Equations Of Order Research Articles

Elzaki residual power series method to solve fractional diffusion equation.

The time-fractional order differential equations are used in many different contexts to analyse the integrated scientific phenomenon. Hence these equations are the point of interest of the researchers. In this work, the diffusion equation for a one-dimensional time-fractional order is solved using a combination of residual power series method with Elzaki transforms. The residual power series approach is a useful technique for finding approximate analytical solutions of fractional differential equations that needs the residual function's (n-1)α derivative. Since it is challenging to determine a function's fractional-order derivative, the traditional residual power series method's application is somewhat constrained. The Elzaki transform with residual power series method is an attempt to get over the limitations of the residual power series method. The obtained numerical solutions are compared with the exact solution of this equation to discuss the method's applicability and efficiency. The results are also graphically displayed to show how the fractional derivative influences the behaviour of the solutions to the suggested method.

Investigation of a class of differential equations of complex order with boundary conditions

Investigation of a class of differential equations of complex order with boundary conditions

A Nonlinear Mathematical Model for the Effect of Diabetes Population on a Community

This work presents a compartmental-based mathematical model of susceptible, diabetes without complications, diabetes with minor and major complications to study the effect of diabetes population on the population dynamics of a community. The model is a system of four nonlinear differential equations of first order. The solution of the model was found to exist and is positive by positivity analysis using a contradiction method. The diabetes-free and the diabetes-endemic equilibrium points were also found to be locally asymptotically stable using the Routh-Hurwitz Stability Criterion for a degree n-polynomial. The numerical simulation of the model was carried out using various scenarios and the results show that diabetes population in a community has great effect on the population dynamics of the community either positively or negatively. The results here represent a real-life scenario thereby making the proposed model realistic. Diabetes, Mathematical Model, Simulation, Stability

Representation formulae for linear hyperbolic curvature flows

In [19], Smoczyk showed that parabolic expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we obtain representation formula for the corresponding linear hyperbolic evolution equation for the support function of locally convex curves of general nonzero rotation number, finding some natural similarities and differences in behaviour of solutions as compared with the parabolic case. As applications, we consider linear hyperbolic flows that preserve length or generalised enclosed area of the evolving curve, flow of curves with Neumann boundary conditions inside cones and hyperbolic approaches to the Yau problem of flowing one curve to another. We also consider radial graphs evolving by similar linear equations and linear higher order hyperbolic equations. Our results may be compared with the corresponding parabolic cases consider by the first author, Schrader and Wheeler, in [17].

Non-existence results for fourth order Hardy–Hénon equations in dimensions 2 and 3

We consider the fourth order Hardy–Hénon equationΔ2u=|x|σupin Rn with n≥2, σ>−4, and p>1. This is the fourth order analogy of the second order equation −Δu=|x|σup, known as the Hardy–Hénon equation, which was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above fourth order equation, the assumption n≥4 is often assumed. In this work, we are interested in classical solutions to the equation in the case of low dimensions, namely 2≤n≤3. Here by a classical solution u we mean u belongs to the class C(Rn)∩C4(Rn∖{0}) if σ<0 and C4(Rn) if σ≥0. We show that if 2≤n≤3 then the equation admits no classical solution. In fact, we are able to provide a single treatment for 2≤n≤4.

On the qualitative behaviors of stochastic delay integro-differential equations of second order

In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE).To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature.

Dynamical Analysis of Thirtieth-Order Difference Equations

The main goal of this paper is to determine exact solutions of a family of thirtieth-order difference equations with variable coefficients. We use similarity variables obtained via symmetries to lower the order of the equations. We then reverse the transformations and obtain closed form solutions. We compare our solutions to those found in the literature for special cases. We investigate the periodic nature of the solutions and present some numerical examples to confirm the results. Finally, we analyze the stability of the equilibrium points. The method employed in this work can be applied to equations of higher order provided that they admit non zero characteristics.

On the Hill Discriminant of Lamé's Differential Equation

Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function ${\rm sn}$ depending on the modulus $k$, and two additional parameters $h$ and $\nu$. This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lamé's equations is determined by the value of its Hill discriminant $D(h,\nu,k)$. The Hill discriminant is compared to an explicitly known quantity including explicit error bounds. This result is derived from the observation that Lamé's equation with $k=1$ can be solved by hypergeometric functions because then the elliptic function ${\rm sn}$ reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of $D(h,\nu,k)$ when the modulus $k$ tends to $1$.

Estimating Baryon Density through the CAMB and Assessing its Reliability

The Cosmic Microwave Background Radiation and its anisotropies allow us to analyze many properties and phenomena of the early universe. One such important property is the Baryon Density(Ω_b), a cosmological constant for the density of Baryons in the universe in comparison to its critical density. Baryon Density affects the interactions of matter in the early universe and therefore the analysis of CMB Anisotropies to study how matter interacts allows for the estimation of the Baryon Density in the Early Universe. Understanding Ω_b, is crucial as it reveals information about the composition of the universe, such as the amount of dark matter and dark energy, the early formation of celestial bodies and matter-antimatter asymmetry. This paper aimed to test the reliability of CAMB, a simulation algorithm for CMB anisotropies, by estimating Ω_b h^2. The CAMB mainly utilizes a combination of field equations, the Friedmann-Lemaître-Robertson-Walker metric, Fluid Equations, the Boltzmann Equation and Linear Perturbation theory in order to compute the CMB power spectra. We estimated the value of Ω_b h^2, maintaining other cosmological parameters constant and changing Ω_b h^2 from 0.01 to 0.03 in increments of 0.000625. The peaks, troughs, positions, damping scale and amplitudes of the resultant TT and TE power spectra were compared with data from the Planck Satellite. We use chi-square minimization and find the best fit value and uncertainty forΩ_b h^2 to be 0.02325 ± 0.00015. The estimated value of the baryon density from our study was compared to existing estimates to evaluate the reliability of CAMB as a simulation and source of information for further CMB Anisotropy related research, where it was confirmed to be accurate. However, further developments using wider sets of data was acknowledged with deep learning being a potential step forward.

Penyelesaian khusus persamaan diferensial biasa ordo dua linier tak homogen dengan koefisien konstan untuk fungsi bagian demi bagian

Spring mechanical vibration motion system with a damped degree of freedom and influenced by external forces, is mathematically expressed as an ordinary differential equation of order of two linear constant coefficients that are not homogeneous. If an external force acts on a stationary system expressed as a continuous function f(t) for any time t, then the system will experience mechanical vibrational motion which mathematically the equation of motion can be expressed as a superposition. The equation consists of as a solution to a homogeneous form with mechanical vibrations as a solution to a particular form. In terms of the particular solution this article will show a mathematical way when f(t) is a continuous function section by part which is defined at an interval, such that the mechanical vibration motion equation is at the same time a special solution of the equation the mechanical vibration system.&#x0D; Keywords: Vibration, Impulse Functions, ConvolutionMSC2020: 34A37

MATHIMATICAL MODEL TO PREDICT THE DESIGN ANI) PERFORMANCE OF HIGH.SPEED TURBOPUMPS

In this work a high-speed turbopumps of (22000 rpm) rotational speed, which are oxidizer and fuel turbopumps, are designed. The design requirement for mass flow rate and outlet pressure are (12.5 kg/s) and (113×10 Pa) respectively for the oxidizer turbopump, and (3.5 kg/s) and (95x10 Pa) respectively for the fuel turbopump. The design work of centrifugal pump is divided into impeller and volute design works. The impeller design work is performed by the streamline curvature and quasi-orthogonal technique based on hub-to-shroud mean stream surface between two blades with assistance of experimental design coefficients and relations for first prediction. The volute is designed by using the method of constant angular momentum, where in this method the tangential and radial velocities are considered uniform around the impeller. The numerical results are compared with experimental data of the actual pumps, and it is found that these results agree well with the real designs. The output of the design program is used as an input to the performance program. The performance prediction is achieved by three dimensional, steady, incompressible and viscous flow analyses through the blade passage. The ANSYS program has been employed to solve these equation in order to predict the flow characteristics at design and off-design conditions. The analysis is done at many operation conditions, which have the range of flow coefficient ratio (ФФ) from (0.4) to (1.8), and the results are compared with experimental results for previous work to make sure the validity of the analysis program at design and off-design conditions.

New abundant analytical solutions of coupled nonlinear Schrödinger (FNSE) equation in fractal order arising in quantum mechanics

New abundant analytical solutions of coupled nonlinear Schrödinger (FNSE) equation in fractal order arising in quantum mechanics

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.