Solution Of Certain Integral Equations Research Articles

Existence and Uniqueness Solution of Certain Integral Equation

The aim of this work is to study the existence and uniqueness of solutions of certain integral equations by using the Picard approximation method and Banach fixed point theorem. The study of integral equations is more general and leads us to improve and extend some results of Butris.

On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems* *The work of the first author is supported by long term structural funding-Methusalem grant of the Flemish Government, by the Fonds Wetenschappelijk Onderzoek-Vlaanderen (FWO) and the Fonds de la Recherche Scientifique-FNRS under EOS Project No. 30889451, and by FWO Flanders projects G.0864.16 and

We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states u(z) in the spectral theory of soliton gases for the one dimensional integrable focusing nonlinear Schrödinger equation (fNLS) and for the Korteweg–de Vries (KdV) equation. Our proofs are based on ideas and methods of potential theory. In particular, we show that the minimising (positive) measure for a certain energy functional is absolutely continuous and its density u(z) ⩾ 0 solves the required integral equation. In a similar fashion we show that v(z), the temporal analog of u(z), is the difference of densities of two absolutely continuous measures. Together, the integral equations for u, v represent nonlinear dispersion relation for the fNLS soliton gas. We also discuss smoothness and other properties of the obtained solutions. Finally, we obtain exact solutions of the above integral equations in the case of a KdV condensate and a bound state fNLS condensate. Our results is a step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis (2020 Phys. Rev. E 101 052207). It is expected that the presented ideas and methods will be useful for studying similar classes of integral equation describing, for example, breather gases for the fNLS, as well as soliton gases of various integrable systems.

SOLUTION OF INTEGRAL EQUATIONS BY BESSEL WAVELET TRANSFORM

The integral equations of the first kind arise in many areas of science and engineering fields such as image processing and electromagnetic theory. The wavelet transform technique to solve integral equation allows the creation of very fast algorithms when compared with known algorithms. Various wavelet methods are used to solve certain type of integral equations. To find the most accurate and stable solution of the integral equation Bessel wavelet is the appropriate method. To study the properties of solution of integral equations on distribution spaces Bessel wavelet transform is also used. In this paper, we accomplished the concept of Hankel convolution and continuous Bessel wavelet transform to solve certain types of integral equations (Volterra integral equation of first kind, Volterra integral equation of second kind and Abel integral equation). Also distributional wavelet transform and generalized convolution will be applied to find the solution of certain Integral equations.

On the Oscillatory and Asymptotic Behavior of Solutions of Certain Integral Equations

We present the conditions under which every positive solution x(t) of the integral equation $$x(t) = a(t) + \int\nolimits_c^t(t - s)^{\alpha-1}[e(s) + k(t, s)f(s, x(s))]\,{\rm d}s,\quad c > 1,\,\alpha > 0$$ Satisfies $$x(t) = O(a(t)) \quad {\rm as}\quad t \to \infty,\quad {\rm i.e.,}\,{\mathop {\rm lim \,sup}\limits_{t \to \infty}} \frac {x(t)}{a(t)} < \infty,$$ From the obtained results, we derive a technique which can be applied to some related integral equations that are equivalent to certain fractional differential equations of Caputo derivative of any order. We also establish new oscillation criteria for such fractional differential equations.

Bounded, asymptotically stable, and L1 solutions of Caputo fractional differential equations

The existence of bounded solutions, asymptotically stable solutions, and L 1 solu- tions of a Caputo fractional dierential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional dierential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point the- orem and Liapunov's method have been employed. The existence of bounded solutions are obtained employing Schauder's theorem, and then it is shown that these solutions are asymp- totically stable by a definition found in (C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solution of certain integral equations, Nonlinear Anal. 66 (2007), 472-483). Finally, the L 1 properties of solutions are obtained using Liapunov's method.

On the asymptotic behavior of positive solutions of certain integral equations

We present the conditions under which every positive solution x of the integral equation x ( t ) = a ( t ) + ∫ c t ( t − s ) α − 1 k ( t , s ) f ( s , x ( s ) ) d s , c > 1 , α > 0 satisfies x ( t ) = O ( a ( t ) ) as t → ∞ , i.e. , lim sup t → ∞ x ( t ) a ( t ) < ∞ . From the obtained results, we derive a technique which can be applied to some related integral equations that are equivalent to certain fractional differential equations of Caputo derivative of any order.

Some generalized Gronwall-Bellman type nonlinear delay integral inequalities for discontinuous functions

In this work, some new generalized Gronwall-Bellman type nonlinear delay integral inequalities for discontinuous functions are established, which can be used as a handy tool in the quantitative and qualitative analysis of solutions of certain integral equations. The established results generalize the main results of Gao and Meng in (Math. Pract. Theory 39(22):198-203, 2009).

On the existence of asymptotically stable solutions of certain integral equations

In this paper two existence results of asymptotically stable solutions of certain integral equations are presented.

The Inversion of an Integral Equation Involving a General Class of Polynomials

The object of the present paper is to derive the solution of a certain convolution integral equation of Fredholm type whose kernel involves a generalization of numerous polynomial systems including, for example, the classical Hermite, Laguerre, and Bessel polynomials, and the Konhauser biorthogonal polynomials. Some interesting applications of the main inversion theorem are also mentioned. One of these applications is shown to lead to the correct solutions of certain integral equations which were considered recently.

Finitely-generated solutions of certain integral equations

Recent work has shown that the solutions of the second-kind integral equation arising from a difference kernel can be expressed in terms of two particular solutions of the equation. This paper establishes analogous results for a wider class of integral operators, which includes the special case of those arising from difference kernels, where the solution of the general case is generated by a finite number of particular cases. The generalisation is achieved by reducing the problem to one of finite rank. Certain non-compact operators, including those arising from Cauchy singular kernels, are amenable to this approach.

The Solution of Certain Integral Equations by Means of Operators of Arbitary Order

(1990). The Solution of Certain Integral Equations by Means of Operators of Arbitrary Order. The American Mathematical Monthly: Vol. 97, No. 6, pp. 498-503.

The Solution of Certain Integral Equations by Means of Operators of Arbitrary Order

The Solution of Certain Integral Equations by Means of Operators of Arbitrary Order

Exact analytic reconstruction in multidimensional inverse problems

This presentation will review some recent mathematical breakthroughs in inverse scattering theory. Consider the pressure wave p(r,s,ω) generated by a point source s oscillating harmonically with frequency ω ∇r⋅{[1/ρ(r)]∇rp(r,s,ω)} + ω2K(r)p(r,s,ω) = −δ(r−s). The problem of determining the density ρ(r) and the compressibility K(r) from knowledge of p(r,s,ω) for sources s and receivers r located on a given surface S has been considered notoriously difficult. Traditional approaches rely on Born or Rytov approximations. A new constructive method for recovering ρ and K from measurements on a surface S with arbitrary geometry, at two frequencies ω1, ω2, will be presented. The procedure involves the solution of certain integral equations on S. No simplifying approximations are made, and the reconstruction is, in principle, exact.

Syntactic methods for recursive equations and applications to differential equations

The objectives of this paper are two‐fold: (1) to show that a proper appreciation of the roles that syntax and semantics play in the solution of equations can not only simplify the solution but also lead to greater generality, and (2) to demonstrate the utility of recursion as a practical tool in analysis. The use of recursion in analysis is not new; Liouville [8] and Neumann [9] used it to obtain formal series solutions of certain integral equations, and Poincare [10] used iterated mappings in his study of periodic motions. This paper is based on the more general interpretation of iteration as a transformation of the recursive equation u = P(u). Use is made of the notions of ‘string’, ‘binary tree’ and ‘formal language’, and those unfamiliar with these may consult [12]. The method of solving recursive equations (and inequalities) developed in this article is based on the scheme u = Pn(u) = Pn_1(P(u)),P0(u) = P(u). The generality of the scheme permits the exploitation of both the syntax and the semantics ...

The magnetic field dependence of the specific heat in quantum spin chains with axial symmetry

The magnetic specific heat in a class of spin-1/2 chains with uniaxial anisotropy is determined accurately by a numerical solution of certain integral equations derived earlier through the Bethe ansatz. The specific heat exhibits a characteristic double peak as a function of an applied magnetic field which cannot be accounted for by the spin wave approximation. This illustrates the importance of non-linearity for the thermodynamics of quantum spin chains even at low temperature.

Stabilized approximate solutions of certain integral equations of first kind arising in contact problems of elasticity

The paper presents a method of constructing, by successive approximation, stabilized approximate solutions of certain integral equations of first kind in contact problems of elasticity. In these equations, the unknown function is a surface stress under the stamp, which is allowed to have a square root singularity under the edges of the stamp, and is therefore not square integrable. The author shows that it is nevertheless possible to apply the theory of positive operators to the present problem by constructing an appropriate L2-space which includes functions with square root singularities.

On the solution of certain integral equations of mixed problems of plate bending theory

The problem of the bending of a Kirchhoff-Love plate in the shape of a strip under the impression of a thin linear rigid inclusion fastened at one of the edges of the plate when the other edge of the plate is rigidly clamped is considered. The problem is reduced by a Fourier integral transform to the solution of a convolution-type integral equation of the first kind in a finite segment with a regular kernel. The exact inversion of the principal part of the corresponding integral operator is constructed in the class of functions with non-integrable singularities on the segment edges. An effective asymptotic solution is given for the integral equation under investigation in this class of functions in the whole range of variation of the characteristic parameter λ. The results obtained are verified numerically. Analogous integral equations were examined in /1, 2/. The mode of investigation is similar to that proposed in /3/.

Storage processes with general release rule and additive inputs

A construction is given of a process in which X(t) represents the content at time t of a dam whose cumulative input process is a Lévy process with measure v and whose release rate at time t is r(X(t)). It is assumed only that r(0) = 0 and that r is strictly positive and left-continuous with strictly positive finite right limits on (0,∞). The sample-paths of X are shown to satisfy the storage equation The process X is analyzed using renewal theory and stochastic comparison techniques, and necessary and sufficient conditions are found in terms of v and r for X to have a stationary distributionπ. These generalize previous results which were obtained under the assumption that v is finite. Conditions for Πto have an atom at 0 are considered in some detail, and related results on the positivity of the expected occupation time of level 0 are given.Necessary and sufficient conditions for the existence of Πare expressed in terms of the existence of non-negative integrable solutions of certain integral equations and conditions are given under which such solutions are necessarily stationary densities for X. A simple sufficient condition for X to have a stationary distribution is found in terms of and in the case when r is non-decreasing the condition is shown to be also necessary. Finally some examples are considered; these show that the results described above unify various known conditions in special cases, and confirm several conjectures in the related literature.

Review of Some Exact Methods for the Solution of the One-Dimensional Inverse Problems

The inverse problem in geophysics is the determination of the inhomogeneous structure of an elastic medium from experimental observations. In this presentation, we first briefly review the Gelfand-Levitan and the Marchenko methods for the solution of the related quantum inverse problem. In these approaches, as well as in the methods for the geophysical problem we will discuss, the desired information is recovered from the solution of certain integral equations. Next, we consider the construction of a computational scheme for the inversion of geophysical data based on one of the methods. We will further show that the resulting algorithm is stable under small perturbations to the data. We will finally demonstrate the inversion scheme by inverting noiseless and noise-corrupted synthetic seismograms.

Storage processes with general release rule and additive inputs

A construction is given of a process in which X(t) represents the content at time t of a dam whose cumulative input process is a Lévy process with measure v and whose release rate at time t is r(X(t)). It is assumed only that r(0) = 0 and that r is strictly positive and left-continuous with strictly positive finite right limits on (0,∞). The sample-paths of X are shown to satisfy the storage equation The process X is analyzed using renewal theory and stochastic comparison techniques, and necessary and sufficient conditions are found in terms of v and r for X to have a stationary distributionπ. These generalize previous results which were obtained under the assumption that v is finite. Conditions for Πto have an atom at 0 are considered in some detail, and related results on the positivity of the expected occupation time of level 0 are given. Necessary and sufficient conditions for the existence of Πare expressed in terms of the existence of non-negative integrable solutions of certain integral equations and conditions are given under which such solutions are necessarily stationary densities for X. A simple sufficient condition for X to have a stationary distribution is found in terms of and in the case when r is non-decreasing the condition is shown to be also necessary. Finally some examples are considered; these show that the results described above unify various known conditions in special cases, and confirm several conjectures in the related literature.