A direct approach to the mellin transform

The aim of this paper is to present an approach to the Mellin transform that is fully independent of Laplace or Fourier transform theory, in a systematic, unified form, containing the basic properties and major results under natural, minimal hypotheses upon the functions in questions. Cornerstones of the approach are two definitions of the transform, a local and global Mellin transform, the Mellin translation and convolution structure, in particular approximation-theoretical methods connected with the Mellin convolution singular integral enabling one to establish the Mellin inversion theory. Of special interest are the Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame. These two operators are different than those considered thus far and more general. They are of particular importance in solving differential and integral equations. As applications, the wave equation onℝ ₊ × ℝ₊ and the heat equation in a semi-infinite rod are considered in detail. The paper is written in part from an historical, survey-type perspective.     

A New defy for iteration methods

The work presents an adaptation of iteration method for solving a class of thirst order partial nonlinear differential equation with mixed derivatives.The class of partial differential equations present here is not solvable with neither the method of Green function, the most usual iteration methods for instance variational iteration method, homotopy perturbation method and Adomian decomposition method, nor integral transform for instance Laplace,Sumudu, Fourier and Mellin transform. We presented the stability and convergence of the used method for solving this class of nonlinear chaotic equations.Using the proposed method, we obtained exact solutions to this kind of equations.     

The collocation method for mixed boundary value problems on domains with curved polygonal boundaries

Summary. We consider an indirect boundary integral equation formulation for the mixed Dirichlet-Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. A complete stability and solvability analysis of the transformed integral equations is given by use of a Mellin transform technique, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space. Received April 15, 1995 / Revised version received April 10, 1996     

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries.

     

A note on the inversion of Laplace transforms

A direct method, using a Mellin transform technique, is presented to derive the solution of a special class of first kind integral equations over the positive real axis, and as a particular case, an inversion formula is deduced for the Laplace transform F(p) of a function f(x) (x>0), when F(p) is known only for p>0.     

Optimization of Functional Integral Equations: A Spectral Approach

Abstract

The spectral method of Elnagar and Kazemi (J. Comp. Appl. Math. 76(1–2):147–158, 1996), which yields spectral convergence rate for the approximate solutions of Volterra-Hammerstein integral equations, is generalized in order to solve the larger class of functional integral equation control systems with spectral accuracy. The proposed method is based on the idea of relating spectrally constructed grid points to the structure of projection operators. These operators will be used to approximate the control vector and the associated state vector. Numerical examples are included to demonstrate the accuracy of the proposed method.     

Estimates for imaginary powers of Laplace operator in variable Lebesgue spaces and applications

In this paper we study some estimates of norms in variable exponent Lebesgue spaces for singular integral operators that are imaginary powers of the Laplace operator in ? ⁿ . Using the Mellin transform argument, fromthese estimates we obtain the boundedness for a family of maximal operators in variable exponent Lebesgue spaces, which are closely related to the (weak) solution of the wave equation.     

A new approach for admissibility analysis of the direct discontinuous Galerkin method through Hilbert matrices

This article continues the study of the so‐called direct discontinuous Galerkin (DDG) method for diffusion problems as developed in [Liu and Yan, SIAM J Numer Anal 47 (2009), 475–698;, Liu and Yan, Commun Comput Phys 8 (2010), 541–564; C. Vidden and J. Yan, J Comput Math 31 (2013), 638–662; H. Liu, Math Comp (in press)]. A key feature of the DDG method lies with the numerical flux design which includes two (or more) free parameters. This article identifies the class of all admissible numerical flux choices (Theorem 2.2) for degree n polynomial approximations for the symmetric DDG method [C. Vidden and J. Yan, J Comput Math 31 (2013), 638–662], guaranteeing stability of the resulting method. Our main contribution is the new technique of analysis for the DDG admissibility condition. The strategy is to directly evaluate the admissibility condition (Lemma 2.4) by choosing a simple polynomial basis. The admissibility condition is then transformed into an eigenvalue problem resulting in showing needed properties of inverse Hilbert matrices (Lemma 2.3, Appendix). Numerical tests are provided to confirm theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 350–367, 2016     

Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem

In this paper, two results concerning the global attractivity and global asymptotic attractivity of the solutions for a nonlinear functional integral equation are proved via a variant of the Krasnoselskii fixed point theorem due to Dhage [B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J. 44 (2004) 145–155]. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A couple of examples are indicated for demonstrating the natural realizations of the abstract results presented in the paper. Our results generalize the attractivity results of Banas and Rzepka [J. Banas, B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003) 1–6] and Banas and Dhage [J. Banas, B.C. Dhage, Global asymptotic stability of solutions of a functional integral equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.07.038], under weaker conditions with a different method.     

Generalized Integral Transforms with the Homotopy Perturbation Method

This paper applies He’s homotopy perturbation method to compute a large variety of integral transforms. The Esscher, Fourier, Hankel, Laplace, Mellin and Stieljes integrals transforms are particular cases of our generalized integral transform. Our method is of practical importance in order to derive new integration formulae, to approximate certain difficult integrals as well as to calculate the expectation of certain nonlinear functions of random variable.     

On the finite Mellin transform in quantum calculus and application

The aim of the present paper is to introduce and study a new type of q-Mellin transform [11], that will be called q-finite Mellin transform. In particular, we prove for this new transform an inversion formula and q-convolution product. The application of this transform is also earlier proposed in solving procedure for a new equation with a new fractional differential operator of a variational type.     

An asymptotic method of solving transient dynamic contact problems of the theory of elasticity for a strip

An asymptotic method is proposed for solving transient dynamic contact problems of the theory of elasticity for a thin strip. The solution of problems by means of the integral Laplace transformation (with respect to time) and the Fourier transformation (with respect to the longitudinal coordinate) reduces to an integral equation in the form of a convolution of the first kind in the unknown Laplace transform of contact stresses under the punch. The zeroth term of the asymptotic form of the solution of the integral equation for large values of the Laplace parameter is constructed in the form of the superposition of solutions of the corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation on the entire axis. In solving the Wiener-Hopf integral equations, the symbols of the kernel of the integral equation in the complex plane is presented in special form — in the form of uniform expansion in terms of exponential functions. The latter enables integral equations of the second kind to be obtained for determining the Laplace-Fourier transform of the required contact stresses, which, in turn, is effectively solved by the method of successive approximations. After Laplace inversion of the zeroth term of the asymptotic form of the solution of the integral equations, the asymptotic solution of the transient dynamic contact problem is determined. By way of example, the asymptotic solution of the problem of the penetration of a plane punch into an elastic strip lying without friction on a rigid base is given. Formulae are derived for the active elastic resistance force on the punch of a medium preventing the penetration of the punch, and the law of penetration of the punch into the elastic strip is obtained, taking into account the elastic stress wave reflected from the strip face opposite the punch and passing underneath it.     

Teichmüller Extensibility of Circle Homeomorphisms

In this paper a condition is obtained in terms of Dirichlet's integral, for a sense-preserving homeomorphism between the unit circumferences to be prolonged into the interior of disk quasiconformally or as extremal Teichmüller mapping, which sharpens and simplifies the widely known theorems by Teichmüller [ Abh. Preuss. Akad. Wiss. Math. Naturw. Kl. 22 (1939) 1-197], Ahlfors [ J. d'Anal. Math ., 3 (1953/54) 1-98], Hamilton [ Trans. Amer. Math. Soc ., 138 (1969) 399-406], Reich [ Ann. Acad. Sci. Fenn. Ser. A. I. Math . 10 (1985) 469-475], Strebel [ Comment. Math. Helv. , 39 (1964) 77-89], Beurling and Ahlfors [ Acta Math ., 96 (1956) 125-142].     

On the Global Relation and the Dirichlet‐to‐Neumann Correspondence

The recently developed Fokas method for solving two‐dimensional Boundary Value Problems (BVP) via the use of global relations is utilized to solve axisymmetric problems in three dimensions. In particular, novel integral representations for the interior and exterior Dirichlet and Neumann problems for the sphere are derived, which recover and improve the already known solutions of these problems. The BVPs considered in this paper can be classically solved using either the finite Legendre transform or the Mellin‐sine transform (which can be derived from the classical Mellin transform in a way similar to the way that the sine transform can be derived from the Fourier transform). The Legendre transform representation is uniformly convergent at the boundary, but it involves a series that is not useful for many applications. The Mellin‐sine transform involves of course an integral but it is not uniformly convergent at the boundary. In this paper: (a) The Legendre transform representation is rederived in a simpler approach using algebraic manipulations instead of solving ODEs. (b) An integral representation, different that the Mellin‐sine transform representation is derived which is uniformly convergent at the boundary. Furthermore, the derivation of the Fokas approach involves only algebraic manipulations, instead of solving an ordinary differential equation.     

A perturbation result for bounded imaginary powers

We give a new proof of a perturbation result due to J. Prüss and H. Sohr [11]: if an operator A has bounded imaginary powers, then so does A+w (w ≧ 0). Instead of Mellin transform on which the proof in [11] is based, we use the functional calculus for sectorial operators developed in particular by A. McIntosh ([8], [3] and [1]). It turns out that our method gives a more general result than the one used in [11].     

Fractional differential equations solved by using Mellin transform

In this paper, the solution of the multi-order differential equations, by using Mellin transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by solving a linear set of algebraic equations, whose inverse transform gives the solution of the fractional differential equation at hands.     

Application of high order numerical quadratures to numerical inversion of the Laplace transform

In this paper, new algorithms are proposed for Fredholm integral equations of the first kind corresponding to the inverse Laplace transform. We apply high order numerical quadratures to the truncated integral equation and apply regularization to the discretized linear systems. The resulted regularized least square problems are then solved by the reduced QR factorization method. Several examples taken from the literature are tested. Numerical results show that the approximate inverse Laplace transform obtained by our approach can be very accurate.     

Comments on “Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method”

Tari et al. [A. Tari, M.Y. Rahimi, S. Shahmorad, F. Talati, Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method, J. Comput. Appl. Math. 228 (2009) 70–76], presented some fundamental properties of TDTM for the kernel functions in two-dimensional Volterra integral equations. Here, we suggest simple proofs of those fundamental properties by using the basic properties of TDTM. Furthermore, we present some fundamental properties of TDTM for the kernel functions of a quotient type in two-dimensional Volterra integral equations. Numerical illustrations are demonstrated to show the effectiveness of the TDTM for solving two-dimensional Volterra integral equations.     

L-functions for Jacobi forms à la Hecke

Hecke's method to associate anL-function to a modular form by a Mellin transform is applied here to Jacobi forms. One comes up with a functional equation and some connection to Shimura's theory for modular forms of half integral weight.