Operational calculus

Brychkov, Yu. A.; Prudnikov, A. P.; Shishov, V. S.

doi:10.1007/bf01377044

Cited by 10 publications

(6 citation statements)

References 160 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we consider a type of generalised convolution operation between two functions, which can be called Ψ‐convolution, and which ties together naturally with the Ψ‐Laplace transform. This generalised convolution was also studied by Jarad and Abdeljawad 19 and earlier by the Kyrgyz mathematicians referenced in Brychkov et al 36 Again, the operational calculus approach enables easy proofs for many properties of the Ψ‐convolution operation.…”

Section: The Generalised Laplace Transformmentioning

confidence: 79%

“…Let 𝑓 ∶ [0, ∞) → R be a real-valued function and Ψ be a nonnegative increasing function such that Ψ(0) = 0. 19,36 Then the Laplace transform of f with respect to Ψ is defined by…”

Section: The Generalised Laplace Transformmentioning

confidence: 99%

“…Let f and g be of Ψ-exponential order, piecewise continuous functions over each finite interval [0, T]. 19,36 Then, the Ψ-convolution of f and g is the function f * Ψ g defined by…”

Section: Combination With ψ-Convolutionmentioning

confidence: 99%

“…In this section, we discuss a generalised integral transform, studied by Jarad and Abdeljawad 19 but also previously introduced by Kyrgyz mathematicians in the 1960s as referenced in Brychkov et al, 36 which can be used to solve linear FDEs involving Ψ‐RL, Ψ‐C and Ψ‐Hilfer fractional derivatives. This integral transform is the natural generalisation of the classical Laplace transform for the setting of fractional operators with respect to functions.…”

Section: The Generalised Laplace Transformmentioning

confidence: 99%

“…Definition Let

f : false[0, \infty false) \to ℝ

be a real‐valued function and Ψ be a nonnegative increasing function such that

normalΨ false(0 false) = 0

19,36 . Then the Laplace transform of f with respect to Ψ is defined by

{scriptL}_{normalΨ} ({}, f false(t false)) = F false(s false) = \int_{0}^{\infty} e^{- s normalΨ false(t false)} {normalΨ}^{'} false(t false) f false(t false) normald t

for all

s \in ℂ

such that this integral converges.…”

Section: The Generalised Laplace Transformmentioning

confidence: 99%

See 4 more Smart Citations

On Laplace transforms with respect to functions and their applications to fractional differential equations

Fahad

Rehman

Fernandez

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called Ψ‐fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.

show abstract

Section: The Generalised Laplace Transformmentioning

confidence: 79%

Section: The Generalised Laplace Transformmentioning

confidence: 99%

“…Let f and g be of Ψ-exponential order, piecewise continuous functions over each finite interval [0, T]. 19,36 Then, the Ψ-convolution of f and g is the function f * Ψ g defined by…”

Section: Combination With ψ-Convolutionmentioning

confidence: 99%

Section: The Generalised Laplace Transformmentioning

confidence: 99%

“…Definition Let