scholarly journals B-spline Wavelet Method for Solving Fredholm Hammerstein Integral Equation Arising from Chemical Reactor Theory

2018 ◽  
Vol 7 (3) ◽  
pp. 163-169 ◽  
Author(s):  
P. K. Sahu ◽  
A. K. Ranjan ◽  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Integral Equation ◽  
Chemical Reactor ◽  
Steady State Solution ◽  
Algebraic Equations ◽  
Hammerstein Integral Equation ◽  
Wavelet Method ◽  
Spline Wavelet ◽  
B Spline

Abstract Mathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction has been considered. For steady state solution for an adiabatic tubular chemical reactor, the model can be reduced to ordinary differential equation with a parameter in the boundary conditions. Again the ordinary differential equation has been converted into a Hammerstein integral equation which can be solved numerically. B-spline wavelet method has been developed to approximate the solution of Hammerstein integral equation. This method reduces the integral equation to a system of algebraic equations. The numerical results obtained by the present method have been compared with the available results.

New wavelet method for solving boundary value problems arising from an adiabatic tubular chemical reactor theory

2020 ◽  
Vol 13 (07) ◽  
pp. 2050059
Author(s):  
Mohamed R. Ali ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Solutions ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Chemical Reactor ◽  
Steady State Solution ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Newton’S Iterative Method ◽  
Present Technique ◽  
Exothermic Chemical Reaction

This paper displays an efficient numerical technique of realizing mathematical models for an adiabatic tubular chemical reactor which forms an irreversible exothermic chemical reaction. At a steady-state solution for an adiabatic rounded reactor, the model can be diminished to a conventional nonlinear differential equation which converts into a system of the nonlinear equation that can proceed numerically utilizing Newton’s iterative method. An operational matrix of coordination is derived and is utilized to decrease the model for an adiabatic tubular chemical reactor to an arrangement of algebraic equations. Simple execution, basic activities, and precise arrangements are the fundamental highlights of the proposed wavelet technique. The numerical solutions attained by the present technique have been contrasted and compared with other techniques.

scholarly journals Initiation of thermal explosion by intense light: criticality dependence on data and parameters

1987 ◽  
Vol 28 (3) ◽  
pp. 287-295 ◽  
Author(s):  
K. K. Tam
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Steady State Solution ◽  
Linear Ordinary Differential Equation ◽  
Critical Parameters ◽  
Linear Parabolic Equation ◽  
Thermal Ignition ◽  
Non Linear ◽  
Intense Light

AbstractA model for thermal ignition by intense light is studied. The governing non-linear parabolic equation is linearized in a two-step manner with the aid of a non-linear ordinary differential equation which captures the salient features of the non-linear parabolic equation. The critical parameters are computed from the steady-state solution of the ordinary differential equation, which can be obtained without actually solving the equation. Comparison with available data shows that the present method yields good results.

scholarly journals A REPRESENTATIVE SOLUTION TO M-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION WITH NONLOCAL CONDITIONS BY GREEN'S FUNCTIONAL CONCEPT

2012 ◽  
Vol 17 (4) ◽  
pp. 571-588 ◽  
Author(s):  
Kemal Ozen ◽  
Kamil Orucoglu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlocal Conditions ◽  
Adjoint System ◽  
Linear Ordinary Differential Equation ◽  
The Other ◽  
Algebraic Equations ◽  
Order Ordinary Differential Equation ◽  
Multipoint Problem ◽  
Nonsmooth Coefficients

In this work, we investigate a linear completely nonhomogeneous nonlocal multipoint problem for an m-order ordinary differential equation with generally variable nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness. A system of m + 1 integro-algebraic equations called the special adjoint system is constructed for this problem. Green's functional is a solution of this special adjoint system. Its first component corresponds to Green's function for the problem. The other components correspond to the unit effects of the conditions. A solution to the problem is an integral representation which is based on using this new Green's functional. Some illustrative implementations and comparisons are provided with some known results in order to demonstrate the advantages of the proposed approach.

A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Suggested Approach ◽  
B Spline ◽  
Riccati Differential Equations ◽  
The Given

AbstractIn this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.

ON THE RELATION BETWEEN ORGANIZATIONS AND LIMIT SETS IN CHEMICAL REACTION SYSTEMS

2011 ◽  
Vol 14 (01) ◽  
pp. 77-96 ◽  
Author(s):  
STEPHAN PETER ◽  
PETER DITTRICH
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Organization Theory ◽  
Dynamical Behavior ◽  
Chemical Reactor ◽  
Deterministic System ◽  
Discrete Problem ◽  
Limit Sets ◽  
Long Term Behavior

Chemical organization theory has been suggested as a new approach to analyze complex reaction networks. Concerning the long-term behavior of the network dynamics we will study its foundations mathematically. Therefore we consider a chemical reactor containing molecules of different species reacting with each other according to a set of reaction rules. We further assume that the dynamical behavior of the concentration of each species is given by a continuous chemical ordinary differential equation. Abstracting from dedicated concentration values we consider a discrete problem: Which species can appear in the reactor after a long time? We define the limit set abstraction, which contains the subsets of species characterizing the long-term behavior. We prove that all these subsets are closed and that for all bounded limit sets, at least one of them is self-maintaining and thus is an organization. This implies for a chemical ordinary differential equation systems that any attractor that does not touch the state space boundaries (in particular, any periodic attractor) lies within one organization, that is, the set of species with positive concentrations in any state of such an attractor is an organization. This in turn explains why in a deterministic system evolving according to reaction rules one can observe species sets that are closed and self-maintaining, thus organizations.

Urn models and differential algebraic equations

2003 ◽  
Vol 40 (02) ◽  
pp. 401-412 ◽  
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Random Environment ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Urn Model ◽  
Recurrent Equation ◽  
Strong Laws ◽  
The Stability ◽  
Generalized Pólya Urn

The aim of this paper is to study the distribution of colours, { X n }, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process { X n } is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process { X n }.

Unusual extension–torsion–inflation couplings in pressurized thin circular tubes with helical anisotropy

2018 ◽  
Vol 24 (9) ◽  
pp. 2694-2712 ◽  
Author(s):  
Raushan Singh ◽  
Pranjal Singh ◽  
Ajeet Kumar
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Radial Displacement ◽  
Applied Pressure ◽  
Nonlinear Ordinary Differential Equation ◽  
Algebraic Equations ◽  
Circular Tubes ◽  
Thin Tube ◽  
Analytical Expressions ◽  
Helical Anisotropy

We present a thin tube formulation for coupled extension–torsion–inflation deformation in helically reinforced pressurized circular tubes. Both compressible and incompressible tubes are considered. On applying the thin tube limit, the nonlinear ordinary differential equation to obtain the in-plane radial displacement is converted into a set of two simple algebraic equations for the compressible case and one equation for the incompressible case. This allows us to obtain analytical expressions, in terms of the tube’s intrinsic twist, material constants, and the applied pressure, which can predict whether such tubes would overwind/unwind on being infinitesimally stretched or exhibit positive/negative Poisson’s effect. We further show numerically that such tubes can be tuned to generate initial overwinding followed by rapid unwinding as observed during finite stretching of a torsionally relaxed DNA. Finally, we demonstrate that such tubes can also exhibit usual deflation initially followed by unusual inflation as the tube is finitely stretched.

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