Analytic solutions of Volterra equations via semigroups

2007 ◽  
Vol 76 (1) ◽  
pp. 142-148 ◽  
Author(s):  
Tomáš Bárta
Keyword(s):  
Analytic Solutions ◽  
Volterra Equations

scholarly journals A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Timon S. Gutleb
Keyword(s):  
Open Source ◽  
Spectral Method ◽  
Jacobi Polynomial ◽  
Numerical Experiments ◽  
Exponential Convergence ◽  
Analytic Solutions ◽  
Convolution Type ◽  
Volterra Equations ◽  
Polynomial Bases ◽  
Sparsity Structure

AbstractWe present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

ON THE ISSUE OF PREDICTING THE ADVANCEMENT OF UNIVERSITIES IN GLOBAL UNIVERSITY RANKINGS

2021 ◽  
Vol 1 (2) ◽  
pp. 15-28
Author(s):  
S. V. Blazhevich ◽  
◽  
V. M. Moskovkin ◽  
He Zhang ◽  
◽  
...  
Keyword(s):  
Population Dynamics ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Original System ◽  
Analytic Solutions ◽  
Volterra Equations ◽  
University Ranking ◽  
Simplified Approach ◽  
Integral Indicator

A simplified approach to solving the equations of population dynamics (Lotka–Volterra equations), which is a nonlinear multidimensional system of ordinary differential equations of the first order, describing the competitive interaction of universities included in some world university ranking, is proposed. The phase variables in these equations are the values of the integral indicator of the university ranking, which is called Overall or Total Score. The simplification consists in reducing this system to a system of independent Verhulst equations with analytic solutions in exponents of time and passing from them to stationary solutions when time tends to infinity. It is shown that with this approach and a given growth rate Overall (Total) Score, it is possible to find symmetric coefficients of interuniversity competition for no more than three competing universities. When finding such coefficients for the first three universities in the THE ranking, numerical solutions of the original system of population dynamics equations were obtained using the Runge–Kutta method in MatLab. It is shown that the development of this approach, based on the equations of population dynamics, can consist in turning to the concept of competitive – cooperative university interactions. The system of differential equations describes the process of changing the integral indicator during the period between two ratings. Using the found values of the coefficients of interuniversity competition, the system is solved sequentially for all stages of the ranking, and the decisions at the previous stage are used as the initial conditions for the next one.

scholarly journals Optimal Parametric Iteration Method for Solving Multispecies Lotka-Volterra Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Excellent Agreement ◽  
Analytical Method ◽  
Iteration Method ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Volterra Equations ◽  
New Approach ◽  
Strongly Nonlinear ◽  
Small Parameters

We apply an analytical method called the Optimal Parametric Iteration Method (OPIM) to multispecies Lotka-Volterra equations. By using initial values, accurate explicit analytic solutions have been derived. The method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement has been demonstrated between the obtained solutions and the numerical ones. This new approach, which can be easily applied to other strongly nonlinear problems, is very effective and yields very accurate results.

Modeling the co-evolution of economic institutions and the economy of resource producing regions

2020 ◽  
Vol 18 (4) ◽  
pp. 122-131
Author(s):  
Vadim F. Islamutdinov ◽  
Sergey P. Semenov
Keyword(s):  
Economic Growth ◽  
Resource Availability ◽  
Regional Economy ◽  
Cumulative Effect ◽  
Volterra Equations ◽  
Economic Institutions ◽  
Predator Prey Model ◽  
Resource Endowment ◽  
Disincentive Effect ◽  
Good Institution

The purpose of the study is to develop a model for the co-evolution of the regional economy and economic institutions. The research methods used: abstract-logical for the study of theoretical aspects and the experience of modeling co-evolution; and economic-mathematical for the development of own model of coevolution. The results of the study: approaches to modeling the evolution of economic institutions, as well as the co-evolution of the regional economy and economic institutions are considered, strengths and weaknesses of existing approaches to modeling co-evolution are identified, on the basis of the logistic model and Lotka-Volterra equations, an own co-evolution model has been developed, which includes three entities: regional economy, “good” institution and “bad” institution. Three versions of the model have been developed: the co-evolution of the regional economy and the “good” institution, the co-evolution of the regional economy and the “bad institution,” and a variant of the co-evolution of all three entities simultaneously, in which the “good” and “bad” institutions interact according to the “predator-prey” model, and their the cumulative effect determines the development of the regional economy. Numerical experiments have been carried out in the MathLab, which have shown the capabilities of the model to reflect the results of the co-evolution of the economy of a resource-producing region and economic institutions. In the first variant, a “good” institution promotes economic growth in excess of the level determined by resource availability. In the second variant, the “bad” institution has a disincentive effect on the GRP, as a result of which the GRP falls below the level determined by the resource endowment. In the third variant, the interaction of “good” and “bad” institutions still contributes to economic growth above the level determined by resource availability, but causes cyclical fluctuations in the GRP.

Analytic Solutions to the Closed Bomb

1988 ◽  
Author(s):  
Frederick W. Robbins ◽  
Franz R. Lynn
Keyword(s):  
Analytic Solutions
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