On fractional matrix exponentials and their explicit calculation

Rodrigo, Marianito R.

doi:10.1016/j.jde.2016.06.023

Cited by 11 publications

(6 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The ML function with matrix arguments is just as valuable as its scalar version and it can be successfully employed in several applications: for the efficient and stable solution of systems of fractional differential equations (FDEs), to determine the solution of certain multiterm FDEs, in control theory and in other related fields (e.g., see [20,21,22,44,46,48,55]).…”

Section: Introductionmentioning

confidence: 99%

Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus

Garrappa

Popolizio

2018

J Sci Comput

View full text Add to dashboard Cite

The computation of the Mittag-Leffler (ML) function with matrix arguments, and some applications in fractional calculus, are discussed. In general the evaluation of a scalar function in matrix arguments may require the computation of derivatives of possible high order depending on the matrix spectrum. Regarding the ML function, the numerical computation of its derivatives of arbitrary order is a completely unexplored topic; in this paper we address this issue and three different methods are tailored and investigated. The methods are combined together with an original derivatives balancing technique in order to devise an algorithm capable of providing high accuracy. The conditioning of the evaluation of matrix ML functions is also studied. The numerical experiments presented in the paper show that the proposed algorithm provides high accuracy, very often close to the machine precision. , avialable at https://doi.org/ 10.1007/s10915-018-0699-5. This work is supported under the GNCS-INdAM 2017 project "Analisi numerica per modelli descritti da operatori frazionari". arXiv:1804.04883v2 [math.NA] 1 Dec 2019where |δ 1 |, |δ 2 |, . . . , |δ J | < and terms proportional to O( 2 ) have been discarded. It is thus immediate to derive the following bound for the round-off errorThe order by which the terms c j are summed is relevant especially for the reliability of the above estimator; clearly, an ascending sorting of |c j | makes the estimate (16) more conservative and hence more useful for practical use. It is therefore advisable, especially in the more compelling cases (namely, for arguments z with large modulus and | arg(z)| > απ/2) to perform a preliminary sorting of the terms in (14) with respect to their modulus.An alternative estimate of the round-off error can be obtained after reformulating (15) asŜ J = S J + J j=1 S j δ j from which one can easily derive S J −Ŝ J ≤ J j=1

show abstract

Section: Introductionmentioning

confidence: 99%

Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus

Garrappa

Popolizio

2018

J Sci Comput

View full text Add to dashboard Cite

show abstract

“…(d) Avoiding entirely the Jordan canonical form of A and only knowing σ(A), Putzer's method (see e.g., [9]) allows to explicitly obtain exp (t α A; α) in fully analytical form.…”

Section: Propertiesmentioning

confidence: 99%

“…In 2016, Rodrigo [9] introduced the fractional exponential matrix of Caputo of order α ≥ 0. Similarly, here we are using the following definition for 0 ≤ α ≤ 1:…”

Section: Introductionmentioning

confidence: 99%

On the Inverse of the Caputo Matrix Exponential

et al. 2019

View full text Add to dashboard Cite

Matrix exponentials are widely used to efficiently tackle systems of linear differential equations. To be able to solve systems of fractional differential equations, the Caputo matrix exponential of the index α > 0 was introduced. It generalizes and adapts the conventional matrix exponential to systems of fractional differential equations with constant coefficients. This paper analyzes the most significant properties of the Caputo matrix exponential, in particular those related to its inverse. Several numerical test examples are discussed throughout this exposition in order to outline our approach. Moreover, we demonstrate that the inverse of a Caputo matrix exponential in general is not another Caputo matrix exponential.Keywords: Caputo matrix exponential; matrix inverse; fractional derivative MSC: 15A09; 15A16 J α f (t) = Γ(α) t 0 (t − τ) α−1 f (τ) dτ, t > 0.

show abstract

“…We will let the Laplace transform indicate what the appropriate initial conditions should be. This is akin to the idea used by Rodrigo (2016Rodrigo ( , 2020 to determine the correct initial conditions for the "fractional" analogues of the matrix exponential.…”

Section: Introductionmentioning

confidence: 99%

A unified way to solve IVPs and IBVPs for the time-fractional diffusion-wave equation

Rodrigo¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The time-fractional diffusion-wave equation is revisited, where the time derivative is of order 2ν and 0 < ν ≤ 1. The behaviour of the equation is "diffusion-like" (respectively, "wave-like") when 0 < ν ≤ 1 2 (respectively, 1 2 < ν ≤ 1). Two types of time-fractional derivatives are considered, namely the Caputo and Riemann-Liouville derivatives. Initial value problems and initial-boundary value problems are investigated and handled in a unified way using an embedding method. A twoparameter auxiliary function is introduced and its properties are investigated. The time-fractional diffusion equation is used to generate a new family of probability distributions, and that includes the normal distribution as a particular case.

show abstract

On fractional matrix exponentials and their explicit calculation

Cited by 11 publications

References 10 publications

Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus

Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus

On the Inverse of the Caputo Matrix Exponential

A unified way to solve IVPs and IBVPs for the time-fractional diffusion-wave equation

Contact Info

Product

Resources

About