scholarly journals Solution of Some Types of Differential Equations: Operational Calculus and Inverse Differential Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
K. Zhukovsky
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Laguerre Polynomial ◽  
Operational Calculus ◽  
Integral Transforms ◽  
Operational Technique ◽  
Mathematical Problems ◽  
Equations Of Mathematical Physics ◽  
Transport Problems ◽  
General Method

We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.

scholarly journals Two-scale homogenization of abstract linear time-dependent PDEs

2020 ◽  
pp. 1-41
Author(s):  
Stefan Neukamm ◽  
Mario Varga ◽  
Marcus Waurick
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Linear Time ◽  
Random Coefficients ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Equations Of Mathematical Physics

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

From the hyper-Bessel operators of Dimovski to the generalized fractional calculus

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Differential Operators ◽  
Integral Transform ◽  
Special Functions ◽  
Operational Calculus ◽  
Integral Transforms ◽  
Bessel Operators ◽  
Generalized Fractional Calculus ◽  
Basic Facts ◽  
Fox’S H Function

AbstractIn 1966 Ivan Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1. He also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey who called them hyper-Bessel differential operators, in relation to the notion of hyper-Bessel functions of Delerue (1953), shown to form a fundamental system of solutions of the IVPs for By(t) = λy(t). We have been able to extend Dimovski’s results on the hyper-Bessel operators and on the Obrechkoff transform due to the happy hint to attract the tools of the special functions as Meijer’s G-function and Fox’s H-function to handle successfully these matters. These author’s studies have lead to the introduction and development of a theory of generalized fractional calculus (GFC) in her monograph (1994) and subsequent papers, and to various applications of this GFC in other topics of analysis, differential equations, special functions and integral transforms.Here we try briefly to expose the ideas leading to this GFC, its basic facts and some of the mentioned applications.

scholarly journals On Professor Whittaker's solution of differential equations by definite integrals: Part I

1931 ◽  
Vol 2 (4) ◽  
pp. 205-219 ◽  
Author(s):  
W. O. Kermack ◽  
W. H. McCrea
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Preceding Paper ◽  
Contact Transformation ◽  
Partial Differential ◽  
Definite Integrals ◽  
General Method

In the preceding paper Professor Whittaker has given a general method for the solution of differential equations by means of definite integrals. It depends on finding a solution χ (q, Q) of an auxiliary pair of simultaneous partial differential equations to be derived from an arbitrary contact transformation by changing the momentum variables into differential operators. The first object of the present paper is to arrive at a method for passing from the contact transformation in its algebraic form to these partial differential equations, in a manner which is unambiguous and which makes them compatible. We show too how to obtain any number of such, pairs of equations from any given contact transformation. Successive transformations are also discussed.

scholarly journals Tides in oceans bounded by meridians

1936 ◽  
Vol 235 (753) ◽  
pp. 273-342 ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Infinite Sequence ◽  
Double Sequence ◽  
Algebraic Equations ◽  
Dynamical Equations ◽  
Mathematical Problems ◽  
Single Sequence ◽  
General Method

This is the first part of a series of papers by A. T. Doodson and myself, in which we intend to publish certain investigations which we have been carrying out intermittently for some years. In 1916 I published an account* of a general method of treating the dynamical equations of the tides in which the ordinary differential equations were transformed into an infinite sequence of algebraic equations. One of the chief features of the treatment is that an attempt was made to deal rigorously with questions of convergence. At that time the determination of the tides in a flat rectangular sea, a flat sectorial sea, and an ocean bounded by two meridians constituted mathematical problems which were completely unsolved, and I pointed out that for basins of these shapes and of uniform depth the coefficients in my algebraic equations could easily be evaluated. It is a disadvantage of the method, however, as applied to these systems, that the algebraic equations are naturally arranged in a double sequence and not in a single sequence.

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

A New Numerical Method for Solving Nonlinear Fractional Fokker–Planck Differential Equations

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
BeiBei Guo ◽  
Wei Jiang ◽  
ChiPing Zhang
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Simple Algorithm ◽  
High Accuracy ◽  
Computational Work ◽  
Reproducing Kernel Theory ◽  
Transport Problems ◽  
Fokker Planck

The nonlinear fractional-order Fokker–Planck differential equations have been used in many physical transport problems which take place under the influence of an external force filed. Therefore, high-accuracy numerical solutions are always needed. In this article, reproducing kernel theory is used to solve a class of nonlinear fractional Fokker–Planck differential equations. The main characteristic of this approach is that it induces a simple algorithm to get the approximate solution of the equation. At the same time, an effective method for obtaining the approximate solution is established. In addition, some numerical examples are given to demonstrate that our method has lesser computational work and higher precision.

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