Equations Of Order Research Articles

Liouville type theorems involving fractional order systems

Abstract In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: ( − Δ ) α / 2 u ( x ) = f ( u ( x ) , v ( x ) ) , x ∈ R n , ( − Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x ∈ R n . $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad \hfill \\ {\left(-{\Delta}\right)}^{\alpha /2}v\left(x\right)=g\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n}.\quad \hfill \end{cases}$ Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang (“A direct method of moving spheres on fractional order equations,” J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu (“Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space,” Potential Anal., vol. 46, pp. 569–588, 2017).

Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method

Abstract One of the efficient and reliable methods for resolving fractional order linear as well as non-linear differential equations is the Laplace transform with residual power series method. This approach is used in the current research to obtain the numerical solutions of the two-dimensional fractional differential equations, namely, the temporal fractional order diffusion equation and the fractional biological population equation. The unknown coefficients of the series solutions to these equations are determined using the proposed approach. The difference between exact and analytical-numerical solutions is presented for these equations in the form of errors. The advantage of the suggested method over alternative approaches is that it requires less computation to solve these two-dimensional differential equations of time-fractional order.

Boundedness of Vector Linéard Equation with Multiple Variable Delays

In this article, we consider a system of ordinary differential equations (ODEs) of second order with two variable time delays. We obtain new conditions for uniform ultimate bounded (UUB) solutions of the considered system. The technique of the proof is based on the Lyapunov–Krasovskii functional (LKF) method using a new LKF. The main result of this article extends and improves a recent result for ODEs of second order with a constant delay to a more general system of ODEs of second order with two variable time delays. In this particular case, we also give a numerical example to verify the application of the main result of this article.

Solving Nonlinear Fractional Differential Equations by Using Shehu Transform and Adomian Polynomials

The current article provides a detailed analysis of the solution of non-linear ordinary differential equations of fractional and non-fractional order in series forms using the Shehu transform (ST) and the Adomian decomposition method (ADM), also known as the Shehu transform Adomian decomposition method (STADM). Previously, these methods were used to solve differential equations of integer order as well as a very small number of ordinary and fractional differential equations (FDEs). The Caputo's operator is used in a number of well-known FDEs, including the logistic equation, the Van der Pole equation, and other non-fractional order differential equations like the nonlinear Bratu type equation. It is noted that all of the example issues had series solutions thanks to STADM. Plotting the graph for several series terms of the series solution demonstrates how the approximate solution tends to the closed form solution. In some example problems the impact of α is shown.

Calculating the acoustic and internal gravity wave dispersion relations in Venus's supercritical lower atmosphere

There is a growing interest in quantifying Venusian seismic events through their infrasonic signatures detected by balloon-borne sensors at ∼ 55 km altitude. The extreme pressure and temperature at Venus’s surface correspond to supercritical conditions in the planet’s deep atmosphere. Therefore, an appropriate real-gas equation of state (EoS) must be used to study the acoustic properties and thermodynamics in the Venusian lower atmosphere. In previous work, the Peng-Robinson (P-R) EoS was used to obtain the acoustic sound speed and attenuation coefficient in the lower atmosphere of Venus. Here, the P-R EoS is coupled with the fluid dynamics equations in order to derive the acoustic and internal gravity wave (IGW) dispersion equations. Results show that in Venus’s deep atmosphere, the acoustic cut-off frequency corresponds to a period of ∼480 s (0.0020 Hz), and the buoyancy frequency corresponds to a period of ∼600 s (0.0016 Hz). By comparison, the values in Earth’s lower atmosphere are ∼310 and ∼340 s, respectively. These differences in acoustic and IGW propagation characteristics will be useful in later efforts to discriminate between the various waves detected by high-altitude sensors.

TOUCHARD POLYNOMIALS APPROACH FOR SOLVING NON- LINEAR FREDHOLM INTEGRO- DIFFERENTIAL EQUATION OF SECOND TYPE

A numerical approach to solve Non- linear Fredholm integro- differential (NLFID) equation of the 1st order and 2nd type has been proposed. The approach was based upon Touchard polynomials. The non-linear Fredholm integro- differential equation was changed into a system of non- linear algebraic equations and solved using Newton repeating approach. The proposed approach was evaluated by displaying three numerical problems, and the approximate numerical solutions were compared with exact solution and four methods in the literature. MATLAB R2018b was used to perform all calculations and graphs.

Efficient Solutions with the LRPS Method for Non-Linear Fractional Order Tuberculosis Models

In this research article, we present a novel Non-Linear Fractional Order Tuberculosis mathematical model (NLFOTB) and introduce an efficient technique to obtain its solution. Fractional Order Models (FOMs) have garnered significant attention in contemporary research due to their widespread applicability. We address the challenge of solving the coupled Initial Value Problems (IVPs) associated with NLFOTB models by utilizing the groundbreaking LRPS method, which combines the RPS approach with the Laplace transform operator. This innovative approach generates approximate solutions in rapidly converging series forms, offering enhanced efficiency and reduced computational effort compared to conventional methods. Through the implementation of the LRPS method, we successfully derive an approximate solution for the NLFOTB model, contributing significantly to the field. Furthermore, our proposed approach demonstrates its efficacy in accurately capturing the dynamics of Tuberculosis (TB) through extensive computations and graphical representations, contributing to a deeper understanding of TB dynamics within a mathematical framework. Additionally, the LRPS method shows promise in tackling real world problems involving differential equations of various orders. Future investigations can extend the application of the LRPS method to explore other Fractional Order Models, further validating its effectiveness in a wide range of epidemic scenarios. Consequently, our study not only provides valuable insights into Tuberculosis dynamics but also introduces a powerful computational tool applicable to various practical problems in diverse disciplines, making a substantial contribution to the field of mathematical modeling and computation. HIGHLIGHTS · The comprehensive study highlights the LRPS strategy’s efficiency and accuracy in approximating solutions for fractional order equations. The research demonstrates its capability to predict compartmental behavior accurately within the specified range · The discussions presented in the article significantly contribute to the field of epidemiology by introducing and showcasing the LRPS approach’s effectiveness. As a valuable tool for investigating and validating epidemic models, the LRPS method offers improved efficiency and convenience, thereby enhancing the understanding of disease dynamics · The researchers anticipate that their findings will inspire further exploration and utilization of the LRPS technique in solving nonlinear models. This, in turn, is expected to contribute to advancements in the broader field of epidemiology, fostering continued innovation and development GRAPHICAL ABSTRACT

Estimation of the Impact of Vaccination Intervention on Recovered Coronavirus Patients

This work estimated the impact of vaccination intervention on coronavirus patients who have recovered from the disease and the vulnerability index of the recovered population due to the impact of vaccine was also investigated. This work adopted a numerical solution to study the continuous dynamical system of linear first order differential equations describing a SEIR (Susceptible, Exposed, Infected, Recovered) model on the spread of Coronavirus Disease – 2019 (COVID-19). To tackle this problem, MATLAB ordinary differential equation of order 45 (ODE45) numerical method was adopted for the analysis. The vulnerability index of the recovered population was low due to the impact of vaccine meaning that the recovered population will gain immunity and they will not be re-infected. The study recommended that coronavirus patients who have recovered from the disease should ensure that they have vaccination administered to them to avoid re-occurrence of the virus attack as an intervention strategy.

Radiation effect on convective boundary layer dusty flow over a stretching surface

The effect of radiation evolved from fluids and particles on a convective balanced boundary layer dusty fluid flow model has been analyzed over a vertical stretching surface. Models are available with the consideration of radiation and viscous dissipation in fluid phase only. However, an equal and opposite reaction/force is also developed in the particle/dust phase, which was not taken into account in the previous literature. So, this paper investigates a model where the radiation and viscous dissipation are coupled through both the fluid and particle phase equations. A linear stretching of surface generates the boundary layer flow. By the help of suitable similar transformation, the prevailing equations are transferred to ordinary non-linear differential equations of order three and are solved numerically by the bvp4c method of MATLAB. An analysis has been performed to show the effectiveness of different thermophysical parameters on different boundary layer characteristics. The numerical results are in good agreement with the past literature, validating the present method and result. Graphs are plotted for different flow characteristics to explore the physics of problem.

Spectral-Galerkin method for second kind VIEs with highly oscillatory kernels of the stationary point

In this paper, we discuss an efficient spectral approach for solving a class of second kind VIEs with highly oscillatory kernels possessing the stationary point. First, we use one variable transform to convert the highly oscillatory problem into the long-time one, and then split the long-time problem into a linear system of integral equations by using a dilation approach. Next on each interval we study the characteristic of the original solution and then propose an efficient spectral-Galerkin method for solving the linear system of integral equations. We prove that the proposed algorithm requires to solve the number O(log2⁡ω) of linear algebra equations of order n+1 and reaches the convergence order O(n−r), where r denotes the degree of the regularity of the original solution and ω denotes the wave number. Moreover, we give the approach of computing the highly oscillatory integrals produced in the spectral-Galerkin method. At last, two numerical examples are provided to verify the efficiency of our proposed method.

Numerical solution of fuzzy fractional volterra integro differential equations with boundary conditions

In this study, the boundary value problem of fuzzy fractional nonlinear Volterra integro differential equations of order 1 < ϱ ≤ 2 is addressed. Fuzzy fractional derivatives are defined in the Caputo sense. To show the existence result, the Krasnoselkii theorem from the theory of fixed points is used, where as the well-known contraction mapping concept is utilized in order to show the solution is unique to the proposed problem. Moreover, a novel Adomian decomposition method is utilized to get numerical solution; the approach behind deriving the solution is from Adomian polynomials, and it is organized according to the recursive relation that is obtained. The proposed method significantly decreases the numerical computations by obtaining solutions without the need of discretization or constrictive assumptions. According to the results, there is substantial agreement between the series solutions produced by the fuzzy Adomian decomposition method. Finally, using MATLAB, the symmetry between the lower and upper-cut representations of the fuzzy solutions is demonstrated in the numerical result.

Maximal regularity for fractional difference equations of order 2&lt;alpha&lt;3 on UMD spaces

In this article, we study the \(\ell^p\)-maximal regularity for the fractional difference equation $$ \Delta^{\alpha}u(n)=Tu(n)+f(n), \quad (n\in \mathbb{N}_0). $$ We introduce the notion of \(\alpha\)-resolvent sequence of bounded linear operators defined by the parameters \(T\) and \(\alpha\), which gives an explicit representation of the solution. Using Blunck's operator-valued Fourier multipliers theorems on \(\ell^p(\mathbb{Z}; X)\), we give a characterization of the \(\ell^p\)-maximal regularity for \(1 &lt; p &lt; \infty\) and \(X\) is a UMD space. For more information see https://ejde.math.txstate.edu/Volumes/2024/20/abstr.html

A simple model for the prediction of CO2 concentrations in the atmosphere, depending on global CO2 emissions

We present a very simple model for estimating time dependent atmospheric CO2 concentrations c(t) from global carbon emission scenarios, serving as single input data. We derive a single linear differential equation of 1st order, based on parameters which are estimated from quantitative data of the global carbon project and Mauna Loa data for CO2 concentrations. The model is tested first by comparing it to the 1960–2021 period with reasonably good quantitative agreement and, second to two of the typical current IPCC scenarios with good qualitative agreement. Finally, some new emission scenarios are modelled. Despite several drawbacks concerning absolute quantitative predictions, there are two important advantages of the model. First, it can be easily executed by students already with simple programmable spreadsheet programs such as Excel. Second input emission scenarios can be changed easily and expected changes are immediately seen for discussion during undergraduate and graduate courses on the carbon cycle and climate change.

Efficiently solving fractional delay differential equations of variable order via an adjusted spectral element approach

This paper presents a new approach for solving fractional delay differential equations of variable order using the spectral element method. The proposed method overcomes the limitations of traditional spectral methods, such as poor approximation in long intervals and inefficiency in high degrees. By introducing a variable order differentiation matrix and using basic Lagrangian functions to approximate the solution in each element, the method achieves high accuracy and efficiency. A penalty method is also applied to minimize the jump of fluxes at interface points, and the effectiveness of this approach is analyzed. Finally, three benchmark problems are solved, and the convergence analysis demonstrates the effectiveness and efficiency of the proposed method. In essence, this paper offers a significant contribution to the literature on fractional differential equations and their numerical solution methodologies.

A refinement of the Hille–Wintner comparison theorem and new nonoscillation criteria for half-linear differential equations

A refinement of the Hille–Wintner comparison theorem is obtained for two half-linear differential equations of the second order. As a consequence, some new nonoscillation tests for such equations are derived by means of this improved comparison technique. In most of our results coefficients and their integrals do not need to be nonnegative and are allowed to oscillate in any neighborhood of infinity.

Second–order discontinuous ODEs and billiard problems

We present an existence principle for boundary value problems involving discontinuous ordinary differential equations of the second order using the Krasovskii regularization technique. Especially we obtain sufficient conditions of transversality type for Krasovskii solutions to be also Carathéodory solutions of the original problem. This result is applied on a certain billiard problem, which can be thought as an ordinary differential equation with state-dependent impulses that is equivalent to certain discontinuous differential equation. In particular, we obtain new existence and multiplicity results for Dirichlet problems in billiard spaces with time-varying boundaries.

Arbitrary coefficient assignment to the characteristic function of the scalar type for linear systems with state delays by linear static delayed output feedback

We consider a control system defined by a system of linear differential equations in real or complex space with multiple lumped and distributed delays in the state variable. For this system, we study the problem of arbitrary coefficient assignment for the characteristic function of the scalar type by linear static delayed output feedback with lumped and distributed delays. We obtain necessary and sufficient conditions for solvability of the arbitrary coefficient assignment problem by the static delayed output feedback controller when the system coefficients have a special form. Corollaries on arbitrary finite spectrum assignment and on stabilization of the system are obtained. We prove that the obtained results generalize the corresponding theorem proved earlier for a control system defined by a linear delay-differential equation of nth order. Examples are presented illustrating the results obtained.

Simulation of Fractional Order 2D-Mathematical Model Using α-Fractional Differential Transform Method

In this paper, we will introduce a well-known transformation technique, the modified α-fractional differential transform, to the differential equation of fractional order. We derive some new results with proof using new techniques that never existed before. By using this new technique, we are attempting to solve the nonlinear fractional-order mathematical epidemic model. Furthermore, the fractional epidemic model’s solution obtained by using this new technique is correlated with the solution of the same model calculated for a different fractional order by the modified α-fractional differential transform method. Moreover, using the Python software, we can numerically and graphically represent the solution of fractional differential equations.

Analytical Solution of Generalized Bratu-Type Fractional Differential Equations Using the Homotopy Perturbation Transform Method

In this study, we present the generalized form of the higher-order nonlinear fractional Bratu-type equation. In this generalization, we deal with a generalized fractional derivative, which is quite useful from an application point of view. Furthermore, some special cases of the generalized fractional Bratu equation are recognized and examined. To solve these nonlinear differential equations of fractional order, we employ the homotopy perturbation transform method. This work presents a useful computational method for solving these equations and advances our understanding of them. We also plot some numerical outcomes to show the efficiency of the obtained results.

The solvability of inhomogeneous boundary-value problems in Sobolev spaces

The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be overdetermined or underdetermined. They may contain derivatives, of the unknown vector-valued function, whose integer or fractional orders exceed the order of the differential equation. Similar problems arise naturally in various applications. The theory introduces the notion of a rectangular number characteristic matrix of the problem. The index and Fredholm numbers of this matrix coincide, respectively, with the index and Fredholm numbers of the inhomogeneous boundary-value problem. Unlike the index, the Fredholm numbers (i.e., the dimensions of the problem kernel and co-kernel) are unstable even with respect to small (in the norm) finite-dimensional perturbations. We give examples in which the characteristic matrix can be explicitly found. We also prove a limit theorem for a sequence of characteristic matrices. Specifically, it follows from this theorem that the Fredholm numbers of the problems under investigation are semicontinuous in the strong operator topology. Such a property ceases to be valid in the general case.