The Difference-Differential Equation F′(x) = eαx + β F(x−1) II

1953 ◽  
Vol 56 ◽  
pp. 459-464 ◽  
Author(s):  
N.G. de Bruijn
2012 ◽  
Vol 517 ◽  
pp. 797-800
Author(s):  
Zhi Yong Yang ◽  
Shun Hu Liu ◽  
Song Zhao ◽  
Jun Hu ◽  
Zeng Chan Lu

The difference existed between results of silos pressure calculation and the actual case, because the influence of density stratification was not taken into consideration. The aim of this paper was to obtain silo pressure calculating formula by consider of storage materials density stratified. To this end, we assume that the density was continuous changed along the height and differential equation of the storage materials pressure was established. By compared the results calculated from the equation with the results calculated from the code, it is found that the maximum pressure increased. The results showed density stratified is an import factor for silo pressure calculation and the equation obtained in this paper can provide references for the theory of silo pressure calculation.


1987 ◽  
Vol 101 (2) ◽  
pp. 323-342
Author(s):  
W. B. Jurkat ◽  
H. J. Zwiesler

In this article we investigate the meromorphic differential equation X′(z) = A(z) X(z), often abbreviated by [A], where A(z) is a matrix (all matrices we consider have dimensions 2 × 2) meromorphic at infinity, i.e. holomorphic in a punctured neighbourhood of infinity with at most a pole there. Moreover, X(z) denotes a fundamental solution matrix. Given a matrix T(z) which together with its inverse is meromorphic at infinity (a meromorphic transformation), then the function Y(z) = T−1(z) X(z) solves the differential equation [B] with B = T−1AT − T−1T [1,5]. This introduces an equivalence relation among meromorphic differential equations and leads to the question of finding a simple representative for each equivalence class, which, for example, is of importance for further function-theoretic examinations of the solutions. The first major achievement in this direction is marked by Birkhoff's reduction which shows that it is always possible to obtain an equivalent equation [B] where B(z) is holomorphic in ℂ ¬ {0} (throughout this article A ¬ B denotes the difference of these sets) with at most a singularity of the first kind at 0 [1, 2, 5, 6]. We call this the standard form. The question of how many further simplifications can be made will be answered in the framework of our reduction theory. For this purpose we introduce the notion of a normalized standard equation [A] (NSE) which is defined by the following conditions:(i) , where r ∈ ℕ and Ak are constant matrices, (notation: )(ii) A(z) has trace tr for some c ∈ ℂ,(iii) Ar−1 has different eigenvalues,(iv) the eigenvalues of A−1 are either incongruent modulo 1 or equal,(v) if A−1 = μI, then Ar−1 is diagonal,(vi) Ar−1 and A−1 are triangular in opposite ways,(vii) a12(z) is monic (leading coefficient equals 1) unless a12 ≡ 0; furthermore a21(z) is monic in case that a12 ≡ 0 but a21 ≢ 0.


Author(s):  
Michael Wibmer

Abstract We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.


2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Yunjiao Bai ◽  
Quan Zhang ◽  
Hong Shangguan ◽  
Zhiguo Gui ◽  
Yi Liu ◽  
...  

The traditional fourth-order nonlinear diffusion denoising model suffers the isolated speckles and the loss of fine details in the processed image. For this reason, a new fourth-order partial differential equation based on the patch similarity modulus and the difference curvature is proposed for image denoising. First, based on the intensity similarity of neighbor pixels, this paper presents a new edge indicator called patch similarity modulus, which is strongly robust to noise. Furthermore, the difference curvature which can effectively distinguish between edges and noise is incorporated into the denoising algorithm to determine the diffusion process by adaptively adjusting the size of the diffusion coefficient. The experimental results show that the proposed algorithm can not only preserve edges and texture details, but also avoid isolated speckles and staircase effect while filtering out noise. And the proposed algorithm has a better performance for the images with abundant details. Additionally, the subjective visual quality and objective evaluation index of the denoised image obtained by the proposed algorithm are higher than the ones from the related methods.


Geophysics ◽  
1970 ◽  
Vol 35 (1) ◽  
pp. 161-161
Author(s):  
M. K. Paul

I thank Mr. N. F. Uren for his interest in my paper. Mr. Uren’s discussion mainly concerns the appropriateness of the difference / differential equation that the regional gravity field, [Formula: see text], has been implied in my paper to satisfy. I think that this point has been discussed in considerable detail in my reply to an earlier discussion on this paper (Geophysics, June 1969, p. 483–485), which can, however, be summarized as follows.


2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskiĭ

It is well known the differential equation−u″(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the positive operatorAis ill-posed in the Banach spaceC(E)=C((−∞,∞),E)of the bounded continuous functionsϕ(t)defined on the whole real line with norm‖ϕ‖C(E)=sup⁡−∞<t<∞‖ϕ(t)‖E. In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions inC(τ,E)of these difference schemes is established.


1920 ◽  
Vol 39 ◽  
pp. 58-62 ◽  
Author(s):  
Bevan B. Baker

1. The Pincherle polynomials are defined as the coefficients in the expansion of {1 − 3 tx + t3}−½ in ascending powers of t. If the coefficient of tn be denoted by Pn(x), then the polynomials satisfy the difference equationand Pn(x) satisfies the differential equation


Author(s):  
K.B. Alkhan ◽  
◽  
Z.E. Shaimova ◽  

The article discusses examples of the differential equation problem in Python with a graph for high school students. The basic characteristics of some tools for solving problems in mathematics using information technology are given. The difference between the two modern-known computer programs Python and Pascal is briefly explained. The article uses illustrative examples of built-in and manually entered functions that can be repeated by readers in Python to recreate graphs of trigonometric, differential, and other functions. In conclusion, it describes how the program can save time and energy for solving graphic problems in mathematics using the Python program. The article concludes that it is important for students to use logic to solve problems with improvised tools, where there are built-in functions, compared to remembering programming language algorithms for solving problems.


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