scholarly journals Lie Symmetry and Lie Bracket in Solving Differential Equation Models of Functional Materials: A Survey

2020 ◽  
Vol 20 (2) ◽  
Author(s):  
Edi Kurniadi ◽  
Asep Supriatna
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Materials ◽  
Lie Symmetry ◽  
Complex Behavior ◽  
Mathematical Methods ◽  
New Approach ◽  
Lie Bracket ◽  
Differential Equation Models ◽  
Alternative Approach

Functional materials are becoming an increasingly important part of our daily life, e.g. they used for sensing, actuation, computing, energy conversion. These materials often have  unique physical, chemical, and structural characteristic involving very complex phase.  Many mathematical model have been devised to study the complex behavior of functional materials. Some of the models have been proven powerful in predicting the behavior of new materials built upon the composites of existing materials. One of  mathematical methods used to model the behavior of the materials is the differential equation. Very often the resulting differential equations are very complicated so that most methods failed in obtaining the exact solutions of the problems. Fortunately, a relatively new approach via Lie symmetry gives a new hope in obtaining or at least understanding the behavior of the solutions, which is needed to understand the behavior of the materials being modeled. In this paper we present a survey on the use of Lie symmetry and related concepts (such as  Lie algebra, Lie group, etc) in modeling the behavior of functional materials and discuss some fundamental results of the Lie symmetry theory which often used in solving differential equations. The survey shows that the use of Lie symmetry and alike have been accepted in many field and gives an alternative approach in studying the complex behavior of functional materials.

scholarly journals New Bi-modular Material Approach to Buckling Problem of Reinforced Concrete Columns

2016 ◽  
Vol 6 (1) ◽  
pp. 19 ◽  
Author(s):  
Ahmad Salah Edeen Nassef ◽  
Mohammed A. Dahim
Keyword(s):  
Differential Equation ◽  
Reinforced Concrete ◽  
Differential Equations ◽  
Concrete Columns ◽  
Neutral Axis ◽  
Reinforced Concrete Columns ◽  
New Approach ◽  
Codes Of Practice ◽  
Governing Differential Equation ◽  
Transverse Deflection

<p class="1Body">This paper was investigating the buckling problem of reinforced concrete columns considering the reinforced concrete as bi – modular material. Governing differential equations was driven. The relation between the non-dimensional transverse deflection and non-dimensional distance between centroid axis and the neutral axis "eccentricity" was drawn to enable the solution of the governing differential equation. The new approach was verified with different experimental results and different codes of practice.<strong></strong></p>

Adaptive Diversification and Speciation as Pattern Formation in Partial Differential Equation Models

2011 ◽  
Author(s):  
Michael Doebeli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Pattern Formation ◽  
Differential Equations ◽  
Adaptive Dynamics ◽  
Partial Differential ◽  
Adaptive Diversification ◽  
Differential Equation Models ◽  
Phenotype Distribution

This chapter discusses partial differential equation models. Partial differential equations can describe the dynamics of phenotype distributions of polymorphic populations, and they allow for a mathematically concise formulation from which some analytical insights can be obtained. It has been argued that because partial differential equations can describe polymorphic populations, results from such models are fundamentally different from those obtained using adaptive dynamics. In partial differential equation models, diversification manifests itself as pattern formation in phenotype distribution. More precisely, diversification occurs when phenotype distributions become multimodal, with the different modes corresponding to phenotypic clusters, or to species in sexual models. Such pattern formation occurs in partial differential equation models for competitive as well as for predator–prey interactions.

scholarly journals A New Approach to Approximate Solutions for Nonlinear Differential Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Safia Meftah
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Perturbation Method ◽  
Periodic Solutions ◽  
Asymptotic Expansions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Approximation Methods ◽  
New Approach

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

scholarly journals Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1483
Author(s):  
Alexander Churkin ◽  
Stephanie Lewkiewicz ◽  
Vladimir Reinharz ◽  
Harel Dahari ◽  
Danny Barash
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Viral Hepatitis ◽  
Antiviral Treatment ◽  
Boundary Values ◽  
Partial Differential ◽  
Differential Equation Models ◽  
Age Structured

Parameter estimation in mathematical models that are based on differential equations is known to be of fundamental importance. For sophisticated models such as age-structured models that simulate biological agents, parameter estimation that addresses all cases of data points available presents a formidable challenge and efficiency considerations need to be employed in order for the method to become practical. In the case of age-structured models of viral hepatitis dynamics under antiviral treatment that deal with partial differential equations, a fully numerical parameter estimation method was developed that does not require an analytical approximation of the solution to the multiscale model equations, avoiding the necessity to derive the long-term approximation for each model. However, the method is considerably slow because of precision problems in estimating derivatives with respect to the parameters near their boundary values, making it almost impractical for general use. In order to overcome this limitation, two steps have been taken that significantly reduce the running time by orders of magnitude and thereby lead to a practical method. First, constrained optimization is used, letting the user add constraints relating to the boundary values of each parameter before the method is executed. Second, optimization is performed by derivative-free methods, eliminating the need to evaluate expensive numerical derivative approximations. The newly efficient methods that were developed as a result of the above approach are described for hepatitis C virus kinetic models during antiviral therapy. Illustrations are provided using a user-friendly simulator that incorporates the efficient methods for both the ordinary and partial differential equation models.

scholarly journals New Approach for Solving Partial Differential Equations Based on Collocation Method

2020 ◽  
Vol 18 ◽  
pp. 118-128
Author(s):  
Alaa Almosawi ◽  
Luma N. M. Tawfiq
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Collocation Method ◽  
Linear Differential Operator ◽  
Partial Differential ◽  
Restrictive Assumption ◽  
New Approach ◽  
Linear Differential

In this paper, a new approach for solving partial differential equations was introduced. The collocation method based on LA-transform and proposed the solution as a power series that conforming Taylor series. The method attacks the problem in a direct way and in a straightforward fashion without using linearization, or any other restrictive assumption that may change the behavior of the equation under discussion. Five illustrated examples are introduced to clarifying the accuracy, ease implementation and efficiency of suggested method. The LA-transform was used to eliminate the linear differential operator in the differential equation.

scholarly journals Differential equation model of financial market stability based on Internet big data

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hongling Chen ◽  
Bahjat Fakieh ◽  
Bishr Muhamed Muwafak
Keyword(s):  
Differential Equation ◽  
Big Data ◽  
Differential Equations ◽  
Financial Market ◽  
Equation Model ◽  
Differential Equation Model ◽  
Nonlinear Characteristics ◽  
Differential Equation Models ◽  
Financial Market Stability ◽  
The Relationship

Abstract In the context of Internet big data, the market characteristics of the financial market can be used to feed back its stability with the help of differential equation models. China's financial market is roughly divided into three main markets: stocks, currency and foreign exchange. The interaction of the three has promoted the development of the financial market. With this as a background, the paper aims at these three financial markets and selects relevant indicators that can reflect the indications of the financial market to construct differential equations to analyse the relationship between the three. The paper uses the nonlinear characteristics of ordinary differential equations and related algorithms to solve the three types of market models. It uses an example to demonstrate that the differential equation model proposed in this paper can feed back the evolutionary characteristics of the three, and this model can help investors produce more correct investment decisions.

scholarly journals Massive computational acceleration by using neural networks to emulate mechanism-based biological models

2019 ◽  
Author(s):  
Shangying Wang ◽  
Kai Fan ◽  
Nan Luo ◽  
Yangxiaolu Cao ◽  
Feilun Wu ◽  
...  
Keyword(s):  
Neural Network ◽  
Differential Equation ◽  
Neural Networks ◽  
Differential Equations ◽  
System Dynamics ◽  
Mechanistic Model ◽  
Biological Models ◽  
Partial Differential ◽  
Parametric Space ◽  
Differential Equation Models

AbstractMechanism-based mathematical models are the foundation for diverse applications. It is often critical to explore the massive parametric space for each model. However, for many applications, such as agent-based models, partial differential equations, and stochastic differential equations, this practice can impose a prohibitive computational demand. To overcome this limitation, we present a fundamentally new framework to improve computational efficiency by orders of magnitude. The key concept is to train an artificial neural network using a limited number of simulations generated by a mechanistic model. This number is small enough such that the simulations can be completed in a short time frame but large enough to enable reliable training of the neural network. The trained neural network can then be used to explore the system dynamics of a much larger parametric space. We demonstrate this notion by training neural networks to predict self-organized pattern formation and stochastic gene expression. With this framework, we can predict not only the 1-D distribution in space (for partial differential equation models) and probability density function (for stochastic differential equation models) of variables of interest with high accuracy, but also novel system dynamics not present in the training sets. We further demonstrate that using an ensemble of neural networks enables the self-contained evaluation of the quality of each prediction. Our work can potentially be a platform for faster parametric space screening of biological models with user defined objectives.

Akbari-Ganji's Method

2017 ◽  
pp. 246-273
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Chemical Engineering ◽  
Solid Mechanics ◽  
Nonlinear Partial Differential Equation ◽  
The Other ◽  
Partial Differential ◽  
New Approach

The nonlinear partial differential equation governing on the mentioned system has been investigated by a simple and innovative method which we have named it Akbari-Ganji's Method or AGM. It is notable that this method has been compounded by Laplace transform theorem in order to covert the partial differential equation governing on the afore-mentioned system to an ODE and then the yielded equation has been solved conveniently by this new approach (AGM). One of the most important reasons of selecting the mentioned method for solving differential equations in a wide variety of fields not only in heat transfer science but also in different fields of study such as solid mechanics, fluid mechanics, chemical engineering, etc. in comparison with the other methods is as follows: Obviously, according to the order of differential equations, we need boundary conditions so in the case of the number of boundary conditions is less than the order of the differential equation, this method can create additional new boundary conditions in regard to the own differential equation and its derivatives. Therefore, a solution with high precision will be acquired. With regard to the afore-mentioned explanations, the process of solving nonlinear equation(s) will be very easy and convenient in comparison with the other methods.

scholarly journals Partial differential equation models in macroeconomics

2014 ◽  
Vol 372 (2028) ◽  
pp. 20130397 ◽  
Author(s):  
Yves Achdou ◽  
Francisco J. Buera ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions ◽  
Benjamin Moll
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Future Research ◽  
Partial Differential ◽  
Open Questions ◽  
Differential Equation Models

The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research.

scholarly journals A Backward Technique for Demographic Noise in Biological Ordinary Differential Equation Models

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1204
Author(s):  
Margherita Carletti ◽  
Malay Banerjee
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stochastic Systems ◽  
Intrinsic Noise ◽  
Extrinsic Noise ◽  
Differential Equation Models ◽  
Demographic Noise

Physical systems described by deterministic differential equations represent idealized situations since they ignore stochastic effects. In the context of biomathematical modeling, we distinguish between environmental or extrinsic noise and demographic or intrinsic noise, for which it is assumed that the variation over time is due to demographic variation of two or more interacting populations (birth, death, immigration, and emigration). The modeling and simulation of demographic noise as a stochastic process affecting units of populations involved in the model is well known in the literature, resulting in discrete stochastic systems or, when the population sizes are large, in continuous stochastic ordinary differential equations and, if noise is ignored, in continuous ordinary differential equation models. The inverse process, i.e., inferring the effects of demographic noise on a natural system described by a set of ordinary differential equations, is still an issue to be addressed. With this paper, we provide a technique to model and simulate demographic noise going backward from a deterministic continuous differential system to its underlying discrete stochastic process, based on the framework of chemical kinetics, since demographic noise is nothing but the biological or ecological counterpart of intrinsic noise in genetic regulation. Our method can, thus, be applied to ordinary differential systems describing any kind of phenomena when intrinsic noise is of interest.

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