scholarly journals MODELLING FOOD PREFERENCES AND VIABILITY CONSTRAINTS

1995 ◽  
Vol 03 (02) ◽  
pp. 313-322
Author(s):  
ALOIS KASTNER-MARESCH ◽  
VLASTIMIL KŘIVAN
Keyword(s):  
Differential Equations ◽  
Population Biology ◽  
Differential Inclusions ◽  
Food Preferences ◽  
Limited Resource ◽  
Numerical Approach ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Two Populations ◽  
Competing Populations

Differential inclusions arising in population biology are analysed and a numerical approach to solve the corresponding initial value problems is proposed. These differential inclusions originate from optimal myopic strategies and from projecting differential equations onto a viability set. We show that such differential inclusions may piecewise be considered either as ordinary differential equations or as differential-algebraic equations with Index 2. The numerical method is based on this observation. Simulations for one population of predators feeding on two populations of prey and for a system of two competing populations whose growth is constrained by a limited resource are given.

Numerical methods for differential algebraic equations

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 141-198 ◽  
Author(s):  
Roswitha März
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Image Position

Differential algebraic equations (DAE) are special implicit ordinary differential equations (ODE)where the partial Jacobian f′y(y, x, t) is singular for all values of its arguments.

scholarly journals Second-Order Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yongtao Xuan ◽  
Rohul Amin ◽  
Fakhar Zaman ◽  
Zohaib Khan ◽  
Imad Ullah ◽  
...  
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Approach ◽  
Haar Wavelet ◽  
Second Order ◽  
Algebraic Equations ◽  
Elimination Algorithm ◽  
Collocation Points ◽  
Delay Differential ◽  
Mean Square Errors

In this article, an efficient numerical approach for the solution of second-order delay differential equations to deal with the experimentation of the Internet of Industrial Things (IIoT) is presented. With the help of the Haar wavelet technique, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for various collocation points. The results show that the Haar wavelet method is an effective method for solving delay differential equations of second order. The convergence rate is also measured for various collocation points, which is almost equal to 2.

scholarly journals Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Qingxue Huang ◽  
Fuqiang Zhao ◽  
Jiaquan Xie ◽  
Lifeng Ma ◽  
Jianmei Wang ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Principal Characteristic ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1

2012 ◽  
Vol 12 (04) ◽  
pp. 1250002 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
NGUYEN THI THE
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lyapunov Exponents ◽  
Algebraic Equation ◽  
Differential Algebraic Equations ◽  
Lyapunov Spectrum ◽  
Stochastic Flow ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Two Parameter

We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

Simple Recipe for Accurate Solution of Fractional Order Equations

2012 ◽  
Vol 8 (3) ◽  
Author(s):  
Sambit Das ◽  
Anindya Chatterjee
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Initial Conditions ◽  
Special Functions ◽  
Differential Algebraic Equations ◽  
Accurate Solution ◽  
Algebraic Equations ◽  
Matlab Program ◽  
Fractional Differential

Fractional order integrodifferential equations cannot be directly solved like ordinary differential equations. Numerical methods for such equations have additional algorithmic complexities. We present a particularly simple recipe for solving such equations using a Galerkin scheme developed in prior work. In particular, matrices needed for that method have here been precisely evaluated in closed form using special functions, and a small Matlab program is provided for the same. For equations where the highest order of the derivative is fractional, differential algebraic equations arise; however, it is demonstrated that there is a simple regularization scheme that works for these systems, such that accurate solutions can be easily obtained using standard solvers for stiff differential equations. Finally, the role of nonzero initial conditions is discussed in the context of the present approximation method.

Waveform relaxation method for fractional differential-algebraic equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Xiao-Li Ding ◽  
Yao-Lin Jiang
Keyword(s):  
Integrated Circuits ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Relaxation Method ◽  
Differential Algebraic Equations ◽  
Iteration Scheme ◽  
Waveform Relaxation ◽  
Algebraic Equations ◽  
Waveform Relaxation Method ◽  
Fractional Differential

AbstractThe waveform relaxation method has been successfully applied into solving fractional ordinary differential equations and fractional functional differential equations [11, 5]. In this paper, the waveform relaxation method is further used to solve fractional differential-algebraic equations, which often arise in integrated circuits with new memory materials. We give the iteration scheme of the waveform relaxation method and analyze the convergence of the method under linear and nonlinear conditions for the right-hand of the equations. Numerical examples illustrate the feasibility and efficiency of the method.

scholarly journals Maximal Regularity for Non-autonomous Evolutionary Equations

2021 ◽  
Vol 93 (3) ◽  
Author(s):  
Sascha Trostorff ◽  
Marcus Waurick
Keyword(s):  
Differential Equations ◽  
Maximal Regularity ◽  
Normal Operator ◽  
Time Derivative ◽  
Parabolic Type ◽  
Regularity Result ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Evolutionary Equations

AbstractWe discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

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