The linear canonical Hankel type transformations associated with translation and convolution

This paper is the study of Hankel type translation and Hankel type convolution for linear canonical Hankel type transformations. In this paper, I have studied some inequalities associated with Hankel type translation and Hankel type convolution. Also, I have studied some applications of linear canonical Hankel transformation to a canonical convolution integral equation and a generalized nonlinear parabolic equation.

On the regularity of curvature fields in stress-driven nonlocal elastic beams

Elastostatic problems of Bernoulli–Euler nanobeams, involving internal kinematic constraints and discontinuous and/or concentrated force systems, are investigated by the stress-driven nonlocal elasticity model. The field of elastic curvature is output by the convolution integral with a special averaging kernel and a piecewise smooth source field of elastic curvature, pointwise generated by the bending interaction. The total curvature is got by adding nonelastic curvatures due to thermal and/or electromagnetic effects and similar ones. It is shown that fields of elastic curvature, associated with piecewise smooth source fields and bi-exponential kernel, are continuously differentiable in the whole domain. The nonlocal elastic stress-driven integral law is then equivalent to a constitutive differential problem equipped with boundary and interface constitutive conditions expressing continuity of elastic curvature and its derivative. Effectiveness of the interface conditions is evidenced by the solution of an exemplar assemblage of beams subjected to discontinuous and concentrated loadings and to thermal curvatures, nonlocally associated with discontinuous thermal gradients. Analytical solutions of structural problems and their nonlocal-to-local limits are evaluated and commented upon.

Convolution Integral: How a Graphical Type of Solution Can Help Minimize Misconceptions

A physical system, Low Pass Filter (LPF) RC Circuit, which serves as an impulse response and a square wave input signal are utilized to derive the continuous time convolution (convolution integrals). How to set up the limits of integration correctly and how the excitation source convolves with the impulse response are explained using a graphical type of solution. This in turn, help minimize the students’ misconceptions about the convolution integral. Further, the effect of varying the circuit elements on the shape of the convolution output plot is presented allowing students to see the connection between a convolution integral and a physical system. PSpice simulation and experiment results are incorporated and are compared with those of the analytical solution associated with the convolution integral.

Asymptotic solution to convolution integral equations on large and small intervals

We consider convolution integral equations on a finite interval with a real-valued kernel of even parity, a problem equivalent to finding a Wiener–Hopf factorization of a notoriously difficult class of 2 × 2 matrices. The kernel function is assumed to be sufficiently smooth and decaying for large values of the argument. Without loss of generality, we focus on a homogeneous equation and we propose methods to construct explicit asymptotic solutions when the interval size is large and small. The large interval method is based on a reduction of the original equation to an integro-differential equation on a half-line that can be asymptotically solved in a closed form. This provides an alternative to other asymptotic techniques that rely on fast (typically exponential) decay of the kernel function at infinity, which is not assumed here. We also consider the problem on a small interval and show that finding its asymptotic solution can be reduced to solving an ODE. In particular, approximate solutions could be constructed in terms of readily available special functions (prolate spheroidal harmonics). Numerical illustrations of the obtained results are provided and further extensions of both methods are discussed.

An Inverse Model for Two-dimensional Reflectors With Scattering

We present a novel approach of modelling surface light scattering in the context of freeform optical design. Using energy conservation, we derive an integral relation between the scattered and specular distributions. This integral relation reduces to a convolution integral in the case of isotropic scattering in the plane of incidence for cylindrically and rotationally symmetric problems.

A stochastic convolution integral inequality

We prove a stochastic Gronwall lemma of the convolution type. Our results extend that of Scheutzow [A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013) 1350019], and the related results established in the non-convolution case. The proofs of the present results are essentially based on the Métivier–Pellaumail inequality for semimartingales.

INVERSE PROBLEM FOR INTEGRO-DIFFERENTIAL HEAT EQUATION WI TION WITH A VARIABLE COEFFICIEN ABLE COEFFICIENT OF THERM T OF THERMAL CONDUCTIVITY

Background. The inverse problem of finding a multidimensional memory kernel of a time convolution integral depending on a time variable t and (n-1)-dimensional spatial variable. 2-dimensional heat equation with a time-dependent coefficient of thermal conductivity is studied. Methods. The article is used Cauchy problems for the heat equation, resolvent methods for Volterra type integral equation and contraction mapping prinsiple.