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.

scholarly journals Basic principles of hp virtual elements on quasiuniform meshes

2016 ◽  
Vol 26 (08) ◽  
pp. 1567-1598 ◽  
Author(s):  
L. Beir ao da Veiga ◽  
A. Chernov ◽  
L. Mascotto ◽  
A. Russo
Keyword(s):  
Numerical Experiments ◽  
Exponential Convergence ◽  
Mesh Size ◽  
Initial Study ◽  
Analytic Solutions ◽  
Basis Functions ◽  
Basic Principles ◽  
Polynomial Degree ◽  
Convergence Results ◽  
Convergence Estimates

In the present paper we initiate the study of [Formula: see text] Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size [Formula: see text] and in the polynomial degree [Formula: see text] in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included.

scholarly journals HOS-ocean: Open-source solver for nonlinear waves in open ocean based on High-Order Spectral method

2016 ◽  
Vol 203 ◽  
pp. 245-254 ◽  
Author(s):  
Guillaume Ducrozet ◽  
Félicien Bonnefoy ◽  
David Le Touzé ◽  
Pierre Ferrant
Keyword(s):  
Open Source ◽  
Spectral Method ◽  
Nonlinear Waves ◽  
Open Ocean ◽  
High Order ◽  
High Order Spectral Method

scholarly journals Numerical Solutions to Fractional Perturbed Volterra Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
B. Bandrowski ◽  
A. Karczewska ◽  
P. Rozmej
Keyword(s):  
Wave Equations ◽  
Numerical Solutions ◽  
Kernel Functions ◽  
Convolution Type ◽  
Volterra Equations

In the paper, a class of perturbed Volterra equations of convolution type with three kernel functions is considered. The kernel functions , , , correspond to the class of equations interpolating heat and wave equations. The results obtained generalize our previous results from 2010.

scholarly journals Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

2011 ◽  
Vol 43 (4) ◽  
pp. 1005-1026 ◽  
Author(s):  
Tuğrul Dayar ◽  
Werner Sandmann ◽  
David Spieler ◽  
Verena Wolf
Keyword(s):  
Chemical Kinetics ◽  
Numerical Experiments ◽  
Analytic Solutions ◽  
Lyapunov Theory ◽  
Stationary Probability ◽  
Stochastic Chemical Kinetics ◽  
Stationary Probability Distribution ◽  
Qbd Processes ◽  
Level Number ◽  
Probability Mass

Systems of stochastic chemical kinetics are modeled as infinite level-dependent quasi-birth-and-death (LDQBD) processes. For these systems, in contrast to many other applications, levels have an increasing number of states as the level number increases and the probability mass may reside arbitrarily far away from lower levels. Ideas from Lyapunov theory are combined with existing matrix-analytic formulations to obtain accurate approximations to the stationary probability distribution when the infinite LDQBD process is ergodic. Results of numerical experiments on a set of problems are provided.

Analytic solutions of Volterra equations via semigroups

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

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.

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