# All questions with tag [math: integral-equations]

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## Integral equation $\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}}f(R \cos(x)) d x = 1$

Can we confirm that there does not exist a function $f$, which pleases this formula for all $R&gt;0$: $$\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}} f(R \cos(x))\, dx= 1.$$
2022-07-25 17:47:10
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## Volterra integral equation of second type solve using resolvent kernel

Solve the integral equation $$y(t)= f(t) + \lambda \int_{0}^{t} (t-s) y(s) ds$$ where $f$ is continuous using the method of finding the resolvent kernel and Newmann series. Here it is what I did: $K_1 (t,s) \equiv K(t,s) =t-s$ $K_2 (t,s) = \int_{s}^{t} K(t, \xi) K_1 (\xi ,s) d \xi= \frac{1}{2} (t+s)^2(t-s)-ts(t-s) +\frac{1}{3} (s^3 -t^3)$...
2022-07-24 06:35:18
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## Volterra integral equation of second type

Solve the Volterra integral equation of second kind : $$y(t)= 1 + 2 \int_{0}^{t} \frac{2s+1}{(2t+1)^2} y(s) ds$$ I know two methods for such integral equations: Picard's method The method of finding the resolvent kernel and the Neumann series I tried using both of these methods but I couldn't solve it. Which of the these methods is bette...
2022-07-24 02:48:27
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## Partial integro-differential equation

I do not recognize if there is a method to address this adhering to integro - differential equation: $$\partial_t{u(x,t)}\int_{0}^{x}u(\zeta,t)d\zeta=\partial_{xx}u(x,t)$$ Can a person offer me some tip? Many thanks.
2022-07-22 14:57:37
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## can we have $u=0$ from the integral value 0?

If u is a bivariate function and also we have $\int_\theta^{\theta+1}{\int_\theta^y{u(x,y)(y-x)^{n-2}}dx}dy=0$ for all $\theta\in\mathbb R$, below $n&gt;2$ is a constant, can we presume that $u=0$ a.e. on the location in between 2 straight lines $y=x$ and also $y=x+1$? If of course, please offer an evidence ; if no, please offer a countere...
2022-07-22 12:45:09
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## Maximal solution of differential or integral equation. Apply it to $\frac{d u}{d x}=\sqrt{u}$

What is the definition of topmost remedy to an indispensable formula and also differential formula. As an example, just how should I recognize the topmost remedy of the adhering to differential formula $\frac{d u}{d x}=\sqrt{u}$?
2022-07-22 12:24:59
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The trouble is connected to this inquiry: How to find eigenfunctions of a linear operator (follow-up question) I uploaded previously. Intend I intend to address the adhering to indispensable formula: $$\int_0^1 K(x,t)y(t)dt=\sqrt{2x^2-2x+1}$$ where $$K(x,t)=\max((1-x)t,(1-t)x),0&lt;x&lt;1,0&lt;t&lt;1.$$ Eigenfunctions of $K(x,... 2022-07-22 12:12:57 1 ## How to find eigenfunctions of a linear operator I ask yourself if there is a basic means of locating particular values and also eigenfunctions of an offered linear driver defined by an indispensable. As a grandfather clause intend I want this function: $$g(x,t)=\min((1-x)t,(1-t)x), 0&ltx&lt1, 0&ltt&lt1$$ and also I intend to locate$\lambda_i$and also$y_i(x)$such that $$y... 2022-07-21 06:13:03 0 ## Integral equation solution hint. I am seeking the family members of circulations that please the adhering to condition:$$\int_{-1}^{+\infty}f(x)x d x=0$$and also with this various other problems on f(x):$$f(x)\ge 0 \text{ in }(-1,+\infty]\int_{-1}^{+\infty}f(x) d x =1$$Any suggestion on just how to address it? 2022-07-20 14:51:45 1 ## How to find eigenfunctions of a linear operator (follow-up question) This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely. I am interested in calculating characteristic values and eigenfunctions of$$K(x,t)=\max((1−x)t,(1−t)x),0&lt;x&lt;1,0&lt;t&lt;1$$and find λ_i and y_i(x) such that$$y_i(x)−λ... 2022-07-20 14:50:16 0 ## Finding a series of functions orthogonal to all$x^{2n}$but one We require to locate a collection of functions$f_m$validating the building $$\int_{-\infty}^{+\infty}\! x^{2n} f_m\!(x) \;{\text d}x=\delta_{nm}.$$ We currently have actually located $$f_0(x) = \frac{e^{-\sqrt{|x|}}\sin(\sqrt{|x|})}{|x|}$$ (within a$\pi/2$normalisation variable) making use of $$\int_0^{+\infty}\!t^{4n+3} e^{-(1+i)t}\;{\text ... 2022-07-19 21:44:38 2 ## homogeneous linear differential equation question I was wondering if there is an analytical solution to the following homogeneous linear differential equation$$\dfrac {dM} {dt}=\dfrac {M} {\alpha \left( t\right) }e^{\beta\left( t\right) t}$$which could obviously be rewritten as$$\int dM =\int \dfrac {M} {\alpha \left( t\right) }e^{\beta\left( t\right) t}dt$$Where both \alpha and \beta... 2022-07-14 02:20:50 1 ## Hammerstein stochastic integral equation I'm in problem with the adhering to indispensable equation:$$\phi(t)=\rho\int_0^1 t^2 s \phi(s)^2 ds+\nu(t)$$where \nu(t) is a white gaussian sound with difference \sigma and also suggest value \mu. Is it feasible to address this formula in a shut kind? Conversely, can you get some building of the range of \phi(t) without address it?... 2022-07-12 11:26:43 1 ## Anomalous integral equation I'm attempting to address the list below formula:$$\int_0^{f(x)}f(t)dt=g(x)$$Differentiating under indispensable I get:$$f[f(x)]\frac{d}{dx}f(x)=\frac{d}{dx}g(x)$$I recognize the function g(x). Exists a straightforward means to find the function f(x)? Is it feasible to find it without the regulation of the by-product under indispensable?... 2022-07-12 11:05:37 1 ## Integral equation and existence: g(x)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du I'd like to know how one would go about showing that the following function, f, that is almost everywhere positive exists:$$g(x_1,\cdots,x_n)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$$where g:\mathbb{R}^n:\rightarrow \mathbb{R} satisfies: (1) g(x_1,\cdots,x_n) is the derivative is the nth order partial derivative of \frac{\part... 2022-07-11 03:22:08 0 ## Solving integral equation a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt I've got this nasty-looking integral equation involving taking two minimums:$$a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$$where \delta(\cdot) is the Dirac delta function and a, b, c, and d are constants. I am trying to find f(x). I recognize that this is a convolution of sorts and that t... 2022-07-11 02:34:41 1 ## How does one solve this integral equation 1+ax=\int_{-\infty}^xf(x-t)dt I've run right into needing to address this formula for f(x):$$1+ax=\int_{-\infty}^xf(x-t)dt$$Unfortunately, I am not accustomed to addressing integral equations. Can any person aid? Is is also soluble? Edit : Fixed a typo in the ceiling in the indispensable. 2022-07-10 05:02:42 1 ## Comparison between solutions of ODE Could any person aid on the adhering to trouble? Allow R (t) be the remedy to the indispensable formula: R(t)=1+\int_{0}^{t}\frac{1}{R(s)}ds, particularly R(t)=\sqrt{2t+1}. Think that X is continual and also favorable on[0,\infty) and also satisfies: X(t) \leq 1+\int_{0}^{t}\frac{1}{X(s)}ds for t\geq0. Does X(t) \leq R(t) adhere to?... 2022-07-08 03:44:20 0 ## Eigenvalues and eigenfunctions for the Fredholm integral operator K(g) = \int_0^1 e^{x t} g(t) \, dt. I would love to calculate the eigenvalues and also eigenfunctions for the Fredholm indispensable driver$$K(g) = \int_0^1 e^{xt} g(t) \,dt.$$The resources I've examined * appear to claim that the procedure is rather entailed. Has anything been released on this bit? Or, otherwise, am I deal with that it is mosting likely to be a tough point to... 2022-07-08 02:56:02 2 ## Solution of inhomogeneous Fredholm integral equation of the first kind with symmetric rational kernel Please clarify just how to address this inhomogeneous Fredholm indispensable formula of the first kind:$$f(x)=\frac{1}{\pi}\int_{0}^{\infty}\frac{g(y)}{x+y}dy$\$
2022-07-03 01:43:30