scholarly journals In–out intermittency in partial differential equation and ordinary differential equation models

2001 ◽  
Vol 11 (2) ◽  
pp. 404-409 ◽  
Author(s):  
Eurico Covas ◽  
Reza Tavakol ◽  
Peter Ashwin ◽  
Andrew Tworkowski ◽  
John M. Brooke
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential ◽  
Differential Equation Models

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.

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

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.

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