scholarly journals Modeling Networks of Evolving Populations

2019 ◽  
Vol 16 (1) ◽  
pp. 49-64
Author(s):  
Sean Elliott
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Dynamic Behavior ◽  
Fitness Function ◽  
Scale Up ◽  
Equilibrium Solutions ◽  
Partial Differential ◽  
Phenotype Space ◽  
Eigen Model

The goal of this research is to devise a method of differential equation based modeling of evolution that can scale up to capture complex dynamics by enabling the inclusion of many—potentially thousands—of biological characteristics. Towards that goal, a mathematical model for evolution based on the well-established Fisher-Eigen process is built with a unique and efficient structure. The Fisher-Eigen partial differential equation (PDE) describes the evolution of a probability density function representing the distribution of a population over a phenotype space. This equation depends on the choice of a fitness function representing the likelihood of reproductive success at each point in the phenotype space. The Fisher-Eigen model has been studied analytically for simple fitness functions, but in general no analytic solution is known. Furthermore, with traditional numerical methods, the equation becomes exponentially complex to simulate as the dimensionality of the problem expands to include more phenotypes. For this research, a network model is synthesized and a set of ordinary differential equations (ODEs) is extracted based on the Fisher-Eigen PDE to describe the dynamic behavior of the system. It is demonstrated that, when juxtaposed with full numerical PDE simulations, this ODE model finds well-matched transient and precise equilibrium solutions. This prototype method makes modeling of high-dimensional data possible, allowing researchers to examine and even predict complex dynamic behavior based on a snapshot of a population. KEYWORDS: Evolutionary Modeling; Mathematical Biology; Network Dynamics; Ordinary Differential Equations; Partial Differential Equations; Fisher-Eigen model; Phenotype; Fitness Function

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

scholarly journals Galois theory of aigebraic and differential equations

1996 ◽  
Vol 144 ◽  
pp. 1-58 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Galois Theory ◽  
Linear Partial Differential Equation ◽  
Linear Algebraic Group ◽  
Partial Differential ◽  
Linear Ordinary Differential Equations ◽  
Infinite Dimensional ◽  
Linear Partial Differential Equations

This paper will be the first part of our works on differential Galois theory which we plan to write. Our goal is to establish a Galois Theory of ordinary differential equations. The theory is infinite dimensional by nature and has a long history. The pioneer of this field is S. Lie who tried to apply the idea of Abel and Galois to differential equations. Picard [P] realized Galois Theory of linear ordinary differential equations, which is called nowadays Picard-Vessiot Theory. Picard-Vessiot Theory is finite dimensional and the Galois group is a linear algebraic group. The first attempt of Galois theory of a general ordinary differential equations which is infinite dimensional, is done by the thesis of Drach [D]. He replaced an ordinary differential equation by a linear partial differential equation satisfied by the first integrals and looked for a Galois Theory of linear partial differential equations. It is widely admitted that the work of Drach is full of imcomplete definitions and gaps in proofs. In fact in a few months after Drach had got his degree, Vessiot was aware of the defects of Drach’s thesis. Vessiot took the matter serious and devoted all his life to make the Drach theory complete. Vessiot got the grand prix of the academy of Paris in Mathematics in 1903 by a series of articles.

scholarly journals Equivalent Lagrangians: Generalization, Transformation Maps, and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
N. Wilson ◽  
A. H. Kara
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Wave Equations ◽  
Symmetry Algebra ◽  
Lie Point Symmetries ◽  
Partial Differential

Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as theKummer equationand thecombined gravity-inertial-Rossbywave equationand certain classes of partial differential equations related to multidimensional wave equations.

Reduced Order Model of Nanoelectromechanical Systems to Include Casimir Effect

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Roberto J. Zapata ◽  
Martin W. Knecht
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Second Order ◽  
Reduced Order Model ◽  
Order Partial Differential Equation ◽  
Order Model ◽  
Partial Differential ◽  
Reduced Order

This paper deals with electrostatically actuated nanoelectromechanical (NEMS) cantilever resonators. The dynamic behavior is described by a second order partial differential equation. The NEMS cantilever resonator device is actuatedby an AC voltage resulting in a vibrating motion of the cantilever. At nano scale, squeeze film damping, Casimir force, and fringing effects significantly influence the dynamic behavior or the cantilever beam. The second order partial differential equation is solved using the Reduced Order Model (ROM) method. The resulting time dependent second order differential equations system is then transformed into a first order differential equations system. Numerical simulations were conducted using Matlab solver ode15s.

Numerical Reservoir Simulation Using an Ordinary-Differential-Equations Integrator

1975 ◽  
Vol 15 (03) ◽  
pp. 255-264 ◽  
Author(s):  
R.F. Sincovec
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Reservoir Simulation ◽  
Nonlinear Partial Differential Equations ◽  
Method Of Lines ◽  
Variable Order ◽  
Partial Differential

Abstract The method of lines used in conjunction with a sophisticated ordinary-differential-equations integrator is an effective approach for solving nonlinear, partial differential equations and is applicable to the equations describing fluid flow through porous media. Given the initial values, the integrator is self-starting. Subsequently, it automatically and reliably selects the time step and order, solves the nonlinear equations (checking for convergence, etc.), and maintains a user-specified time-integration accuracy, while attempting to complete the problems in a minimal amount of computer time. The advantages of this approach, such as stability, accuracy, reliability, and flexibility, are discussed. The method is applied to reservoir simulation, including high-rate and gas-percolation problems, and appears to be readily applicable to problems, and appears to be readily applicable to compositional models. Introduction The numerical solution of nonlinear, partial differential equations is usually a complicated and lengthy problem-dependent process. Generally, the solution of slightly different types of partial differential equations requires an entirely different computer program. This situation for partial differential equations is in direct contrast to that for ordinary differential equations. Recently, sophisticated and highly reliable computer programs for automatically solving complicated systems of ordinary differential equations have become available. These computer programs feature variable-order methods and automatic time-step and error control, and are capable of solving broad classes of ordinary differential equations. This paper discusses how these sophisticated ordinary-differential-equation integrators may be used to solve systems of nonlinear partial differential equations. partial differential equations.The basis for the technique is the method of lines. Given a system of time-dependent partial differential equations, the spatial variable(s) are discretized in some manner. This procedure yields an approximating system of ordinary differential equations that can be numerically integrated with one of the recently developed, robust ordinary-differential-equation integrators to obtain numerical approximations to the solution of the original partial differential equations. This approach is not new, but the advent of robust ordinary-differential-equation integrators has made the numerical method of lines a practical and efficient method of solving many difficult systems of partial differential equations. The approach can be viewed as a variable order in time, fixed order in space technique. Certain aspects of this approach are discussed and advantages over more conventional methods are indicated. Use of ordinary-differential-equation integrators for simplifying the heretofore rather complicated procedures for accurate numerical integration of systems of nonlinear, partial differential equations is described. This approach is capable of eliminating much of the duplicate programming effort usually associated with changing equations, boundary conditions, or discretization techniques. The approach can be used for reservoir simulation, and it appears that a compositional reservoir simulator can be developed with relative ease using this approach. In particular, it should be possible to add components to or delete components possible to add components to or delete components from the compositional code with only minor modifications. SPEJ P. 255

Practical Applications of Optimal-Control Theory to History-Matching Multiphase Simulator Models

1975 ◽  
Vol 15 (04) ◽  
pp. 347-355 ◽  
Author(s):  
M.L. Wasserman ◽  
A.S. Emanuel ◽  
J.H. Seinfeld
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
History Matching ◽  
Single Phase ◽  
Continuous Functions ◽  
Partial Differential

Abstract This paper applies material presented by Chen et al. and by Chavent et al to practical reservoir problems. The pressure history-matching algorithm used is initially based on a discretized single-phase reservoir model. Multiphase effects are approximately treated in the single-phase model by multiplying the transmissibility and storage terms by saturation-dependent terms that are obtained from a multiphase simulator run. Thus, all the history matching is performed by a "pseduo" single-phase model. The multiplicative factors for transmissibility and storage are updated when necessary. The matching technique can change any model permeability thickness or porosity thickness value. Three field examples are given. Introduction History matching using optimal-control theory was introduced by two sets of authors. Their contributions were a major breakthrough in attacking the long-standing goal of automatic history matching. This paper extends the work presented by Chen et al. and Chavent et al. Specifically, we focus on three areas.We derive the optimal-control algorithm using a discrete formulation. Our reservoir simulator, which is a set of ordinary differential equations, is adjoined to the function to be minimized. The first variation is taken to yield equations for computing Lagrange multipliers. These Lagrange multipliers are then used for computing a gradient vector. The discrete formulation keeps the adjoint equations consistent with the reservoir simulator.We include the effects of saturation change in history-matching pressures. We do this in a fashion that circumvents the need for developing a full multiphase optimal-control code.We show detailed results of the application of the optimal-control algorithm to three field examples. DERIVATION OF ADJOINT EQUATIONS Most implicit-pressure/explicit-saturation-type, finite-difference reservoir simulators perform two calculation stages for each time step. The first stage involves solving an "expansivity equation" for pressure. The expansivity equation is obtained by summing the material-balance equations for oil, gas, and water flow. Once the pressures are implicitly obtained from the expansivity equation, the phase saturations can be updated using their respective balance equations. A typical expansivity equation is shown in Appendix B, Eq. B-1. When we write the reservoir simulation equations as partial differential equations, we assume that the parameters to be estimated are continuous functions of position. The partial-differential-equation formulation is partial-differential-equation formulation is generally termed a distributed-parameter system. However, upon solving these partial differential equations, the model is discretized so that the partial differential equations are replaced by partial differential equations are replaced by sets of ordinary differential equations, and the parameters that were continuous functions of parameters that were continuous functions of position become specific values. Eq. B-1 is a position become specific values. Eq. B-1 is a set of ordinary differential equations that reflects lumping of parameters. Each cell has three associated parameters: a right-side permeability thickness, a bottom permeability thickness, and a pore volume. pore volume.Once the discretized model is written and we have one or more ordinary differential equations per cell, we can then adjoin these differential equations to the integral to be minimized by using one Lagrange multiplier per differential equation. The ordinary differential equations for the Lagrange multipliers are now derived as part of the necessary conditions for stationariness of the augmented objective function. These ordinary differential equations are termed the adjoint system of equations. A detailed example of the procedure discussed in this paragraph is given in Appendix A. The ordinary-differential-equation formulation of the optimal-control algorithm is more appropriate for use with reservoir simulators than the partial-differential-equation derivation found in partial-differential-equation derivation found in Refs. 1 and 2. SPEJ P. 347

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