Solutions with concentration for conservation laws with discontinuous flux and its applications to numerical schemes for hyperbolic systems

Measure-valued weak solutions for conservation laws with discontinuous flux are proposed and explicit formulae have been derived. We propose convergent discontinuous flux-based numerical schemes for the class of hyperbolic systems that admit nonclassical -shocks, by extending the theory of discontinuous flux for nonlinear conservation laws to scalar transport equation with a discontinuous coefficient. The article also discusses the concentration phenomenon of solutions along the line of discontinuity, for scalar transport equations with a discontinuous coefficient. The existence of the solutions for transport equation is shown using the vanishing viscosity approach and the asymptotic behavior of the solutions is also established. The performance of the numerical schemes for both scalar conservation laws and systems to capture the -shocks effectively is displayed through various numerical experiments.     

On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law

The present article deals with a fractional extension of the vibration equation for very large membranes with distinct special cases. The fractional derivative is considered in Atangana-Baleanu sense. A numerical algorithm based on homotopic technique is employed to examine the fractional vibration equation. The stability analysis is conducted for the suggested scheme. The maple software package is utilized for numerical simulation. In order to illustrate the effects of space, time, and order of Atangana-Baleanu derivative on the displacement, the outcomes of this study are demonstrated graphically. The results revel that the Atangana-Baleanu fractional derivative is very efficient in describing vibrations in large membranes.     

On the impulsive implicit Ψ-Hilfer fractional differential equations with delay

In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam-Hyers-Mittag-Leffler stability results for impulsive implicit Ψ-Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam-Hyers and generalized Ulam-Hyers stability are the specific cases of Ulam-Hyers-Mittag-Leffler stability. Extended version of the Gronwall inequality, abstract Gronwall lemma, and Picard operator theory are the primary devices in our investigation. We provide an example to illustrate the obtained results.     

一致分数阶非线性微分方程的精确解

同时考虑了Kudryashov方法和Khalil一致分数阶变换,构造了求解一致分数阶非线性微分方程精确解的新方法,并将其用于求解时间-空间一致分数阶Whitham-Boroer-Kaup方程,得到了Whitham-Boroer-Kaup方程新的精确解,验证了该方法的有效性和可行性.     

Tensor method for optimal control problems constrained by fractional three-dimensional elliptic operator with variable coefficients

We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two- (2D) and three-dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D $n\times n\times n$ spacial grids. Using the diagonalization of the arising matrix-valued functions in the eigenbasis of the one-dimensional Sturm–Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right-hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low-rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an $\epsilon$-threshold. This method reduces the numerical cost for solving the control problem to $O(n\log n)$ (plus the quadratic term $O(n^2)$ with a small weight), which outperforms traditional approaches with $O(n^3\log n)$ complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by $O(n\log n)$. This essentially outperforms the traditional methods operating with fully populated $n^3\times n^3$ matrices and vectors in ${?}^{n^3}$. Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.     

Modeling and character analyzing of multiple fractional-order memcapacitors in parallel connection

Recently,the memory elements-based circuits have been addressed frequently in the nonlinear circuit theory due to their unique behaviors.Thus,the modeling and characterizing of the mem-elements become essential.In this paper,the analysis of the multiple fractional-order voltage-controlled memcapacitors model in parallel connection is studied.Firstly,two fractional-order memcapacitors are connected in parallel,the equivalent model is derived,and the characteristic of the equivalent memcapacitor is analyzed in positive or negative connection.Then a new understanding manner according to different rate factor K and fractional orderαis derived to explain the equivalent modeling structure conveniently.Additionally,the negative order appears,which is a consequence of the combination of memcapacitors in different directions.Meanwhile,the equivalent parallel memcapacitance has been drawn to determine that multiple fractional-order memcapacitors could be calculated as one composite memcapacitor.Thus,an arbitrary fractional-order equivalent memcapacitor could be constructed by multiple fractional-order memcapacitors.     

Dynamic stability of a class of fractional-order nonlinear systems via fixed point theory

In this paper, the problem of the uniform stability for a class of fuzzy fractional-order genetic regulatory networks with random discrete delays, distributed delays, and parameter uncertainties is studied. Although there is a portion of literature on using fixed point theorems to study the stability of fractional neural networks, most of them required the fractional order to be in $\left(\frac{1}{2},1\right)$. However, the case of the fractional-order belonging to $(0,\frac{1}{2})$ has not been discussed. To solve it, this work proposes a novel idea of using fixed point theory to study the stability of fuzzy (0,1) order neural networks, the problem of the uniqueness of the solution of the considered genetic regulatory networks is resolved, and a novel sufficient condition to guarantee the uniform stability of above genetic regulatory networks is also derived. Eventually, an example is given to demonstrate that the obtained result is effective.     

On a Lyapunov-type inequality and the zeros of a certain Mittag–Leffler function

In this work, a Lyapunov-type inequality is obtained for the case when one is dealing with a fractional differential boundary value problem. We then use that result to obtain an interval where a certain Mittag–Leffler function has no real zeros.     

Ecofriendly synthesis and biological evaluation of 4-(4-nitro-phenyl)-2-phenyl-1,4-dihydro-benzo[4,5]imidazo[1,2-a]pyrimidine-3-carboxylic acid ethyl ester derivatives as an antitubercular agents

An ecofriendly route has been investigated for the synthesis of 4-(4-nitro-phenyl)-2-phenyl-1,4-dihydro-benzo[4,5]imidazo[1,2-a]pyrimidine-3-carboxylic acid ethyl ester derivatives by one-pot, three-component condensation of ethyl benzoylacetate, aromatic aldehydes, and 2-amino benzimidazole using 260?mol% of citric acid as reaction mediator. Citric acid is an inexpensive, nontoxic, and green medium with smoothly activates the rate of reaction. The synthesized compounds were assessed for in vitro antimycobacterial activity against Mycobacterium tuberculosis H₃₇RV strain using the microplate alamar blue assay (MABA). The results indicate that among all the synthesized compound series, P-4 and P-9 compounds illustrate effective activity with a minimum inhibitory concentration of 25?µg/ml.