Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels

Bologna, Mauro

doi:10.1088/1751-8113/43/37/375203

Cited by 23 publications

(16 citation statements)

References 22 publications

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“…Polynomial coefficients in differential equations are widely studied in the literature and there are many works on this topic (here a non exhaustive list of references [19][20][21][22][23][24][25]). For the sake of brevity we limit ourselves to noting that in the case when the matrix elements are polynomials satisfying opportune requirements we can find a closed form expression for eq.…”

Section: Formal Solution For Linear N-th-order Differential Equationsmentioning

confidence: 99%

Exact analytical approach to differential equations with variable coefficients

Bologna

2016

Eur. Phys. J. Plus

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Abstract. This paper shows how to build an analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the analytical expression presented in the paper. The formalism can be easily extended to the infinite dimensional case such as the quantum time-dependent Hamiltonian problem and used for the eigenvalues problems.

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Section: Formal Solution For Linear N-th-order Differential Equationsmentioning

confidence: 99%

Exact analytical approach to differential equations with variable coefficients

Bologna

2016

Eur. Phys. J. Plus

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“…Following the procedure demonstrated in [1] withK 0 (0) = γ from (2.16), it can be shown that, for large μ and t,…”

Section: Solution Of (28)mentioning

confidence: 99%

Time Correlation Functions of Brownian Motion and Evaluation of Friction Coefficient in the Near-Brownian-Limit Regime

Kim¹,

Karniadakis²

2014

Multiscale Model. Simul.

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The exponentially decaying behavior of the momentum-momentum and momentumforce time correlation functions of Brownian motion at large times has been extensively used for the numerical evaluation of the friction coefficient γ from molecular dynamics simulations. We perform numerical analysis on these methods and address issues related to the appropriate choice of large time and the rate of convergence of these methods. To this end, we obtain asymptotic expansions of the time correlation functions with respect to the reduced mass μ of the Brownian particle. For two important limit procedures of achieving the Brownian limit, certain forms of the asymptotic expansion of the Mori memory function K(t) are introduced by physical arguments, and then the asymptotic expansions of the time correlation functions are expressed in terms of the limit of K(t), i.e., K 0 (t) = limμ→∞ K(t), and the next-order correction term K 1 (t). For the infinite mass limit case, we show that the numerical methods of estimating γ from the exponential decay of the time correlation functions produce γ + O(μ −1), where the first-order correction depends on the microscopic nature of K 0 (t) (i.e., deviation of K 0 (t) from the Dirac delta function γδ(t)) as well as the contribution of K 1 (t) (i.e., deviation of K(t) from K 0 (t)). We also analyze the ratio of the momentum-force correlation function to the momentum-momentum correlation function, which gives instantaneous exponential decay rate. For the thermodynamic limit case, we consider the Rayleigh gas system to investigate the finite-volume effect due to the boundary conditions and to demonstrate that the lowest-order terms of the asymptotic expansions may fail to describe some characteristic behavior of the time correlation functions. We perform systematic molecular dynamics simulations of the Rayleigh gas system to confirm the theoretical predictions.

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“…wherẽ(2 ) is the Laplace transform of ( ) evaluated in 2 and ( , , , ) is the hypergeometric function (for more details, see, e.g., [39]). Expressing the result in terms of ⟨ ( )⟩,…”

Section: Mean Value Of ( )mentioning

confidence: 99%

Effects of a Fluctuating Carrying Capacity on the Generalized Malthus-Verhulst Model

Calisto¹,

Chandía²,

Bologna³

2014

Advances in Mathematical Physics

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We consider a generalized Malthus-Verhulst model with a fluctuating carrying capacity and we study its effects on population growth. The carrying capacity fluctuations are described by a Poissonian process with an exponential correlation function. We will find an analytical expression for the average of a number of individuals and show that even in presence of a fluctuating carrying capacity the average tends asymptotically to a constant quantity.

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Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels

Cited by 23 publications

References 22 publications

Exact analytical approach to differential equations with variable coefficients

Exact analytical approach to differential equations with variable coefficients

Time Correlation Functions of Brownian Motion and Evaluation of Friction Coefficient in the Near-Brownian-Limit Regime

Effects of a Fluctuating Carrying Capacity on the Generalized Malthus-Verhulst Model

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