Convergence Analysis of Legendre-Collocation Methods for Nonlinear Volterra Type Integro Equations

AbstractA Legendre-collocation method is proposed to solve the nonlinear Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicate that the numerical errors inL2-norm andL∞-norm will decay exponentially provided that the kernel function is sufficiently smooth. Numerical results are presented, which confirm the theoretical prediction of the exponential rate of convergence.

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Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

Advances in Mathematical Physics ◽

10.1155/2013/821327 ◽

2013 ◽

Vol 2013 ◽

pp. 1-14 ◽

Cited By ~ 20

Author(s):

Yin Yang ◽

Yunqing Huang

Keyword(s):

Approximate Solution ◽

Error Analysis ◽

Fractional Derivative ◽

Integrodifferential Equations ◽

Collocation Methods ◽

Volterra Type ◽

Numerical Examples ◽

Rigorous Error ◽

Spectral Collocation Methods ◽

Theoretical Results

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially inL∞norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.

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The exponential rate of convergence of the distribution of the maximum of a random walk. Part II

Journal of Applied Probability ◽

10.1017/s0021900200104395 ◽

1976 ◽

Vol 13 (04) ◽

pp. 733-740

Author(s):

N. Veraverbeke ◽

J. L. Teugels

Keyword(s):

Random Walk ◽

Rate Of Convergence ◽

Real Line ◽

Exponential Convergence ◽

Exponential Rate ◽

The Real ◽

Distribution Of The Maximum ◽

Exponential Rate Of Convergence ◽

Successive Maximum

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn (x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G ∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

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The exponential rate of convergence of the distribution of the maximum of a random walk. Part II

Journal of Applied Probability ◽

10.2307/3212528 ◽

1976 ◽

Vol 13 (4) ◽

pp. 733-740 ◽

Cited By ~ 16

Author(s):

N. Veraverbeke ◽

J. L. Teugels

Keyword(s):

Random Walk ◽

Rate Of Convergence ◽

Real Line ◽

Exponential Convergence ◽

Exponential Rate ◽

The Real ◽

Distribution Of The Maximum ◽

Exponential Rate Of Convergence ◽

Successive Maximum

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn(x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

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The exponential rate of convergence of the distribution of the maximum of a random walk

Journal of Applied Probability ◽

10.1017/s0021900200047963 ◽

1975 ◽

Vol 12 (02) ◽

pp. 279-288 ◽

Cited By ~ 7

Author(s):

N. Veraverbeke ◽

J. L. Teugels

Keyword(s):

Distribution Function ◽

Random Walk ◽

Rate Of Convergence ◽

Exponential Convergence ◽

Random Variables ◽

Limiting Distribution ◽

Partial Sums ◽

Exponential Rate ◽

Distribution Of The Maximum ◽

Exponential Rate Of Convergence

Let Gn (x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn (x) — G(x) is asymptotically equal to c.H(x)n −3/2 γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

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An Adaptive Observer with Exponential Rate of Convergence

Transactions of the Society of Instrument and Control Engineers ◽

10.9746/sicetr1965.18.343 ◽

1982 ◽

Vol 18 (4) ◽

pp. 343-348 ◽

Cited By ~ 2

Author(s):

Zenta IWAI ◽

Makoto SATO ◽

Akira INOUE ◽

Kazuo MANO

Keyword(s):

Rate Of Convergence ◽

Adaptive Observer ◽

Exponential Rate ◽

Exponential Rate Of Convergence

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Convergence Analysis of Legendre Pseudospectral Scheme for Solving Nonlinear Systems of Volterra Integral Equations

Advances in Mathematical Physics ◽

10.1155/2014/307907 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Emran Tohidi ◽

O. R. Navid Samadi ◽

S. Shateyi

Keyword(s):

Nonlinear Systems ◽

Theoretical Prediction ◽

Numerical Solution ◽

Integral Equations ◽

Volterra Integral Equations ◽

Volterra Equations ◽

Exponential Rate ◽

Exponential Rate Of Convergence ◽

Legendre Spectral Method ◽

Pseudospectral Scheme

We are concerned with the extension of a Legendre spectral method to the numerical solution of nonlinear systems of Volterra integral equations of the second kind. It is proved theoretically that the proposed method converges exponentially provided that the solution is sufficiently smooth. Also, three biological systems which are known as the systems of Lotka-Volterra equations are approximately solved by the presented method. Numerical results confirm the theoretical prediction of the exponential rate of convergence.

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