scholarly journals Fourier Approximation for Integral Equations on the Real Line

2009 ◽  
Vol 2009 ◽  
pp. 1-10
Author(s):  
S. M. Hashemiparast ◽  
H. Fallahgoul
Keyword(s):  
Integral Equation ◽  
Periodic Solution ◽  
Fourier Series ◽  
Integral Equations ◽  
Real Line ◽  
Finite Interval ◽  
High Accuracy ◽  
Fourier Series Expansion ◽  
Fourier Approximation ◽  
The Real

Based on Guass quadradure method a class of integral equations having unknown periodic solution on the real line is investigated, by using Fourier series expansion for the solution of the integral equation and applying a process for changing the interval to the finite interval (; 1), the Chebychev weights become appropriate and examples indicate the high accuracy and very good approximation to the solution of the integral.

scholarly journals Integral equations of the first kind of Sonine type

2003 ◽  
Vol 2003 (57) ◽  
pp. 3609-3632 ◽  
Author(s):  
Stefan G. Samko ◽  
Rogério P. Cardoso
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Lebesgue Space ◽  
Inverse Operator ◽  
Orlicz Spaces ◽  
Hypersingular Integral ◽  
The Real ◽  
Integrable Kernel

A Volterra integral equation of the first kindKφ(x):≡∫−∞xk(x−t)φ(t)dt=f(x)with a locally integrable kernelk(x)∈L1loc(ℝ+1)is called Sonine equation if there exists another locally integrable kernelℓ(x)such that∫0xk(x−t)ℓ(t)dt≡1(locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversionφ(x)=(d/dx)∫0xℓ(x−t)f(t)dtis well known, but it does not work, for example, on solutions in the spacesX=Lp(ℝ1)and is not defined on the whole rangeK(X). We develop many properties of Sonine kernels which allow us—in a very general case—to construct the real inverse operator, within the framework of the spacesLp(ℝ1), in Marchaud form:K−1f(x)=ℓ(∞)f(x)+∫0∞ℓ′(t)[f(x−t)−f(x)]dtwith the interpretation of the convergence of this “hypersingular” integral inLp-norm. The description of the rangeK(X)is given; it already requires the language of Orlicz spaces even in the case whenXis the Lebesgue spaceLp(ℝ1).

scholarly journals An integral equation associated with linear homogeneous differential equations

1986 ◽  
Vol 9 (2) ◽  
pp. 405-408 ◽  
Author(s):  
A. K. Bose
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Real Line ◽  
Homogeneous Differential Equation ◽  
The Real ◽  
Linear Homogeneous Differential Equation

Associated with each linear homogeneous differential equationy(n)=∑i=0n−1ai(x)y(i)of ordernon the real line, there is an equivalent integral equationf(x)=f(x0)+∫x0xh(u)du+∫x0x[∫x0uGn−1(u,v)a0(v)f(v)dv]duwhich is satisfied by each solutionf(x)of the differential equation.

scholarly journals On the Solvability of Second Kind Integral Equations on the Real Line

2000 ◽  
Vol 245 (1) ◽  
pp. 28-51 ◽  
Author(s):  
Simon N. Chandler-Wilde ◽  
Bo Zhang ◽  
Chris R. Ross
Keyword(s):  
Integral Equations ◽  
Real Line ◽  
The Real

A UNIQUENESS THEOREM FOR A NONLINEAR SINGULAR INTEGRAL EQUATION ARISING IN $ p $-ADIC STRING THEORY

2019 ◽  
Vol 53 (1 (248)) ◽  
pp. 17-22 ◽  
Author(s):  
A.Kh. Khachatryan ◽  
Kh.A. Khachatryan
Keyword(s):  
Integral Equation ◽  
String Theory ◽  
Singular Integral Equation ◽  
Uniqueness Theorem ◽  
Singular Integral ◽  
Real Line ◽  
Nonlinear Integral Equation ◽  
The Real

We study a singular nonlinear integral equation on the real line that appear in $ p $-adic string theory. A uniqueness theorem for this equation in certain class of odd functions is proved. At the end of the paper we give examples, satisfying the conditions of the formulated theorem.

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