A New Operator Theory of Linear Partial Differential Equations

“…When acting on elementary functions, the abstract operators cos(h∂ x ) and sin(h∂ x ) have complete basic formulas as differential operations, now we are going to use the algorithms of abstract operators in reference [1] or [4] to establish these formulas.…”

Section: Basic Formulas Of Abstract Operatorsmentioning

confidence: 99%

“…According to the following theorem: Theorem 1. [4] If y = f (bx) ∈ J (set of analytic functions) is the inverse function of bx = g(y), namely g(f (bx)) = bx, then sin(h∂ x )f (bx)(denoted by Y ) and cos(h∂ x )f (bx)(denoted by X) can be determined by the following set of equations:…”

Section: Basic Formulas Of Abstract Operatorsmentioning

confidence: 99%

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New Properties of Fourier Series and Riemann Zeta Function

Bi,

2010

Preprint

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We establish the mapping relations between analytic functions and periodic functions using the abstract operators cos(h∂x) and sin(h∂x), including the mapping relations between power series and trigonometric series, and by using such mapping relations we obtain a general method to find the sum function of a trigonometric series. According to this method, if each coefficient of a power series is respectively equal to that of a trigonometric series, then if we know the sum function of the power series, we can obtain that of the trigonometric series, and the non-analytical points of which are also determined at the same time, thus we obtain a general method to find the sum of the Dirichlet series of integer variables, and derive several new properties of ζ(2n + 1).

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“…Thus, it leaves us a matter of mathematical techniques. Therefore, we introduce the mathematical method in reference [1]- [4]. Power functions and exponential functions play special roles in this method, which are called the base functions as we can establish mapping relations between them and arbitrary functions in a certain range.…”

Section: Significance Of Their Stationary Solutionsmentioning

confidence: 99%

Stationary Solutions of the Klein-Gordon Equation in a Potential Field

Bi,

2010

Preprint

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We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r, t) = ψ(r)e −iEt/h , and define m = E/c 2 , which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schrödinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.

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A New Operator Theory of Linear Partial Differential Equations

Cited by 3 publications

References 1 publication

The Cauchy problem for higher-order linear partial differential equation

The Cauchy problem for higher-order linear partial differential equation

New Properties of Fourier Series and Riemann Zeta Function

Stationary Solutions of the Klein-Gordon Equation in a Potential Field

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