have 6 variables in my model and all variables are co integrated of order one I (1). but when i run Johansen co integration test i get Trace test indicating 2 cointegrating equation and Max eigen value test indicates no co integrating equation at level 0.05 so does it means there is no long term relation? Appreciate your help

Hello all.

I am working on Deuteron Bound-State; Numerical Integral Equations and I need help finding a fortran code that calculates eigenvalues of Lippmann-Schwinger equation for deuteron bound states, prefarably one that follows Lanczos iterative scheme. Thank you

Since this point is the intersection of the airfoil and the wake, it is difficult to deal with it in the Boundary Integral Equation?

Integration of tan(x) from 0 to pi/2, can the one use Jordan-contour to calculate it?

can help me

Numerical solution of the Volterra equations of the first kind that appear in an inverse boundary-value problem of heat conduction

i want new efficient technique( from numerical method) to solve fuzzy volterra integral equation of the second kind and fuzzy initial value problem.

please , can you help me and suggest to me?

I need researchers of Fredholm integro differential equation of first kind

AIEM is used for soil moisture retrievals

How do I calculate this integral in terms of Bessel functions? 

I am trying to solve the following integral equation:

P[b]=N\int_{-\infty}^{\infty} d \tilde{b} P[\tilde{b}] \exp(-(n\beta b^2)/2)\cosh^n(\beta(b+\tilde{b})),

where N is a normalisation constant and \beta, n are real positive numbers. For n=1  this equation has a solution  given by N\exp(-(\beta b^2)/2)\cosh(\beta b). Also when  n is a positive integer then it is easy to solve this equation numerically: in this case we can use Binomial theorem to write  cosh^n(x) as a sum and the problem becomes finite dimensional. I would like also to solve the above equation for non-integer n. I don't have much experience with integral equations and would appreciate any advise on how to solve this equation numerically or analytically.  

The equation is of the type 

y''-a*integral(f(t-t1)*y'*dt1)-b*y=0?

where a, b are constants.

Any suggestions (books, articles, links, MatLab examples) are highly appreciated.

Thank you.

I have two systems of coupled integral equations to solve. I know how to solve a single integral equation by reducing it to an eigenvalue problem. How can two systems of coupled integral equations be reduced also to an eigenvalue problem. Can two systems of coupled integral equations be reduced to a single integral equation?

It is necessary to solve integro-differential equations in attachement? Can you help with some references explaining the bicubic spline method applicable in praxis for solving the integral equation with variable upper integration limit?

I am working on charged colloidal one component system theoretically. Can anyone help me how can I maintain electroneutrality condition.