scholarly journals Probabilistic initial value problem for cellular automaton rule 172

2010 ◽  
Vol DMTCS Proceedings vol. AL,... (Proceedings) ◽  
Author(s):  
Henryk Fuks
Keyword(s):  
Initial Value Problem ◽  
Level Sets ◽  
Topological Structure ◽  
Fibonacci Numbers ◽  
Initial Condition ◽  
Lucas Numbers ◽  
Initial Value ◽  
Time Step ◽  
International Audience ◽  
Combinatorial Methods

International audience We present a method of solving of the probabilistic initial value problem for cellular automata (CA) using CA rule 172 as an example. For a disordered initial condition on an infinite lattice, we derive exact expressions for the density of ones at arbitrary time step. In order to do this, we analyze topological structure of preimage trees of finite strings of length 3. Level sets of these trees can be enumerated directly using classical combinatorial methods, yielding expressions for the number of $n$-step preimages of all strings of length 3, and, subsequently, probabilities of occurrence of these strings in a configuration obtained from the initial one after $n$ iterations of rule 172. The density of ones can be expressed in terms of Fibonacci numbers, while expressions for probabilities of other strings involve Lucas numbers. Applicability of this method to other CA rules is briefly discussed.

scholarly journals Existence and uniqueness of analytic solutions of the Shabat equation

2005 ◽  
Vol 2005 (8) ◽  
pp. 855-862 ◽  
Author(s):  
Eugenia N. Petropoulou
Keyword(s):  
Upper Bound ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Initial Condition ◽  
Initial Value ◽  
First Derivative

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely forz∈ℂ:|z|<T,T>0. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, andT. Furthermore, from these conditions, one can obtain an upper bound forT. Our results are in consistence with some recently found results.

scholarly journals Soution of fuzzy initial value problems by fuzzy Laplace transform

10.29007/pnq2 ◽  
2018 ◽  
Author(s):  
Komal Patel ◽  
Narendrasinh Desai
Keyword(s):  
Laplace Transform ◽  
Initial Value Problem ◽  
Level Sets ◽  
Initial Value Problems ◽  
Initial Value ◽  
Generalized Differentiability ◽  
Strongly Generalized Differentiability ◽  
Fuzzy Laplace Transform ◽  
Mathematica Software

In this paper we propose a fuzzy Laplace transform to solve fuzzy initial value problem under strongly generalized differentiability concept. The fuzzy Laplace transform of derivative was used to solve Nth-order fuzzy initial value problem. To illustrate applicability of proposed method we plot graphs for different values of r -level sets by using Mathematica Software.

HOPF SOLUTIONS TO A FLUID-ELASTIC INTERACTION MODEL

2008 ◽  
Vol 18 (02) ◽  
pp. 215-269 ◽  
Author(s):  
M. GUIDORZI ◽  
M. PADULA ◽  
P. I. PLOTNIKOV
Keyword(s):  
Global Existence ◽  
Existence Theorem ◽  
Initial Value Problem ◽  
Interaction Model ◽  
Elastic Interaction ◽  
Initial Condition ◽  
Two Dimensional ◽  
Initial Value ◽  
Global Existence Theorem ◽  
Model Equations

In this paper, we give a global existence theorem of weak solutions to model equations governing interaction fluid structure in a two-dimensional layer, cf. Refs. 8 and 14. To our knowledge this is the first existence theorem of global in time solutions for such model. The interest of our result is double because, first, we change the original initial value problem by deleting one initial condition, second, we construct a solution through the classical Galerkin method for which several computing codes have been constructed.

scholarly journals The Picard-Lindel of’s theorem at a regular singular point

2013 ◽  
Vol 29 (2) ◽  
pp. 167-178
Author(s):  
CHELO FERREIRA ◽  
◽  
JOSE L. LOPEZ ◽  
ESTER PEREZ SINUSIA ◽  
◽  
...  
Keyword(s):  
Singular Point ◽  
Initial Value Problem ◽  
Lipschitz Constant ◽  
Regular Singular Point ◽  
Initial Condition ◽  
Taylor Polynomial ◽  
System Of Differential Equations ◽  
Initial Value ◽  
Lipschitz Continuous ◽  
Open Set

We consider initial value problems of the form..., where f : [−a, b] × U → Cn is a continuous function in its variables and U ⊂ Cn is an open set. D(x) is an n × n diagonal matrix whose first n − m diagonal entries are 1 and the last m diagonal entries are x, with m = 0, 1, 2, . . . or n. This is an initial value problem where the initial condition is given at a regular singular point of the system of differential equations. The main result of this paper is an existence and uniqueness theorem for the solution of this initial value problem. It is shown that this problem has a unique solution and the Picard-Lindelof’s expansion converges to that solution if the function ¨ F(y, x) := xD−1 (x)f (x, y) is Lipschitz continuous in the variables y with Lipschitz constant L of the form L = N + Mxp for a certain p > 0, M > 0 and 0 ≤ N < 1. When we add the condition y (s) ∈ C[−a, b], s ∈ N, to the formulation of the problem and the Taylor polynomial of y at x = 0 and degree s − 1 is available from the differential equation, then the same conclusion is true with a less restrictive condition upon N: 0 ≤ N < s + 1. The standard Picard-Lindelof’s ¨ theorem is the particular case of the problem studied here obtained for m = 0 (D(x) is the identity matrix), N = 0, p = 1 and M is the Lipschitz constant of f (x, y).

Asymptotic Speed of Propagation for Fisher-Type Degenerate Reaction-Diffusion-Convection Equations

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Luisa Malaguti ◽  
Stefano Ruggerini
Keyword(s):  
Initial Value Problem ◽  
Reaction Diffusion ◽  
Initial Condition ◽  
Initial Value ◽  
Convection Equation ◽  
All Solutions ◽  
Asymptotic Speed ◽  
Speed Of Propagation ◽  
Diffusion Convection

AbstractThe paper deals with the initial value problem for the degenerate reaction-diffusion-convection equationuwhere h is continuous, m > 1, and f is of Fisher-type. By means of comparison type techniques, we prove that the equilibrium u ≡ 1 is an attractor for all solutions with a continuous, bounded, non-negative initial condition u

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

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