Ram P. Kanwal's research while affiliated with Pennsylvania State University and other places

Publications (107)

Chapter
The purpose of this chapter is to give the basic ideas of the theory of distributions. Distributions or generalized functions, as they are also known, have proved to be very useful in many branches of pure and applied mathematics. Many textbooks, monographs and articles have been written on their theory and their applications [23], [42], [110], [12...
Chapter
The purpose of this chapter is to study the behavior of distributions at infinity in an average sense, which corresponds to the idea of Cesàro summability studied in classical analysis. Indeed, following [69] it is shown that the notion of Cesàro summability of divergent series and integrals admits a generalization to distributions and that this ge...
Chapter
In this chapter we discuss several properties of series of Dirac delta functions of the type $$ \sum\limits_{n = 0}^\infty {a_n \delta ^{(n)} (x)} . $$ (7.1)
Chapter
In many problems of engineering and physical sciences we attempt to write the solutions as infinite series of functions. The simplest series representation is the power series.
Chapter
The aim of the present chapter is to study the asymptotic behavior as ε → O of series of the type $$ \sum\limits_{n = 1}^\infty {a_n \varphi (n\varepsilon )} . $$ (5.1)
Book
An integral equation is called singular if either the range of integration is infinite or the kernel has singularities within the range of integration. Such equations occur rather frequently in mathematical physics and possess very unusual properties. This chapter discusses singular integral equations, Cauchy principal value for integrals, and the...
Chapter
Singular integral equations with logarithmic kernels arise in analysis and in many two-dimensional problems in mathematical physics, mechanics and engineering such as potential and scattering theories.
Chapter
In this chapter we give the solutions of Abel’s and some related integral equations. In the first section we present two methods for the solution of Abel’s equation and by using similar techniques solve some integral equations that can be reduced to Abel’s equation in the next section. The range of the parameter α appearing in Abel’s equation is ex...
Chapter
In this chapter we study the distributional solutions of singular integral equations. The distributional framework is rather convenient for the study of several singular operators, including the singular integrals that we have studied in the previous chapters. Therefore, the study of singular integral equations in spaces of distributions is a natur...
Chapter
In this chapter we study various singular integral equations with kernels of the Cauchy type, starting from the most basic Cauchy type integral equation.
Chapter
Many problems in physics and engineering which can be reduced to the integral equation $$ \alpha (\xi )g(\xi ) - \lambda \beta (\xi ){\text{p}}{\text{.v}}{\text{.}}\int\limits_C {\frac{{\gamma (\omega )g(\omega )}}{{\omega - \xi }}} d\omega = f(\xi ) $$ (4.1), where α(ξ),β(ξ), γ(ξ) and f (ξ) are prescribed functions of a real or complex variable ξ....
Article
Many physical problems that are usually solved by differential equation techniques can be solved more effectively by integral equation methods. This work focuses exclusively on singular integral equations and on the distributional solutions of these equations. A large number of beautiful mathematical concepts are required to find such solutions, wh...
Chapter
In this chapter we continue the study of the distributional solutions of integral equations. Our aim is to present the solution of singular integral equations of the type where $$a(x)g(x) + b(x)H\{ g(t);x\} = f(x), - \infty < x < \infty$$ (6.1), where H is the Hilbert transform $$H(g) = - \frac{1}{\pi }{{x}^{{ - 1}}}*g(x) = \frac{1}{\pi }p.v.\int_{...
Chapter
The purpose of this chapter is to study the distributional solution of the integral equations of the type $$g(x) + \lambda \int_{0}^{\infty } {k(x - y)g(y)dy = f(x), x \geqslant 0}$$ (8.1), as well as the corresponding equations of the first kind, the so-called Wiener-Hopf integral equations. Observe that the kernel k(x–y) is a difference kernel an...
Chapter
In this chapter we give a brief summary of several topics and results that are used throughout the book.
Chapter
In this chapter we shall consider dual and triple integral equations with trigonometric and Bessel-type kernels.
Article
We apply our recently developed distributional technique [2, 3] to study time-domain asymptotics. This enables us to present a rigorous mathematical discussion and extensions of the results given by Chapman [1] and subsequent workers in this field. The present analysis is facilitated by defining functions which are distributionally small at infinit...
Article
The author reviews and demonstrates some techniques based on distributional theory of asymptotic analysis and moment expansion of generalized functions in solving certain differential and integral equations. First, the author studies boundary layer theory and singular perturbation problems. The method is illustrated with the help of simple examples...
Article
The classical Poisson summation formula (1.1) and the corresponding distributional formula (1.2) have found extensive applications in various scientific fields. However, they are not universally valid. For instance, if φ(x) is a smooth function, the left-hand side of (1.1) is generally divergent. Even when both sides of (1.1) converge absolutely, t...
Chapter
Some algebraic operations on the delta function were studied in the last chapter. In subsequent chapters we shall be required to transform this function to certain curvilinear coordinates. For this purpose we devote an entire section to this topic. Let us first study the meaning of the function δ[f (x)] and prove the result $$\delta \left[ {f\left(...
Chapter
Let Rn be a real n-dimensional space in which we have a Cartesian system of coordinates such that a point P is denoted by x = (x1, x2,..., xn ) and the distance r, of P from the origin, is r = |x| = (x 12 + x 22 + ... + x n 2)1/2. Let k be an n-tuple of nonnegative integers, k = (k1, k2,..., kn ), the so-called multiindex of order n; then we define...
Chapter
The Heaviside function H (x) is defined to be equal to zero for every negative value of x and to unity for every positive value of x, that is, $$H(x)=\left\{ \begin{matrix}0, & x<0, \\ 1. & x>0. \\ \end{matrix}\right\}$$ (1) It has a jump discontinuity at x = 0 and is also called the unit step function. Its value at x = 0 is usually taken to be 1/2...
Chapter
In attempting to define the Fourier transform of a distribution t (x), we would like to use the formula (in R1) $$\hat{t}(u)=F(t(x))=\int_{-\infty }^{\infty }{{{e}^{iux}}t(x)dx}$$ (1)
Chapter
Estrada and Kanwal have recently [6, 69–71] developed a distributional approach to asymptotic analysis. Their study is based on the interplay between the generalized functions and the theory of moments. Thus, they have not only succeeded in presenting a simplified approach to various known aspects of asymptotics but have also found many new results...
Chapter
Let Rm and Rn be Euclidean spaces of dimensions m and n respectively, and let x = (x1,..., xm ) and y = (y1,..., yn ) denote the generic points in Rm and Rn , respectively. Then a point in the Cartesian product Rm+n = Rm × Rn is (x, y) = (x1,..., xm , y1,..., yn ). Furthermore, let us denote by Dm , Dn , and Dm+n the spaces of test functions with c...
Chapter
The main applications of the Laplace transform are directed toward problems in which the time t is the independent variable. We shall therefore use this variable in this chapter. Let f(t) be a complex-valued function of the real variable t such that f(t)e−ct is abolutely integrable over 0 < t < ∞, where c is a real number. Then the Laplace transfor...
Chapter
In Chapter 10 we have discussed various properties of the homogeneous and inhomogeneous wave equations in two and three dimensions. We derived their fundamental solutions and studied moving point, line, and surface sources. In Chapter 5 we considered various kinematic and geometrical aspects of the wave propagation in the context of surface distrib...
Chapter
In boundary and initial-value problems relating to the potential, scattering and wave propagation theories, we encounter functions that are defined inside or outside some surface S if the surface is closed, and on both sides of it if it is open. However, these functions or their first- or higher-order derivatives have jumps across S. Classical theo...
Chapter
Recall that we define a function as a rule that maps (transforms) numbers into numbers, whereas a functional maps functions into numbers. An operator maps functions into functions. Let a class of functions be given, all defined for a variable, say, time t, −∞ < t < ∞. Then an operator (transformation) L assigns a member of this class (inputs, excit...
Chapter
In order to present the applications of generalized functions to the theories of probability and random processes let us start with some basic concepts.
Chapter
In the previous chapters we have defined various singular distributions. One of them is Pf(1/x), defined in Example 4 of Section 2.4. The function 1/x is not integrable on any neighborhood of the origin. We succeeded in regularizing this function by defining the functional Pf(1/x) by the principal value of the singular integral defined by the quant...
Article
We introduce the concepts of pre-asymptotic schemes and pre-asymptotic expansions to study the divergent series that formally are solutions of various types of equations.
Article
We present a distributional approach for solving problems in the boundary layer and singular perturbation theories. We discuss the asymptotic expansions in the inner and outer regions and their matching in the overlapping domain by our theory. This technique is illustrated with various examples from the initial value and boundary value problems. Fr...
Chapter
In this chapter we study the asymptotic behavior as ε → 0 of series of the type $$\sum\limits_{{n = 1}}^{\infty } {{a_{n}}\phi (n} \varepsilon )$$ (5.1.1)
Chapter
In this chapter we discuss several properties of series of Dirac delta functions [31] of the type $$\sum^{\infty}_{n=0}{a_{n}\delta^{(n)}}{x}$$ (6.1.1) As we have seen in the previous chapters, not only do such series form the building blocks in the asymptotic expansion of distributions, but they also arise in other contexts.
Chapter
In many problems of engineering and the physical sciences we attempt to write the solutions as infinite series of functions. The simplest series representation is the power series. Given a function f(x) of a real variable x containing a number x0 in its domain of definition, we try to find a power series of the form $$\sum^{\infty}_{j=0}{a_{j}(x-x_...
Chapter
The purpose of this chapter is to present the basic ideas of the theory of distributions. Distributions or generalized functions, as they are also known, have proved to be very useful in many branches of pure and applied mathematics. Many textbooks, monographs and articles have been written on their theory and their applications [12], [23], [53], [...
Article
We present the Taylor asymptotic expansion of a perturbed distribution of the form is a smooth function defined in ℝn. First we present the one-dimensional theory to illustrate the underlying concepts and then we discuss the multi-dimensional case. We find that various known results follow as special limits of our results.
Article
We present various aspects of generalized functions and their applications to wave propagation, asymptotic expansions and singular integral equations. We start with the basic concepts of the sifting property of the delta function and the fact that it is the derivative of the Heaviside function. Then we build up the derivatives of the generalized fu...
Article
We apply regularization of divergent integrals in the derivation of the asymptotic expansion of certain multi-dimensional generalized functions. We further present several illustrations to demonstrate that the asymptotic development of generalized functions provides a lucid formulation of many concepts in asymptotic analysis such as the expansion o...
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We present the theory and technique for obtaining the distributional solutions for the Wiener-Hopf integral and integro-differential equations. This is achieved by identifying a class of kernels for which these equations are well defined and are of the Fredholm type. Consequently, the associated operators and their images are of finite dimensions....
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We present various techniques for the asymptotic expansions of generalized functions. We show that the moment asymptotic expansions hold for a very wide variety of kernels such as generalized functions of rapid decay and rapid oscillations. We do not use Mellin transform techniques as done by previous authors in the field. Instead, we introduce a d...
Article
First, we discuss and correlate the various types of regularizations available in the literature for the singular function , where k is an integer and H(x) is the Heaviside function. Then we present the corresponding regularization for the function r−k, where r is the radial distance in n. Thereafter, we express the recently discovered distribution...
Article
We present an algebraic technique for solving integral equations. This technique is based on first differentiating both sides of the integral equation n times and then substituting the Taylor series for the unknown function in the resulting equation. This results in a linear algebraic system which can be solved approximately by a suitable truncatio...
Article
A consolidated account is presented of the singular integral equations with logarithmic kernels which occur very frequently in applied mathematics. The analysis starts with the simplest integral equation of this type and progresses steadily to more complicated ones. The Fredholm integral equations of the first as well as the second kind are studied...
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The authors study the delta function delta ( zeta , zeta ), where zeta is a point in the Minkowski space and quantity ( zeta , zeta )=t2-x21-x22-x23=t2-r2. This generalised function has its support on the light cone t+or-r=0, which has an interesting singularity at its vertex (0,0). They present the derivatives of the multilayers spread over this c...
Article
It is shown that the formulas for the jumps in various derivatives of a harmonic function across a surface of discontinuity can be given in terms of various differential geometric quantities of this surface such as its mean curvature.
Article
We Present the necessary and sufficient conditions for a sequence to be the moment sequence of a distribution in the space of the distributions of compact support. The analysis is based on a series of lemmas that reduce the problem to that of existence of distributional boundary limits of certain analytic functions defined in the unit disk. These r...
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We present a consolidated account of Carleman type singular integral equations. We have started from the most basic equation of this type and have at first attempted to integrate from the existing literature the progress which has been made so far in this field. Many different techniques are presented to solve these types of integral equations both...
Article
We present the distributional solutions to the hypergeometric differential equation. These solutions are obtained in the form of infinite series of the Dirac Delta functions and its derivatives. We employ these solutions to observe their interesting features. Furthermore, the form of these solutions is the same as the ones required for the weight d...
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We present a general theory of moving and deforming wave fronts by first defining and discussing the higher order fundamental forms for such a surface. These fundamental forms are then used to find the general formula for the jump of the Nth order differential of generalized functions which are discontinuous across this surface. Subsequently, we co...
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It is shown that a series of positive terms that converges on all sets of null density should be convergent. Using this result we construct examples of complete topological vector spaces that are proper subspaces of a Banach space, but whose dual spaces coincide with the dual space of the Banach space.
Article
It is shown that a series of positive terms that converges on all sets of null density should be convergent. Using this result we construct examples of complete topological vector spaces that are proper subspaces of a Banach space, but whose dual spaces coincide with the dual space of the Banach space.
Article
We consider the scattering of low-frequency sound waves by an arbitrary penetrable obstacle whose density and compressibility are different from those of the surrounding infinite medium. We present solutions both inside and outside the scatterer. The physical properties of the scatterer are implicit in the transition conditions at its surface and i...
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We present distributional solutions of three singular dual integral equations. They are of Cauchy, Abel and Titchmarsh types. These solutions are achieved by defining appropriate function spaces, by using the concepts of analytic representation of distributions, and by introducing suitable operators and boundary value problems.
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Our main aim is to present the value of the distributional derivative {partial}^N/partial xk_1_1 partial x^{k_2_2 \cdots partial xk_p_p}(1/r^n), where r = (x^2_1+x^2_2+ \cdots +x^2_p)1/2 in R^p, N = k_1 + k_2 + \cdots + k_p, and p, n, k_1, k_2, \cdots, k_p are positive integers. For this purpose, we first define a regularization of 1/x^n in R^1, wh...
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Our main aim is to present the value of the distributional derivative ∂͞ N /∂ x 1 k 1 ∂ x 2 k 2 . . . ∂ x p k p (1/ r n ), where r = ( x 1 ² + x 2 ² + . . . + x p ² ) ½ in R p , N = k 1 + k 2 + . . . + K p , and p , n , k 1 , k 2 , . . ., k p are positive integers. For this purpose, we first define a regularization of 1/ x n in R ¹ , which in turn...
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We present the solutions for the boundary value problems of elasticity when a homogeneous and isotropic solid of an arbitrary shape is embedded in an infinite homogeneous isotropic medium of different properties. The solutions are obtained inside both the guest and host media by an integral equation technique. The boundaries considered are an oblon...
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A complete and unified approach for finding the distributional solutions for various singular integral equations is presented. The equations considered are the Abel, the Cauchy, the Carleman, and those with logarithmic kernels. Some related singular integral equations are also considered. These solutions are achieved by introducing various interest...
Article
It is advantageous to convert initial and boundary value problems to integral equations because many more analytical, numerical and finite element methods are available for solving the integral equations than are available for the partial differential equations. Moreover, since the integral equations incorporate the initial and boundary value probl...
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We present some integral equation techniques for solving the composite media problems in various physical fields such as potential theory, scattering phenomena, elasticity, quantum mechanics and electromagnetism. These methods enable us to find solutions for many penetrable boundaries such as triangular, rectangular and elliptic cylinders, prisms a...
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A theory for distributional boundary values of harmonic and analytic functions is presented. In this analysis there arise several indicators that measure the growth of these functions near the boundaries. An extension of the Phragmén-Lindelöf maximum principle is derived. Furthermore, the algebraic properties of the space of real periodic distribut...
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We study two‐dimensional problems of elasticity when a homogeneous and isotropic solid of an arbitrary shape is embedded in an infinite homogeneous isotropic medium of different properties. Solutions are obtained both inside the guest and host media. These solutions are derived by first transforming the boundary value problems to the equivalent int...
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First, we present formal aspects of the scattering of plane elastic waves by a single elastic cylindrical inclusion or a cavity of an arbitrary cross section embedded in an isotropic homogeneous elastic medium of different properties. An integral equation is used to derive expressions for the displacement field valid in the entire domain. Thereafte...
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The integral equations derived in Part I of this work are solved for a cylindrical inclusion of an arbitrary symmetrical shape embedded in an infinite medium of different properties in various physical fields. The configurations considered are circular, elliptic, and rectangular cylinders and a prism. Since the integral equations are valid in the e...
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The 2-dimensional problem of scattering of obliquely incident P and SV waves by an infinite rigid elliptic cylinder embedded in an infinite, isotropic and homogeneous elastic medium is solved. Approximate formulas are derived for the displacement field, stress tensor, far-field amplitudes and the scattering cross section when the wave lengths are l...
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Suitable singularites such as a dynamical Kelvin quadropole are defined to study the dynamical displacements set up in an infinite homogeneous and isotropic elastic medium. Approximate solutions are presented up to terms which are of higher order than those known so far.
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Integral equations are derived for boundary value problems in various branches of potential theory when an inclusion of arbitrary shape is embedded in an infinite host medium of different properties. These integral equations are derived in full generality by taking the parameters of the media to be scalar functions of position. Furthermore, they ar...
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The problem of scattering of plane compressional wave by an elastic sphere embedded in an isotropic elastic medium of different material properties is solved. Approximate formulas are derived for the displacement field, stress tensor, stess intensity factors, far-field amplitudes and the scattering cross section. It is assumed that the wave length...
Article
The displacement field is determined when a rigid spheroidal inclusion is present in an infinite, isotropic and homogeneous elastic medium under torsion. The values of the force and moment are derived. The solutions for the limiting cases of a sphere and a circular disk are also presented. The analysis is based on suitable distributions of singular...
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First the concepts of the surface distributions are explained. Thereafter the first and higher-order distributional derivatives are derived for a function of several variables. These concepts are then used to study the propagation of wave fronts in continuum mechanics. Explicit formulas for the jump relations for the first and second-order partial...
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The problems of scattering of plane compressional and shear waves by a class of infinite circular cylindrical flaws and inclusions embedded in an infinite isotropic homogeneous elastic medium are solved. The objects considered are an elastic cylinder, an inviscid-fluid-filled cavity, an empty cavity, and a rigid cylinder. Approximate formulas are d...
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Full-text available
A method of singularities recently developed to solve various displacement-type boundary value problems in elastostatics is extended to elastodynamics. Specifically, a study is made of the dynamical displacements set up in an infinite elastic space when a rigid spheriod, embedded in this space, is subjected to a transverse periodic displacement. Th...
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The solutions are presented for six problems of the scattering of low frequency plane harmonic elastic P and S waves when they impinge on a movable or an immovable rigid spherical inclusion or a spherical cavity. In each problem, the scatterer is embedded in an infinite homogeneous isotropic elastic medium. Formulas are derived for the scalar wave...
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By considering suitable distributions of the Dirac delta function and its derivatives on lines and curves, solutions are obtained in closed form for various boundary value problems in mathematical physics. The fields considered are electrostatics, magnetostatics, potential theory, hydrodynamics, elasticity, scattering of low frequency electromagnet...
Article
By distributing the concentrated singularities such as a Kelvin doublet along the axis of symmetry we describe the displacement field, in an elastic medium, for various modes of rotation and translation for a rigid prolate and oblate spheroid. The limiting cases of a sphere, a slender body and a thin circular disk are also discussed. All the soluti...
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We present a complete and a simplified discussion of the scattering of acoustic, electromagnetic and earthquake SH waves by various shapes such as infinite cylinders, strips, slits, cracks, cavities, and semiinfinite planes. Formulas are derived for the velocity potentials, electromagnetic fields, displacement fields, far-field amplitudes, scatteri...
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This paper is devoted to the applications of an integral equation perturbation technique to the problems of scattering by soft and rigid obstacles in acoustics, scattering by rigid inclusions and cracks in elastic medium and conducting bodies in electromagnetism. Various interesting shapes such as spheres, ellipsoids, disks and spherical caps are c...
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We present the axially symmetric stress distributions in elastic solids containing ring-shaped cracks when the solids are kept under torsion. Two cases are analyzed. In the first case we discuss the crack problem in an infinite elastic medium when the ratio of the inner to the outer radius is small. In the second case this ratio is almost equal to...
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Es wird eine Theorie für die Strömung einer elastoviskosen Flüssigkeit aufgestellt, in der ein axialsymmetrischer Körper Drehschwingungen ausführt. Ziel ist die Behandlung des Scheiben-Elastoviskosimeters. Zu diesem Zweck Lösen wir die Randwertaufgabe für Rotationsellipsoide und leiten daraus durch Grenzübergang die Lösung für die Scheibe her. Es w...
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Integral equations associated with the diffraction of obliquely incident plane waves of various kinds by two coplanar and parallel infinite strips and slits are solved approximately for low frequencies. The solutions are presented as expansions in terms of a suitable perturbation parameter.Certaines quations intgrales en relation avec la diffractio...
Article
The problem of diffraction of normally incident longitudinal and antiplane shear waves by two parallel and coplanar rigid strips embedded in an infinite, isotropic and homogeneous elastic medium is solved. Approximate formulas are derived for the displacement field, stress tensor, far-field amplitudes and the scattering cross section when the wave...
Article
A solution for the electrostatic field due to an inner lamina at a constant potential surrounded by an annular ring at another constant potential is presented. Two cases are considered. The first is a double circular disk and the second is a double spherical cap.
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The problem of diffraction of normally incident longitudinal and antiplane shear waves by two parallel and coplanar Griffith cracks embedded in an infinite, isotropic and homogeneous elastic medium is solved. Approximate formulas are derived for the displacement field, stress tensor, stress intensity factors, farfield amplitudes and scattering cros...
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The problem of the diffraction of time-harmonic axially symmetric acoustic waves by a perfectly rigid annular spherical cap is solved approximately by an integral equation technique. Formulas are derived for the far-field amplitude as well as the scattering cross section when the incident wave is a plane wave traveling along the polar axis. By taki...
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The paper deals with the diffraction of time-harmonic axisymmetric acoustic waves by a perfectly soft annular spherical cap. An integral equation technique is used to solve this three-part boundary value problem. Formulae are presented for the far field amplitude as well as the scattering cross section for the case when the incident wave is a plane...
Article
The representation formula which embodies the solution of a two-part or a three-part mixed boundary value problem leads to a Fredholm integral equation of the first kind which cannot be solved easily. In this paper we present a technique which reduces the Fredholm integral equation of the first kind to Volterra integral equations of the first kind...

Citations

... The Abel integral equation is provided by [31] f (s) = s a g(τ) (s − τ) γ dτ, s > a, 0 < γ < 1 (8) and its solution is ...
... [ϕ (z)] ∈ U ′ . Here p (z) is a polynomial such that ν/ p ∼ O x −1 in the Silva-Cesàro sense and P (z) is an arbitrary polynomial [4,9,10]. ...
... The symbolic function δ(z − a) represents the pressure in a source point located in z = a, and the accumulated pressure is termed as the Heaviside function [32], in such a way ∆F pres = ∆p · ∆s, where, ∆F pres is the pressurization force, [N ]; ∆s is the area of the oil injection port, [m 2 ], and ∆p is the pressure of the oil injection port, [P a]. Consequently, when ∆p → 0, then ∆s → 0 so that: ∆F pres = ∆p · ∆s = q = constant. ...
... Formulas (1.3) are correct distributional identities, and while they have been generalized to any dimension [2,3] they have been questioned by several authors. Indeed, the distribution (3x i x j − ...
... The application of dual integral equations to diffraction problems has been extensively studied by Lur'e [28], Lebedev and Skal'skaya [29][30][31][32]. Recently solutions involving generalised functions have been found by Estrada and Kanwal [33][34]. Various methods of numerical treatment of dual integral equations, namely, reduction to a system of algebraic equations, or to a Fredholm type of equation, the multiplying factor method, the integral representation method, the reduction to a single integral equation of the second kind have been very well reviewed by Sneddon [9] and also by Williams [35]. ...
... Interestingly these elementary formulas are the starting point of the method for solving integral equations on a circle as presented in [7] and a similar method for solving equations over a sphere could be constructed from the formulas of the present study. ...
... We refer to the texts for the basic ideas about distributions [9,10]. Now we present some ideas of the local analysis of distributions [11][12][13][14][15] and of the distributional integral [4,16]. Łojasiewicz [17] defined the value of a distribution f ∈ D (R) at the point x 0 as the limit ...
... Most of the work on compatibility conditions has been concentrated on the first and second order partial derivatives. We have presented the compatibility conditions for the fourth order biharmonic equation[6]. Our analysis, however, contained a large number of terms and a clear pattem did not emerge. ...
... Thus, we obtain (3.5) as required. [13], and Littlejohn and Kanwal [16], we find that the conditions of m in their methods are identical to ours. ...
... to various differential equations in a normal form with singular coefficients was studied by many researchers [9][10][11][12][13]. Furthermore, a brief introduction to these concepts is presented by Kanwal [14]. ...