Integral Representations For Solutions to Some Differential Equations That Arise in Wave Theory

1984 ◽  
pp. 391-408 ◽  
Author(s):  
Robert F. Millar
Keyword(s):  
Differential Equations ◽  
Wave Theory ◽  
Integral Representations

scholarly journals On Existence Theorems to Symmetric Functional Set-Valued Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1219
Author(s):  
Marek T. Malinowski
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Existence Theorems ◽  
Type Theorem ◽  
Integral Representations ◽  
Uniqueness Of The Solution ◽  
Nonempty Compact ◽  
Picard Type Theorem ◽  
Convex Subsets

In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is provided as well. We show the closeness of the unique solutions when the equations differ slightly.

The Wave Theory of the Electron

1928 ◽  
Vol 24 (4) ◽  
pp. 501-505 ◽  
Author(s):  
J. M. Whittaker
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Commutative Algebra ◽  
Wave Theory ◽  
Wave Functions ◽  
Linear Differential Equations ◽  
Wave Mechanics ◽  
First Order ◽  
Linear Differential

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.

scholarly journals The monodromy representation and twisted period relations for Appell’s hypergeometric function F 4

2015 ◽  
Vol 217 ◽  
pp. 61-94
Author(s):  
Yoshiaki Goto ◽  
Keiji Matsumoto
Keyword(s):  
Differential Equations ◽  
Hypergeometric Function ◽  
Hypergeometric Series ◽  
Integral Representations ◽  
Monodromy Representation ◽  
Homology Groups ◽  
Linearly Independent ◽  
Fundamental Systems

AbstractWe consider the systemF4(a, b, c)of differential equations annihilating Appell's hypergeometric seriesF4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric seriesF4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation ofF4(a, b, c)and the twisted period relations for the fundamental systems of solutions ofF4.

A Class of Generalized Hypergeometric Functions in Several Variables

1992 ◽  
Vol 44 (6) ◽  
pp. 1317-1338 ◽  
Author(s):  
Zhimin Yan
Keyword(s):  
Asymptotic Behavior ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Unique Solution ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Gaussian Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Several Variables

AbstractWe study a class of generalized hypergeometric functions in several variables introduced by A. Korânyi. It is shown that the generalized Gaussian hypergeometric function is the unique solution of a system partial differential equations. Analogues of some classical results such as Kummer relations and Euler integral representations are established. Asymptotic behavior of generalized hypergeometric functions is obtained which includes some known estimates.

scholarly journals SIMULATION OF OSCILLATORY PROCESSES USING DIFFERENTIAL EQUATIONS OF LIOUVILLE TYPE

2018 ◽  
pp. 117-123
Author(s):  
Albina Kuandykovna Ilyasova ◽  
Yuliia Vladimirovna Bulycheva
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Differential Equations ◽  
Explicit Form ◽  
Physical Phenomenon ◽  
Integral Representations ◽  
Explicit Solutions ◽  
Cauchy Type ◽  
Partial Derivatives ◽  
Oscillatory Processes

The problems of mathematical modeling lead to the necessity to create computational algorithms directly related to finding solutions of differential equations with partial derivatives in explicit form. In this study, explicit solutions are original tests for approximate methods that reflect the essence of the general solution. Each explicit solution of the differential equation has great importance as an accurate representation of the physical phenomenon under study within the framework of this model, as an analysis of the verification of numerical methods, as a theoretical basis for further modeling of the researched process. There have been considered aspects of the application of mathematical modeling to the study of oscillatory processes. Methods of reducing the solution of differential equations to an explicit form are proposed. Solution is given through functions of real arguments. The possible field of application is the study of wave processes. There is being considered the problem of building a variety of explicit solutions of the nonlinear third-order differential equation with partial derivatives with two boundary singular planes in space and second-order equation of general form with hyper-singular lines in the plane. On the basis of the developed method there has been proved the uniqueness of the obtained integral representations, and the boundary value problem of Cauchy type is posed and solved. The results are formulated in the form of theorems.

A hybrid analytical–numerical method for solving evolution partial differential equations. I. The half-line

2008 ◽  
Vol 464 (2095) ◽  
pp. 1823-1849 ◽  
Author(s):  
N Flyer ◽  
A.S Fokas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Plane ◽  
Complex Analysis ◽  
Variable Coefficients ◽  
Integral Representations ◽  
New Method ◽  
Half Line ◽  
Time Step ◽  
Partial Differential

A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.

The Stokes phenomenon and certain nth-order differential equations II. The Stokes phenomenon

1957 ◽  
Vol 53 (2) ◽  
pp. 419-441 ◽  
Author(s):  
J. Heading
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Integral Representations ◽  
Rational Fraction ◽  
Series Solutions ◽  
Stokes Phenomenon ◽  
Constant Power ◽  
Asymptotic Expressions ◽  
Power Series Solutions ◽  
Image Position

1. Introduction. In paper I(2), solutions have been found for the nth-order differential equations (I, 3), (I, 9), (I, 18), namely,where ϑ = wd/dw, m is any rational fraction and a any constant. Power-series solutions have been found for these equations, together with integral representations and their asymptotic expressions valid for restricted ranges of arg z. The object of this second paper is to consider these asymptotic solutions in more detail, and to extend these expressions to all values of arg z. The Stokes phenomenon will be manifest throughout, and this will be treated in a manner suitable for further application.

scholarly journals Of one over determined system differential equations at private derivative second order with one singular lines in general case

2021 ◽  
pp. 10-15
Author(s):  
Boitura Shoimkulov ◽  
◽  
Р. М. С. Lukmon ◽  
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Compatibility Condition ◽  
Second Order ◽  
Integral Representations ◽  
Singular Line ◽  
Cauchy Type ◽  
Partial Differential

In this paper, an over determined system of second-order partial differential equations with a single singular line in the General case is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with a single singular line in the General case. If the compatibility condition is met, integral representations of the variety of solutions are found explicitly in terms of three arbitrary constants, when the singular line is in the boundaries of the domain for which initial data problems (Cauchy-type Problems) can be set.

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