Order Differential Equations Research Articles

Existence and Uniqueness of Periodic Solutions for a Class of Higher Order Differential Equations

Existence and Uniqueness of Periodic Solutions for a Class of Higher Order Differential Equations

On the qualitative evaluation of the variable-order coupled boundary value problems with a fractional delay

The logical progression from the constant order differential equations is the field of variable-order differential equations. Such equations can frequently give a more succinct description of problems in the real world. In light of this, we therefore take into account a class of coupled boundary value problems under variable-order differentiation. By utilizing the fixed-point techniques of Banach and Schauder, we investigate the existence and uniqueness of solutions to the proposed problem. Also, sufficient results are documented for the necessary needs. Furthermore, some stability results based on the ideas of Ulam, Hyers, and Rassias are elaborated upon. Ultimately, appropriate examples and in-depth analysis are presented to support our results.

The attractor flow for AdS5 black holes in mathcal{N} = 2 gauged supergravity

We study the flow equations for BPS black holes in mathcal{N} = 2 five-dimensional gauged supergravity coupled to any number of vector multiplets via FI couplings. We develop the Noether-Wald procedure in this context and exhibit the conserved charges as explicit integrals of motion, in the sense that they can be computed at any radius on the rotating spacetime. The boundary conditions needed to solve the first order differential equations are discussed in great detail. We extremize the entropy function that controls the near horizon geometry and give explicit formulae for all geometric variables at their supersymmetric extrema. We have also considered a complexification of the near-horizon variables that elucidates some features of the theory from the near-horizon perspective.

Is reinfection negligible effect in COVID‐19? A mathematical study on the effects of reinfection in COVID‐19

Serological studies show that besides the development of suitable antibodies that are evidenced in people who get infected with SARS‐CoV2 or in people who are vaccinated, the possibility of getting reinfected with SARS‐CoV2 remains non‐zero (may be finite). The present article studies how far this possibility of reinfection impacts the transmission dynamics of COVID‐19. Considering a six compartment mathematical model, we have studied the transmission dynamics of the disease and presented the situations that will lead to endemic‐free (also endemic) state. Considering the COVID‐19 waves in India during the spread of SARS‐CoV‐2, delta variant of SARS‐CoV‐2, and Omicron variant, the parameters of the model are estimated corresponding to these three situations. As expected, the obtained parameters lie in the stable region of endemic state. The changes in dynamics of the system with the reinfection is studied, and it shows that its effects are not neglible. Even finite value of reinfection can significantly vary the number of infected individuals, and it could explain secondary (smaller) rise in the COVID‐19 cases after the COVID‐19 wave. We also study the dynamics of the system through fractional order differential equations.

On classification of singular matrix difference equations of mixed order

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

On a family of higher order recurrence relations: symmetries, formula solutions, periodicity and stability analysis

In this paper, we present formula solutions of a family of difference equations of higher order. We discuss the periodic nature of the solutions and we investigate the stability character of the equilibrium points. We utilize Lie symmetry analysis as part of our approach together with some number theoretic functions. Our findings generalize certain results in the literature.

A Numerical Technique Based on Bernoulli Wavelet Operational Matrices for Solving a Class of Fractional Order Differential Equations

In this paper, we present an efficient, new, and simple programmable method for finding approximate solutions to fractional differential equations based on Bernoulli wavelet approximations. Bernoulli Wavelet functions involve advantages such as orthogonality, simplicity, and ease of usage, in addition to the fact that fractional Bernoulli wavelets have exact operational matrices that improve the accuracy of the applied approach. A fractional differential equation was simplified to a system of algebraic equations using the fractional order integration operational matrices of Bernoulli wavelets. Examples are used to demonstrate the technique’s precision.

The Solution of a System of Higher-Order Difference Equations in Terms of Balancing Numbers

In this paper, we are interested in the closed-form solution of the following system of nonlinear difference equations of higher order,un+1 = 1/34-vn-m , vn+1 = 1/34-un-m, n, m ∈ N0,and the initial values u-j and v-j , j∈{0, 1, ..., m} are real numbers do not equal 34. We show that the solutions of this system are associated with Balancing numbers. As consequence, these solutions are also associated with Pell numbers, Pell-Lucas numbers, and Lucas-Balancing numbers. It is shown that the global stability of positive solutions of this system holds. Our results are illustrated via numerical examples.

Painlevé V and confluent Heun equations associated with a perturbed Gaussian unitary ensemble

We discuss the monic polynomials of degree n orthogonal with respect to the perturbed Gaussian weight w(z,t)=|z|α(z2+t)λe−z2,z∈R,t>0,α>−1,λ>0, which arises from a symmetrization of a semi-classical Laguerre weight wLag(z,t)=zγ(z+t)ρe−z,z∈R+,t>0,γ>−1,ρ>0. The weight wLag(z) has been widely investigated in multiple-input multi-output antenna wireless communication systems in information theory. Based on the ladder operator method, two auxiliary quantities, Rn(t) and rn(t), which are related to the three-term recurrence coefficients βn(t), are defined, and we show that they satisfy coupled Riccati equations. This turns to be a particular Painlevé V (PV, for short), i.e., PVλ22,−(1−(−1)nα)28,−2n+α+2λ+12,−12. We also consider the quantity σn(t)≔2tddtlnDn(t), which is allied to the logarithmic derivative of the Hankel determinant Dn(t). The difference and differential equations satisfied by σn(t), as well as an alternative integral representation of Dn(t), are obtained. The asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s ≔ 4nt is fixed, are established. Finally, by using the second order difference equation satisfied by the recurrence coefficients, we obtain the large n full asymptotic expansions of βn(t) with the aid of Dyson’s Coulomb fluid approach. By employing these results, the second differential equations satisfied by the orthogonal polynomials will be reduced to a confluent Heun equation.

Resonant fractional order differential equation with two-dimensional kernel on the half-line

Resonant fractional order differential equation with two-dimensional kernel on the half-line

LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY

In this work, we study the existence and regularity of solutions for some second order differential equations with infinite delay in Banach spaces. We suppose that the undelayed part admits a cosine operator in the sense given by Da Prato and Giusi, [ G. Da Prato and E. Giusi, \emph{Una caratterizzazione dei generatori di funzioni coseno astratte}, Bollettino dell'Unione Matematica Italiana, 22, 357-362, (1967)]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of strict solutions.

A novel approach to the motion equations of null Cartan curves via the compatible Hasimoto map

In this manuscript, we try to understand the links between nonlinear equations of motion and equations attained by taking geometric differential description of Hasimoto map of null Cartan curves and some physical results. First, we obtain the equations of Hasimoto transformation occurred by applying Hasimoto map for null Cartan curves. Then, based on the Hasimoto map of null Cartan curves, we examine both Landau–Lifshitz system and visco Landau–Lifshitz system. Lastly, we portray the links of findings obtained by geometric physical issues. After all, we propose a new method, known as Khater method, to obtain solutions of fractional order differential equations. Also, these solutions are simulated by graphically using physical constructions.

Existence of a variational principle for PDEs with symmetries and current conservation

It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a differential equation, which satisfies sufficiently many symmetries and corresponding conservation laws, leads to a variational functional, whose Euler-Lagrange equation is the given differential equation. Sufficiently many symmetries means that the set of symmetry vector fields satisfy span{Vp:V∈Sym}=TpE, that is, they span the tangent space TpE at each point p of a fiber bundle E, which describes the dependent- and independent coordinates. Higher order coordinates are described by the jet bundle JkE and we require no span-assumptions on TJkE. Our main theorem states that Noether's theorem can be reversed in this sense for second order differential equations, or more precisely, for so-called second order source forms on J2E, which are required to write the differential equation as a weak formulation (every Euler-Lagrange equation is derived from a first variation, that is, from a weak formulation). Counter examples show that our theorem is sharp.

First law of thermodynamics and entropy of FLRW universe in modified gravity

We investigate the first law of thermodynamics and entropy associated to the apparent horizon of (non-flat) FLRW space–time in different theories of modified gravity and in the presence of a perfect fluid of matter. We pose our attention on those theories which lead to second order differential field equations on FLRW background. In this way, we observe that one may obtain a formula for entropy in terms of the radius of the apparent horizon only. Thus, when considering a modification to the area law of General Relativity, it is possible to reconstruct the gravitational lagrangian consistent with the corresponding first law.

Krylov Subspace Residual and Restarting for Certain Second Order Differential Equations

Krylov Subspace Residual and Restarting for Certain Second Order Differential Equations

(ω,ρ)-BVP Solutions of Impulsive Differential Equations of Fractional Order on Banach Spaces

The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω,ρ)-BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer’s fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results.

Existence of a system of fractional order differential equations via generalized contraction mapping in partially ordered Banach space

Existence of a system of fractional order differential equations via generalized contraction mapping in partially ordered Banach space

Numerical simulation of conduction problem with evaporation based on a SPH model improved by a fractional order convection-diffusion equation

Smoothed particle hydrodynamics (SPH) simulations of phase transitions are not accurate because of the decrease in liquid particles at the end of the evaporation phase transition. In this paper, we establish a numerical model of droplet evaporation suitable for SPH by introducing a time fractional-order convection–diffusion equation. This model was used to study the evaporation process of droplets under different environments, and the stationary evaporation of droplets was simulated by combining convection–diffusion equations of different orders to determine the optimal order. The results conformed to the “D2 Law,” confirming the effectiveness and accuracy of our method. The model was then used to study the phase transition process of droplets impacting flat and curved surfaces. Among the tested equations, the 0.1-order convection–diffusion equation had the best correction effect on the evaporation model. This equation improved the accuracy of the calculation and had a good corrective effect for both static evaporation and the evaporation of droplets impacting flat and curved surfaces.

Periodic solutions of discontinuous second order differential equations. The porpoising effect

We construct a mathematical model in order to study the so called porpoising effect in racing cars, and prove that, when adding a small periodic perturbation, large-amplitude subharmonic solutions may arise.

New finite-type multi-indexed orthogonal polynomials obtained from state-adding Darboux transformations

Abstract The Hamiltonians of finite-type discrete quantum mechanics with real shifts are real symmetric matrices of order N + 1. We discuss the Darboux transformations with higher-degree (>N) polynomial solutions as seed solutions. They are state-adding and the resulting Hamiltonians after M steps are of order N + M + 1. Based on 12 orthogonal polynomials ((q-)Racah, (dual, q-)Hahn, Krawtchouk, and five types of q-Krawtchouk), new finite-type multi-indexed orthogonal polynomials are obtained, which satisfy second-order difference equations, and all the eigenvectors of the deformed Hamiltonian are described by them. We also present explicit forms of the Krein–Adler-type multi-indexed orthogonal polynomials and their difference equations, which are obtained from the state-deleting Darboux transformations with lower-degree (≤N) polynomial solutions as seed solutions.