The g-analytic Function Theory and Wave Equation

Chein-Shan Liu

doi:10.5539/jmr.v7n3p85

Abstract

In this paper we develop a $g$-analytic function and a $g$-harmonic function theory for one-dimensional wave equation in the Minkowski space. In terms of the Minkowskian polar coordinates we can derive a set of complete hyperbolic type Trefftz bases, which can be transformed to polynomials as the bases for a trial solution of wave equation. The Cauchy-Riemann equations and the Cauchy theoremfor $g$-analytic functions are proved, and meanwhilethe existence of Cauchy integral formula is disproved from thenon-uniqueness of the Dirichlet problem for wave equation under the boundary conditions on whole boundary, which isalso known as the backward wave problem (BWP).Examples are used to demonstrate these results.

Highlights

The one-dimensional wave equation is given by ∂2u ∂t2 = c2 ∂2u ∂x2 (x, t) ∈ Ω, (1)where c is the speed of wave propagation, and the domain Ω may be bounded or unbounded
In this paper we develop a g-analytic function and a g-harmonic function theory for one-dimensional wave equation in the Minkowski space
The Cauchy-Riemann equations and the Cauchy theorem for g-analytic functions are proved, and the existence of Cauchy integral formula is disproved from the non-uniqueness of the Dirichlet problem for wave equation under the boundary conditions on whole boundary, which is known as the backward wave problem (BWP)

Summary

Introduction

Where c is the speed of wave propagation, and the domain Ω may be bounded or unbounded. X π, t απ, u(0, t) = u(π, t) = 0, 0 ≤ t ≤ απ, u(x, 0) = φ(x), u(x, απ) = φ(x), 0 ≤ x ≤ π, it is a classical ill-posed problem, which has been studied by Bourgin & Duffin (1939); John (1941); Fox & Pucci (1958); Dunninger & Zachmanoglou (1967); Abdul-Latif & Diaz (1971); Papi Frosali (1979); Levine & Vessella (1985); Vakhania (1994); Kabanikhin & Bektemesov (2012) They asserted that when α is a rational number the solution is not unique.

The Laplace Equation and Analytic Function

The Trefftz Bases

The g-Analytic Function and Cauchy-Riemann Equations

The Cauchy Theorem

Conclusions