Kernel Of Convolution Type Research Articles

Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃

Abstract The Laplace transform method is applied in this article to study the semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order n, with convolution-type kernel. This kind of stability extends the original Hyers-Ulam stability whose study originated in 1940. A general integral equation is formulated first, and then some particular cases (polynomial function and exponential function) for the function from the kernel are considered.

Solution of Linear Volterra Integral Equation of Second Kind via Rishi Transform

The solution of various problems of engineering and science can easily determined by representing these problems in integral equations. There are numerous analytical and numerical methods which can be used for solving different kinds of integral equations. In this paper, authors used recently developed integral transform “Rishi Transform” for obtaining the analytical solution of linear Volterra integral equation of second kind (LVIESK). For this, the kernel of LVIESK has assumed a convolution type kernel. Five numerical examples are considered for demonstrating the complete procedure of determining the solution. Results of these problems suggest that Rishi transform provides the exact analytical solution of LVIESK without doing complicated calculation work.

Inhomogeneous affine Volterra processes

We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel K(t,s) and inhomogeneous drift and diffusion coefficients b(s,Xs) and σ(s,Xs). In the case of affine b and σσT we show how the conditional Fourier–Laplace functional can be represented by a solution of an inhomogeneous Riccati–Volterra integral equation. For a kernel of convolution type K(t,s)=K¯(t−s) we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition b and σσT are affine, we prove that the conditional Fourier–Laplace functional is exponential–affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance.

Usage of the Fuzzy Laplace Transform Method for Solving One-Dimensional Fuzzy Integral Equations

In this paper, we propose the solution of fuzzy Volterra and Fredholm integral equations with convolution type kernel using fuzzy Laplace transform method (FLTM) under Hukuhara differentiability. It is shown that FLTM is a simple and reliable approach for solving such equations analytically. Finally, the method is illustrated with few examples to show the ability of the proposed method.

About nonlinear integro-differential Volterra and Fredholm equations

Two nonlinear problems in terms of abstract operator equations of the form Bx = f are investigated in this paper. In the first problem the operator B contains a linear differential operator A , the Volterra operator K with kernel of convolution type and the inner product of vectors g ( x )Ф( u ) with nonlinear bounded functionals Φ. The first problem is given by equation Bu ( x ) = Аu ( x ) - Ku ( x ) - g ( x )Ф( u ) = f ( x ) with boundary condition D ( B ) = D ( A ). In the second problem the operator B contains a linear differential operator A and the inner product of vectors g ( x )F( Аu ) with nonlinear bounded on C [ a , b ] functionals F , where F ( Аu ) denotes the nonlinear Fredholm integral. The second problem is presented by equation Bu = Аu - gF ( Au ) = f with boundary condition D ( B ) = D ( A ). A direct method for exact solutions of nonlinear integro-differential Volterra and Fredholm equations is proposed. Specifically, the three theorems about existing exact solutions are proved in this paper. The first theorem is mean that for nonzero constant α0 Volterra integro-differential equation Аu ( x ) - Ku ( x ) = 0 is reducing to Volterra integral equation and has a unique zero solution. During it the operator A - K is closed and continuously invertible. Also, if the functions u ( t ), g ( t ) and f ( t ) are of exponential order α then nonhomogeneous equation Аu ( x ) - Ku ( x ) = f ( x ) for each f ( x ) has a unique solution, shown in this paper. The second theorem is mean that for the first investigated problem with an injective operator A - K , for f ( x ) and g ( x ) from C [ a , b ], the exact solution is given by equation u = ( A - K )- 1 f +( A - K )- 1 g b* for every vector b* = Ф( u ) that solves nonlinear algebraic (transcendental) system of n equations b = Ф(( A - K )- 1 f +( A - K )- 1 g b). And if the last algebraic system has no solution, then investigated problem also has no solution. The third theorem is means that exact solution of the second investigated problem is given by u = A - 1( f+g d*) for every vector d* = F ( Au ) that solves nonlinear algebraic (transcendental) system of n equations d = F ( f + g d). In this case we have same property - if the last algebraic system has no solution, then investigated problem also has no solution. Two particular examples for each considered problem are shown for illustration of exact solutions giving by perform the suggested in this paper methods. In the first example was considered integro-differential Volterra and Fredholm equation and in the second case was considered equation with nonlinear Fredholm integral.

Semi-Hyers–Ulam–Rassias Stability of a Volterra Integro-Differential Equation of Order I with a Convolution Type Kernel via Laplace Transform

In this paper, we investigate the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: exponential and polynomial functions. An important aspect that appears in the form of the studied equation is the symmetry of the convolution product.

Cones, rectifiability, and singular integral operators

Let \mu be a Radon measure on \mathbb{R}^d . We define and study conical energies \mathcal{E}_{\mu,p}(x,V,\alpha) , which quantify the portion of \mu lying in the cone with vertex x\in\mathbb{R}^d , direction V\in G(d,d-n) , and aperture \alpha\in (0,1) . We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that \mu has polynomial growth, we give a sufficient condition for L^2(\mu) -boundedness of singular integral operators with smooth odd kernels of convolution type.

A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type

Abstract The Legendre-based ultraspherical spectral method for ordinary differential equations (Olver, S. & Townsend, A. (2013) A fast and well-conditioned spectral method. SIAM Rev., 55, 462–489.) is combined with a formula for the convolution of two Legendre series (Hale, N. & Townsend, A. (2014a) An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 36, A1207–A1220.) to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth, then the method is spectrally accurate and the resulting almost-banded linear systems can be solved with linear complexity.

Collocation method with convergence for generalized fractional integro-differential equations

In this paper, we study a numerical approach for some class of generalized fractional integro-differential equations (GFIDEs) defined in terms of the B-operators presented recently. We develop collocation method for linear and nonlinear forms of GFIDEs. The numerical approach uses the idea of collocation methods for solving integral equations. Legendre polynomials are used to approximate the solution in finite dimensional space with convergence analysis. The obtained approximate solution recovers the solution of the fractional integro-differential equation (FIDE) defined using Caputo derivatives in a special case. FIDEs containing convolution type kernels appear in diverse area of science and engineering applications; therefore, some test examples varying the kernel in the B-operator are considered to perform the numerical investigations. The numerical results validate the presented scheme and provide good accuracy using few Legendre basis functions. • Collocation method for a linear and nonlinear Generalized Fractional Integro-differential equations (GFIDEs) defined in terms of B-operators is developed. • The convergence analysis for both forms of GFIDEs is established. • Test examples from the literature are considered to perform the numerical simulations, and the obtained results are presented through figures and tables.

On the Existence of Approximate Solution of Fredholm Integral Equation of the First Kind by Band-Limited Scaling Function

In this paper, we study the existence and uniqueness of the solution of Fredholm integral equation of the first kind with convolution type kernel. These results are based on band-limited scaling function which is generated by a class of band-limited wavelets. Since these band-limited functions are infinitely differentiable and possess rapid decay property, methods based on these functions would be highly accurate. Finally, convergence analysis has been discussed to validate the approximate solution.

Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method

We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.

Periodic Homogenization of Nonlocal Operators with a Convolution-Type Kernel

The paper deals with a homogenization problem for a nonlocal linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behavior of the rescaled operators as the scaling parameter tends to 0. More precisely we show that in the topology of resolvent convergence the family of rescaled operators converges to a second order elliptic operator with constant coefficients. We also prove the convergence of the corresponding semigroups both in $L^2$ space and the space of continuous functions and show that for the related family of Markov processes the invariance principle holds.

AN APPROXIMATE SOLUTION OF FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND BY BAND-LIMITED SCALING FUNCTION

Fredholm integral equation of the second kind appears naturally in different areas of mathematics, physics and engineering. In this paper we consider Fredholm integral equation of the second kind with convolution type kernel. We assume f in the approximation space and prove the existence and uniqueness of approximate solution by band-limited scaling function. Since these functions are infinitely differentiable and possess decay property, methods based on these functions would be very accurate. Convergence analysis has been discussed to validate our approximate solution.

Density Problem and Approximation Error in Learning Theory

We study the density problem and approximation error of reproducing kernel Hilbert spaces for the purpose of learning theory. For a Mercer kernel on a compact metric space (, ), a characterization for the generated reproducing kernel Hilbert space (RKHS) to be dense in is given. As a corollary, we show that the density is always true for convolution type kernels. Some estimates for the rate of convergence of interpolation schemes are presented for general Mercer kernels. These are then used to establish for convolution type kernels quantitative analysis for the approximation error in learning theory. Finally, we show by the example of Gaussian kernels with varying variances that the approximation error can be improved when we adaptively change the value of the parameter for the used kernel. This confirms the method of choosing varying parameters which is used often in many applications of learning theory.

Resolvents for weakly singular kernels and fractional differential equations

The solution R(t,s) of the resolvent equation (1)R(t,s)=B(t,s)+∫stB(t,u)R(u,s)du, where B(t,s) denotes a given weakly singular matrix, is obtained by means of fixed point mappings. The result is a series that begins with some singular terms after which the remainder of the terms defines a continuous function. In particular, the resolvent is calculated for the kernel B(t,s)=λ(t−s)q−1 of the scalar Abel integral equation of the second kind (2)x(t)=a(t)+λ∫0t1(t−s)1−qx(s)ds where 0<q<1. It is then used to derive closed-form formulas for the resolvents corresponding to q=1/2 and 1/3. Furthermore, a closed-form formula for the resolvent for B(t,s)=λs(t2−s2)q−1 is derived thereby demonstrating that the results of this paper apply not only to kernels of convolution type but also to those of non-convolution type as well. Finally, the resolvent for the kernel of (2) is used to find a general solution of a linear fractional differential equation of Caputo type.

A fast solver for integral equations with convolution-type Kernel

This paper studies the data redundancy of the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms of solving the integral equation whose kernel includes a convolution function factor. We develop lossless matrix compression strategies, which reduce the cost of integral evaluations and the storage to linear complexity, i.e., the same order of the approximation space dimensions. We establish that this algorithm preserves the convergence order of the approximate solution. We also propose a hardware-aware parallel algorithm for these strategies.

Convergence of series expansions for some infinite dimensional nonlinear systems

Volterra series expansions have been extensively used to solve and represent the dynamics of weakly nonlinear finite dimensional systems. Such expansions can be recovered by using the regular perturbation method and choosing the input of the system as the perturbation: the state (or the output) is then described by a series expansion composed of homogeneous contributions with respect to the input, from which kernels of convolution type can be deduced. This paper provides an extension (based on this approach) to a class of semilinear infinite dimensional systems, nonlinear in state and affine in input. As a main result, computable bounds of the convergence radius of the series are established. They characterize domains on which the series defines a mild solution of the system. The convergence criterion is established for bounded signals (infinite norms on finite or infinite time intervals) as follows: first, norm estimates of the series expansion terms are derived; second, the singular inversion theorem is used to deduce an easily computable bound of the convergence radius. In the formalism proposed here, non zero initial conditions can be also considered as a perturbation so that no precomputation of nominal trajectories is required in practice. The relevance of the method is illustrated on an academic example.

Sampling theorem, bandlimited integral kernels and inverse problems

We describe a novel approach based on the sampling theorem for studying eigenvalue problems associated with bandlimited integral kernels of convolution type. Two sets of functions biorthogonal to the eigenfunctions, one over the infinite interval and the other over a finite interval, are constructed and several identities satisfied by them are derived. The sampling theorem-based approach to the eigenvalue problem is further extended to construct the singular functions associated with the integral operator. It is shown that for the special case of the sinc-kernel, the eigenfunctions, the two biorthogonal sets and the singular functions reduce to the angular prolate spheroidal functions (or Slepian functions). Two methods are discussed for treating the inverse problem associated with bandlimited kernels—one employing the eigenfunctions and the biorthogonal sets and the other employing the singular functions. Numerical examples are included to illustrate the computation of eigenfunctions, biorthogonal sets and the singular functions and their application to the estimation of inverse solution.

Lattice-Nyström method for Fredholm integral equations of the second kind with convolution type kernels

We consider Fredholm integral equations of the second kind of the form f ( x ) = g ( x ) + ∫ k ( x - y ) f ( y ) d y , where g and k are given functions from weighted Korobov spaces. These spaces are characterized by a smoothness parameter α > 1 and weights γ 1 ≥ γ 2 ≥ ⋯ . The weight γ j moderates the behavior of the functions with respect to the jth variable. We approximate f by the Nyström method using n rank-1 lattice points. The combination of convolution and lattice group structure means that the resulting linear system can be solved in O ( n log n ) operations. We analyze the worst case error measured in sup norm for functions g in the unit ball and a class of functions k in weighted Korobov spaces. We show that the generating vector of the lattice rule can be constructed component-by-component to achieve the optimal rate of convergence O ( n - α / 2 + δ ) , δ > 0 , with the implied constant independent of the dimension d under an appropriate condition on the weights. This construction makes use of an error criterion similar to the worst case integration error in weighted Korobov spaces, and the computational cost is only O ( n log nd ) operations. We also study the notion of QMC-Nyström tractability: tractability means that the smallest n needed to reduce the worst case error (or normalized error) to ɛ is bounded polynomially in ɛ - 1 and d; strong tractability means that the bound is independent of d. We prove that strong QMC-Nyström tractability in the absolute sense holds iff ∑ j = 1 ∞ γ j < ∞ , and QMC-Nyström tractability holds in the absolute sense iff lim sup d → ∞ ∑ j = 1 d γ j / log ( d + 1 ) < ∞ .