A generalization of Cooke's integral inversion formula with application to remote-sensing theory

The motivation of the present work is to propose high order iterative methods with a recurrence formula for approximate matrix inversion and provide regularized solution of Fredholm integral equation of first kind with noisy data by an algorithm using the proposed methods. From the given family of methods of orders p = 7,11,15,19 are applied to solve problems of Fredholm integral equation of first kind. From the literature, iterative methods of same orders are used to solve the considered problems and numerical comparisons are shown through tables and figures.

SOLUTION OF INTEGRAL EQUATIONS BY BESSEL WAVELET TRANSFORM

In this article, an inversion formula is obtained for the spherical transform which integrates functions, defined on the unit sphere \(S^{2}\), on circles. The inversion formula is for the case where the circles of integration are obtained by intersections of \(S^{2}\) with hyperplanes passing through a common point \(\overline{a}\) strictly inside \(S^{2}\). In particular, this yields inversion formulas for two well-known special cases. The first inversion formula is for the special case where the family of circles of integration consists of great circles; this formula is obtained by taking \(\overline{a} = 0\). The second inversion formula is for the special case where the circles of integration pass through a common point \(p\) on \(S^{2}\); this formula is obtained by taking the limit \(\overline{a}\rightarrow p\).

A novel numerical technique to solve 2D Fredholm integral equations (2DFIEs) of first kind is proposed in this study. This technique is based on the discretization of 2DFIEs by replacing the unknown function with two-dimensional Bernstein polynomial basis functions. We formulate the convergence analysis which shows the fast converges of this technique to the actual solution. Some problems of 2D linear Fredholm integral equations are illustrated to show the efficiency of the proposed scheme.

An inverse measurement of the sudden underwater movement of the sea-floor by using the time-history record of the water-wave elevation

The problem of oblique water wave scattering by three thin vertical barriers in deep water is investigated here assuming linear theory. The geometrical configurations of the three barriers are such the inner(middle) barrier is partially immersed and the two outer barriers are completely submerged and extend infinitely downwards. A system of three simultaneous integral equations of first kind involving differences of velocity potentials across the barriers has been obtained in the mathematical analysis by employing Havelock’s expansion of water wave potential alongwith Havelock’s inversion formulae. Approximate numerical solutions of these integral equations are obtained by using single-term Galerkin approximations where the single term is chosen to be the exact solution of the integral equation obtained for a single vertical barrier in deep water for normal incidence of surface waves. Fairely accurate numerical estimates for the reflection and transmission coefficients are obtained by solving this system of linear equations. These estimates satisfy the energy identity, thereby justifying the validity of the present method. Due to energy identity, the behaviour of reflection coefficient is discussed by depicting it graphically against wavenumber. Existence of zeros of reflection coefficient is observed only when the two outer barriers are identical i.e. they are submerged from the same depth below the mean free surface. Some earlier results for two completely submerged barriers and a single barrier are recovered as special cases for a particular set of values of the parameters involved in the problem. This provides another check on the correctness of the present method.

B.D. Smith (ibid., vol.MI-4, p.15-25, 1985; Opt. Eng., vol.29, p.524-34, 1990) and P. Grangeat (These de doctorat, 1987; Lecture Notes in Mathematics 1497, p.66-97, 1991) derived a cone-beam inversion formula that can be applied when a nonplanar orbit satisfying the completeness condition is used. Although Grangeat's inversion formula is mathematically different from Smith's one, they have similar overall structures to each other. The contribution of the present paper is two-fold. First, based on the derivation of Smith, the authors point out that Grangeat's inversion formula and Smith's one can be conveniently described using a single formula (the Smith-Grangeat inversion formula) that is in the form of space-variant filtering followed by cone-beam back projection. Furthermore, the resulting formula is reformulated for data acquisition systems with a planar detector to obtain a new reconstruction algorithm. Second, the authors make two significant modifications to the new algorithm to reduce artifacts and numerical errors encountered in direct implementation of the new algorithm. As for exactness of the new algorithm, the following fact can be stated. The algorithm based on Grangeat's intermediate function is exact for any complete orbit, whereas that based on Smith's intermediate function should be considered as an approximate inverse excepting the special case where almost every plane in 3D space meets the orbit. The validity of the new algorithm is demonstrated by simulation studies.

Given n disjoint intervals on together with n functions , , and an matrix , the problem is to find an L2 solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem on n copies of and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.

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.

The problems of Engineering and Science can easily represent by developing their mathematical models in the terms of integral equations. Various analytical and numerical methods are available that can be used for solving integral equations of different kinds. In this paper, authors have considered recently developed integral transform “Rishi Transform” for obtaining the exact solution of non-linear Volterra integral equation of first kind (NLVIEFK). Four numerical problems have considered for demonstrating the complete procedure of determining the exact solution. Results of these problems depict that Rishi transform is very effective integral transform and it provides the exact solution of NLVIEFK without doing complicated calculation work.

The Pick's theorem is one of the rare gems of elementary mathematics because this is a very innocent sounding hypothesis imply a very surprising conclusion (Bogomolny 1997). Yet the statement of the theorem can be understood by a fifth grader. Call a polygon a lattice polygon if the co-ordinates of its vertices are integers. Pick's theorem asserts that the area of a lattice polygon P is given by A(P) = I(P) + B(P) / 2 - 1 = V(P) - B(P) / 2 - 1 where I(P), B(P) and V(P) are the number of interior lattice points, the number of boundary lattice points and the total number of lattice points of P respectively. It is worth to mention that the I(P) (understand like digital area) is digital mapping standard in USA since decade (Morrison, J. L. 1988 and 1989). Because the Pick's theorem was first published in 1899 therefore our planned presentation had timing its 100 anniversary. Currently it has greater importance than realized heretofore because of the Pick's theorem forms a connection between the old Euclidean and the new digital (discrete) geometry. During this long period lots of proof had been made of Pick's theorem and many trial of its generalization from simple polygons towards complex polygon networks, moreover tried to extend it to the direction of 3D geometrical objects as well. It is also turned out that nowadays the inverse Pick's formulas comes to the front instead of the original ones, consequently of powerful spreading the digital geometry and mapping. Today the question is not the old one: how can we produce traditional area without co-ordinates, using only inside points and boundary points. Just on the contrary: how is it possible to simply determine digital boundary and digital area (namely the number of boundary points and inside points) using known co-ordinates of vertices. The inverse formulas are: B(P)=ΣGCD (AX, AY, AZ) (1D Pick's theorem) and I(P)=A(P)-B(P)/2+1 (2D Pick's theorem) where GCD is the Great Common Divisor of the co-ordinate differences of two-two neighboring vertices. The our main object is not these formulas to present, but we desire to show that the Pick's theorem (after adequate redrafting) indeed valid for every spatial triangle which are determined by three arbitrary points of a 3D lattice. The original planar theorem is only a special case of it. However if it is true then its valid not only for triangles but all irregular polygons also which are lying in space and have its vertices in spatial lattice points. Finally if the extended Pick's theorem is true for all face of a lattice polyhedron then it is true for total surface as well. Consequently we developed so simple and effective algorithms which solve enumeration tasks without the time- and memory-wasting immediate computing. These algorithms make possible that using the vertex-co-ordinate list and the topological description of a convex or non-convex polyhedron (cube, prism, tetrahedron etc.) getting answer many elementary questions. For example, how many vaxels can be found on the complex surface of a polyhedron, how many on its edges or on its individual faces. We succeeded to extend our results also to the surface of non-cornered geometric objects (circle, sphere, cylinder, cone, ellipsoid etc.), but anyway, this have to be object of another presentation.

A method for the numerical resolution of Abel-type integral equations of the first kind

In the paper, we present a new approximate inversion of Volterra integral equation of first kind by using the Taylor expansin of the unknown function about lower limit of the Volterra integral.In this method, this Volterra integral equation is approximately transformed to a system of linear equations for the unknown function together with its derivatives.A desired solution can be determined by solving the resulting system according to the Cramer's rule.This method gives a simple and closed form of approximate Volterra integral equation of first kind, which may be able to use in computation work.Finally, we derive approximate solutions of this Volterra integral equation for special kernels about lower limit of Volterra integral.

We reduce the solution of contact problems in the interaction of rigid bodies (dies) with thin-walled elements (one-dimensional problems) to Volterra integral equations. We study the effect of the model describing the stress-strain state of plates on the type of integral equations and the structure of their solutions. It is shown that taking account of reducing turns the problem into a Volterra integral equation of second kind, which has a unique solution that is continuous and agrees quite well with the results obtained from the three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra integral equation of first kind that has a unique continuous solution; but for dies without corners the Herz condition does not hold (p(a) ≠ 0), and the contact pressure assumes its maximal value at the end of the zone of contact. For thin-walled elements, whose state can be described by the classical Kirchhoff-Love theory, the integral equation of the problem (a Volterra equation of first kind) has a solution in the class of distributions. The contact pressure is reduced to concentrated reactions at the extreme points of the contact zone. We give a comparative analysis of the solutions in all the cases just listed (forces, normal displacements, contact pressures). Three figures, 1 table. Bibliography: 5 titles.

We provide a constructive method of checking whether a linear programming problem (LPP) has a unique feasible or a unique optimal solution. Our method requires the solution of only one extra LPP such that the original problem has alternative solutions if and only if the optimal value of the new LPP is positive. If the original solution is not unique, an alternative solution is displayed. Possible applications are discussed.