Analysis of the Time Fractional-Order Coupled Burgers Equations with Non-Singular Kernel Operators

In this article, we have investigated the fractional-order Burgers equation via Natural decomposition method with nonsingular kernel derivatives. The two types of fractional derivatives are used in the article of Caputo–Fabrizio and Atangana–Baleanu derivative. We employed Natural transform on fractional-order Burgers equation followed by inverse Natural transform, to achieve the result of the equations. To validate the method, we have considered a two examples and compared with the exact results.

Numerical Investigation of the Time-Fractional Whitham–Broer–Kaup Equation Involving without Singular Kernel Operators

This paper aims to implement an analytical method, known as the Laplace homotopy perturbation transform technique, for the result of fractional-order Whitham–Broer–Kaup equations. The technique is a mixture of the Laplace transformation and homotopy perturbation technique. Fractional derivatives with Mittag-Leffler and exponential laws in sense of Caputo are considered. Moreover, this paper aims to show the Whitham–Broer–Kaup equations with both derivatives to see their difference in a real-world problem. The efficiency of both operators is confirmed by the outcome of the actual results of the Whitham–Broer–Kaup equations. Some problems have been presented to compare the solutions achieved with both fractional-order derivatives.

FACTORIZING MULTILINEAR KERNEL OPERATORS THROUGH SPACES OF VECTOR MEASURES

We consider a multilinear kernel operator between Banach function spaces over finite measures and suitable order continuity properties, namely $T:X_{1}(\,\unicode[STIX]{x1D707}_{1})\times \cdots \times X_{n}(\,\unicode[STIX]{x1D707}_{n})\rightarrow Y(\,\unicode[STIX]{x1D707}_{0})$ . Then we define, via duality, a class of linear operators associated to the $j$ -transpose operators. We show that, under certain conditions of $p$ th power factorability of such operators, there exist vector measures $m_{j}$ for $j=0,1,\ldots ,n$ so that $T$ factors through a multilinear operator $\widetilde{T}:L^{p_{1}}(m_{1})\times \cdots \times L^{p_{n}}(m_{n})\rightarrow L^{p_{0}^{\prime }}(m_{0})^{\ast }$ , provided that $1/p_{0}=1/p_{1}+\cdots +1/p_{n}$ . We apply this scheme to the study of the class of multilinear Calderón–Zygmund operators and provide some concrete examples for the homogeneous polynomial and multilinear Volterra and Laplace operators.

Ideal structure and factorization properties of the regular kernel operators

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on $$L^p$$ L p -spaces. We show, in particular, that for any $$L^p(\mu )$$ L p ( μ ) space, with $$\mu $$ μ $$\sigma $$ σ -finite and $$1<p<\infty $$ 1 < p < ∞ , the norm-closure of the ideal of finite-rank operators on $$L^p(\mu )$$ L p ( μ ) , is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the $$L^p$$ L p setting is the fact that every regular kernel operator on an $$L^p(\mu )$$ L p ( μ ) space ($$\mu $$ μ and p as before) factors with regular factors through $$\ell _p$$ ℓ p . We show that a similar but weaker factorization property, where $$\ell _p$$ ℓ p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

Density-operator evolution: Complete positivity and the Keldysh real-time expansion

We study the reduced time-evolution of general open quantum systems by combining insights from quantum-information and statistical field theory. Inspired by prior work [Eur. Phys. Lett.~102, 60001 (2013) and Phys. Rev. Lett.~111, 050402 (2013)] we establish the explicit structure guaranteeing the complete positivity (CP) and trace-preservation (TP) of the real-time evolution expansion in terms of the microscopic system-environment coupling.This reveals a fundamental two-stage structure of the coupling expansion: Whereas the first stage naturally defines the dissipative timescales of the system -before having integrated out the environment completely- the second stage sums up elementary physical processes, each described by a CP superoperator. This allows us to establish the highly nontrivial functional relation between the (Nakajima-Zwanzig) memory-kernel superoperator for the reduced density operator and novel memory-kernel operators that generate the Kraus operators of an operator-sum. We illustrate the physically different roles of the two emerging coupling-expansion parameters for a simple solvable model. Importantly, this operational approach can be implemented in the existing Keldysh real-time technique and allows approximations for general time-nonlocal quantum master equations to be systematically compared and developed while keeping the CP and TP structure explicit.Our considerations build on the result that a Kraus operator for a physical measurement process on the environment can be obtained by `cutting' a group of Keldysh real-time diagrams `in half'. This naturally leads to Kraus operators lifted to the system plus environment which have a diagrammatic expansion in terms of time-nonlocal memory-kernel operators. These lifted Kraus operators obey coupled time-evolution equations which constitute an unraveling of the original Schroedinger equation for system plus environment. Whereas both equations lead to the same reduced dynamics, only the former explicitly encodes the operator-sum structure of the coupling expansion.