Differential Equations Of Integer Order Research Articles

The series solutions of fractional foam drainage and fractional modified regularized long wave problems

In general, integer order differential equations fall short of adequately describing a range of phenomena in many disciplines of science and engineering when compared to fractional order differential equations. In the context of the Caputo fractional derivative, this study examines the nonlinear fractional order foam drainage problem as well as the fractional order modified regularized long wave equation. In this work, we offer two novel techniques by combining the Elzaki transform with the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). We first translate the problem into its differential partner using the Elzaki transform, and then we use He’s polynomials and the Adomian polynomials that are derived from HPM and ADM, respectively. These are two very powerful supports for nonlinear problems. The efficacy and precision of the recommended techniques are illustrated through the use of numerical and graphical findings. The numerical simulations of the derived solutions make use of a number of relevant values of the order of fractional parameter λ. It is important to keep in mind that the recommended techniques can achieve higher overall performance since they need less computational effort than alternative strategies while yet retaining a high degree of numerical precision. The solutions acquired through the suggested techniques have been compared with the classical solutions and the solution obtained by ADM, new iterative method (NIM). The considerable results show that these approaches minimize the heavy calculation without restricting variables while requiring no assumptions. The present study can confirm the applicability of the new generalized Caputo type fractional operator to mathematical real-world problems. This approach shows to be one of the most efficient methods to solve fractional order nonlinear differential equations that arise in related fields of engineering and science. These methods also make clear how to use fractal calculus in real life. Furthermore, the results of this study support the value and significance of fractional operators in real-world applications.

Numerical Technique Based on Operational Matrices of Fractional Integration Using ψ$$ \psi $$‐Shifted Chebyshev Polynomials

ABSTRACTIn this paper, we present a numerical scheme based on a modified form of shifted Chebyshev polynomials to find the numerical solution of a class of fractional differential equations. For this purpose, we work out operational matrices of fractional integration of ‐shifted Chebyshev polynomials obtained from shifted Chebyshev polynomials. Finally, the solution to the problem under consideration is obtained by solving a system of algebraic equations that results from the use of operational matrices of integration. The analysis of integer and non‐integer order differential equations is presented to show the convergence of the solution of fractional order differential equation to the corresponding solution of the integer order differential equation. At the end, we present some linear and non‐linear examples to validate the theoretical analysis. Non‐linear examples are solved using Quasilinearization and proposed numerical technique.

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

Shifted Legendre Gauss‐Lobatto collocation scheme for solving nonlinear coupled space‐time fractional reaction‐advection diffusion equations

AbstractThis work aims to develop and apply an algorithm for solving space‐time fractional reaction‐advection diffusion (STFRAD) nonlinear coupled equations with specified initial and boundary conditions by employing a Shifted Legendre Gauss‐Lobatto collocation (SLGLC) scheme. The fractional derivatives of time and space are defined in Caputo sense. The approximate solutions have been constructed with the help of shifted Legendre polynomials. The proposed numerical scheme reduces the coupled fractional order differential equations into a system of algebraic equations which can then be easily solved. The error analysis through the two test problems is carried out to validate the efficiency and efficacy of the method. The effect of advection and reaction terms, and various space‐time fractional order derivatives on the solution profiles have been studied and are shown graphically for different particular cases. It has been observed that the species whose transport is simulated using STFRADE shows higher concentration at a specific distance in porous media in case of integer order differential equations as compared to fractional order equations demonstrating overestimation of its impact in case of integer order differential equation. This is extremely important observation with respect to applications in the field of groundwater pollution management. The salient feature of the article is the stability and convergence analyses of the proposed scheme on the concerned model, which demonstrates the effectiveness and capabilities of the developed method.

A novel structure grey prediction model with strong compatibility and its application in forecasting the annual average concentration of particulate matter in Beijing

Beijing has long suffered from the events of fine particulate matter pollution. Consequently, it is of paramount importance to be able to scientifically and reasonably predict the concentration of particulate matter. To this end, a novel prediction model with an adaptive grey generation operator and fractional derivative is proposed. The novel model initially expands the range values for the order of the grey generation operator to enhance data preprocessing capabilities. Meanwhile, the introduction of fractional derivatives enhances the model’s adaptability and overcomes the limitations of traditional grey prediction models that use integer-order differential equations. Subsequently, the fractional derivative and operator orders of the novel model are optimized to find the best parameter combination, thereby enhancing the model's performance. Furthermore, the novel model is compatible with seven other models when its parameters take different values, thus expanding the model’s applicability. The novel model is then applied to simulate and predict the particulate matter concentration in Beijing, achieving a comprehensive mean relative percentage error of only 1.86%, significantly outperforming several similar models(17.56%,17.64%,3.42%). Finally, the paper presents a forecast of the particulate matter concentration in Beijing over the next four years.

An improved fractional-order transmission model of COVID-19 with vaccinated population in United States

In this paper, we propose a new fractional-order differential equation model with latent and vaccinated population to describe the dynamics of COVID-19. Firstly, the theoretical mathematical model is established based on the transmission mechanism of COVID-19 in the population. Then, the data of the infected, the recovered and the death are collected from big data report of Baidu’s epidemic situation, and the parameters are estimated by piecewise fitting and nonlinear least square method based on collected data. The correlation coefficients between the infected and model simulation, between the recovered and model simulation, between the death and model simulation are 0.9868, 0.9948 and 0.9994, respectively and the accuracy of prediction are 96.05%, 99.33% and 99.88%, respectively. Additionally, the accuracy of prediction is compared between fractional-order differential equation model and integer-order differential equation model, and the results show fractional-order differential equation model can better predict the development trend of COVID-19. Finally, we analyze the sensitivity of the parameters through numerical simulations, and put forward the corresponding strategies to control the epidemic development according to the screened sensitive parameters.

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

In this paper, the extended (3 + 1)-dimensional Zakharov-Kuznetsov equation, which describes the propagation of ion-acoustic waves in a magnetic environment, is investigated. Due to the exposure of the propagation to unpredictable factors, the stochastic model is assessed including the Brownian process, in addition to including the recent concept of truncated M-fractional derivative. A fractional stochastic transformation is applied to transform the model into an integer-order ordinary differential equation which in turn is equivalent to a conservative Hamiltonian model. Novel solutions, such as hyperbolic, trigonometric, and Jacobian elliptic functions, are established by employing both of the qualitative analysis of dynamical systems and the first integral of the Hamiltonian model. We explore and graphically display the effects of the fractional derivative order and noise intensity on the solutions structures. In the deterministic instance, i.e. in the absence of noise, solitary and cnoidal solutions among other traveling wave solutions of the Zakharov-Kuznetsov equation, are derived. Further, it is found that the curvature of the wave disturbs and the surface turns substantially flat by increasing the value of noise. While the curve in all cases loses its characteristic shape and degenerates into another deterministic shape by changing the fractional derivative order.

Decomposing Method for Space-Time Fractional Order PDEs

Numerical solutions to integer-order differential equations frequently employ decomposition as one of many methods. This paper provides a generalization of integer-order differential equations into fractional order, which is more general, as well as generalizing the decomposing method into the fractional case and then applying it to solve differential equations of fractional order in both space and time. A fractional calculus's theorems and properties to generalize both differential equations and the decomposition method have been utilized. This method to solve different fractional time and space partial differential equations, demonstrating its ability and applicability to various problems have been applied. The method to various cases to highlight its strengths has been used. The graphs and tables of results obtained using Matlab programs have been presented and discussed.

Fractional mathematical model for the transmission dynamics and control of Lassa fever

This paper aims to investigate various epidemiological aspects of Lassa fever viral infection using a fractional-order mathematical model so as to assess the impacts of treatment and vaccination on the spread of Lassa fever transmission dynamics. Initially, the model employs integer-order nonlinear differential equations, incorporating imperfect vaccination and treatment as control measures for the human population. Subsequently, the model is redefined using a fractional order derivative with power law to enhance understanding of disease dynamics. The paper establishes conditions ensuring the existence and uniqueness of the model’s solution in the fractional case, alongside presenting stability analysis of the endemic equilibrium using the Lyapunov function approach. Numerical simulations are performed using the fractional Adams–Bashforth–Moulton method, shedding light on the influence of model parameters and fractional order values on Lassa fever dynamics and control. Additional numerical simulations conducted utilizing surface and contour plots revealed that elevating the contact rates and diminishing the efficacy of vaccines would escalate the prevalence of Lassa fever among the human populace. It was also observed that optimizing treatment and enhancing vaccination strategies would ultimately mitigate the prevalence of Lassa fever within the human population.

On the construction of various soliton solutions of two space-time fractional nonlinear models

Abstract In this article, we investigate a couple of nonlinear fractional models of eminent interests subsequently the conformable derivative sense is used to designate the fractional order derivatives. The given structures are transformed into nonlinear ordinary differential equations of integer order, and the extended simple equation technique is then employed to solve the resulting equations. Initially, the nonlinear space time fractional Klein–Gordon equation is considered emerging from quantum and classical relativistic mechanics, which have application in plasma physics, dispersive wave phenomena, quantum field theory, and optical fibres. Later, the (2 + 1)-dimensional time fractional Zoomeron equation is analysed which is convenient to explore the innovative phenomena related to boomerons and trappons. As a result, various new soliton solutions are successfully established. The reported results offer a key implementation for analysing the soliton solutions of nonlinear fractional models which are extremely encouraging arising in the recent era of science and engineering. The 3D simulations have been carried out to demonstrate dynamics of the various soliton solutions for a given set of parameters.

Developing a fuzzy logic-based carbon emission cost-incorporated inventory model with memory effects

Inventory control is a widely discussed topic in the real world, and recently, it has become closely linked to concerns about carbon emissions and global warming. Global warming is a pressing issue, mainly due to a lack of awareness and action. Traditional inventory models, which typically use integer-order differential equations, overlook the memory aspect of the system. Addressing inventory management is essential in our efforts to combat global warming. This paper introduces a novel approach by incorporating carbon emission costs within a fuzzy environment. Fractional Calculus, a powerful mathematical tool, is employed to capture the memory effect of the system. This approach distinguishes between long memory, characterized by a strong memory effect, and short memory, associated with a poor memory effect, through the use of fractional derivatives and integrals. Numerical results are analyzed based on these memory concepts. Entrepreneurs often find it difficult to determine exact values for known parameters in the real world. Therefore, this study considers the uncertain nature of ordering costs, the rate of deterioration, and demand rates as triangular fuzzy numbers. The optimal average cost and ordering intervals are determined using a solution methodology. A sensitivity analysis is performed to demonstrate how different system parameters influence the outcomes within a fuzzy environment. Notably, it is observed that profit tends to be higher under conditions of strong memory compared to poor memory effects. Moreover, it's worth emphasizing that profits are notably more favorable when employing the signed distance method compared to the graded mean integration method, especially in situations marked by strong memory effects. This study highlights the importance of considering sensitive parameters in the model, especially under conditions of strong memory effects. Such parameters require careful attention in the pursuit of effective inventory management strategies to mitigate carbon emissions and combat global warming.

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.

New chirp soliton solutions for the space-time fractional perturbed Gerdjikov–Ivanov equation with conformable derivative

The space–time perturbed fractional Gerdjikov–Ivanov equation is the main topic of this work, together with quintic nonlinearity and self-steepening, as it involves several intricate physical phenomena including nonlinearity, self-steepening and fractional calculus, where the fractional derivative is described by employing a conformable derivative. In addition, the governing equation is transformed into an integer-order ordinary differential equation by using an appropriate fractional complex transformation. Under certain restrictions, a direct algebraic method is employed to investigate the structures of chirp soliton solutions enfolding hyperbolic functional terms. The dynamic behaviour and bifurcation of equilibria of the system are thoroughly examined; chirp soliton solutions under specified constraints are investigated and the evolving profiles of the obtained solutions are visualized. Moreover, this research offers valuable perspectives on the behaviour of chirp solitons under specific conditions, which have practical applications in nonlinear physical systems and optical communication systems. The significant contribution of this work is the investigation and obtaining of novel chirp soliton solutions with hyperbolic functional terms under particular limitations using a novel approach. It further extends the prior approaches by treating difficult fractional differential equations from a fresh angle, offering new tools, and closely examining soliton solutions.

Fractional-Order Model of Electric Arc Furnace

The paper introduces a fractional-order electric arc furnace (EAF) model. It is based on a power balance equation which, as has been shown, can be represented by a Hammerstein-Wiener model. Such a model includes a block described by a linear integer-order differential equation, in this case the order is equal to one, which can be extended to a fractional-order equation. The introduced fractional order is either constant or variable in time. The variable coefficients of the model have been identified by a genetic algorithm The proposed model has been tested using measurement data from two different EAFs and its accuracy has been compared with the classical integer-order model based on the RMS error between the model output and measurement waveforms. The results show that the fractional-order EAF model enables error reduction and consequently reflects more accurately the phenomena taking place in the furnace.

A Systematic Approach to Delay Functions

We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions.

Unconditional stability analysis of Grünwald Letnikov method for fractional-order delay differential equations

In this paper, we investigate the unconditional stability and the generally unconditional stability of the Grünwald Letnikov method for fractional-order delay differential equations (FDDEs), which is the generalization of P-stability and GP-stability for classical integer-order delay differential equations. Using the Z-transform, an equivalent form of the discrete Laplace transform, we first show the unconditional stability of the Grünwald Letnikov method for any delay and any constraint mesh. Secondly, we also derive the generally unconditional stability of the Grünwald Letnikov method with a linear interpolation for approximating the delay term under a general uniform mesh. It is shown that the Grünwald Letnikov method for FDDEs preserves the stability for the analytical solution and hence naturally inherits the α-dependence. Finally, two numerical examples for FDDEs and time fractional-order diffusion equations with delay are presented to demonstrate the validity and effectiveness of theoretical results.

Modeling and Initialization of Nonlinear and Chaotic Fractional Order Systems Based on the Infinite State Representation

Based on the infinite state representation, any linear or nonlinear fractional order differential system can be modelized by a finite-dimension set of integer order differential equations. Consequently, the recurrent issue of the Caputo derivative initialization disappears since the initial conditions of the fractional order system are those of its distributed integer order differential system, as proven by the numerical simulations presented in the paper. Moreover, this technique applies directly to fractional-order chaotic systems, like the Chen system. The true interest of the fractional order approach is to multiply the number of equations to increase the complexity of the chaotic original system, which is essential for the confidentiality of coded communications. Moreover, the sensitivity to initial conditions of this augmented system generalizes the Lorenz approach. Determining the Lyapunov exponents by an experimental technique and with the G.S. spectrum algorithm provides proof of the validity of the infinite state representation approach.

Enhancing medical ultrasound imaging through fractional mathematical modeling of ultrasound bubble dynamics

The classical mathematical modeling of ultrasound acoustic bubble is so far using to improve the medical imaging quality. A clear and visible medical ultrasound image relies on bubble’s diameter, wavelength and intensity of the scattered sound. A bubble with diameter much smaller than the sound wavelength is regarded as highly efficient source of sound scattering. The dynamical equation for a medical ultrasound bubble is primarily modeled in classical integer-order differential equation. Then a reduction of order technique is used to convert the modeled dynamic equation for the bubble surface into a system of incommensurate fractional-orders. The incommensurate fractional-order values are calculated directly, by using Riemann stability region. On the basis of stability the convergence and accuracy of the numerical scheme is also discussed in detail. It has been found that the system will remain stable and chaotic for the incommensurate values α1<0.737 and α2<2.80, respectively.

Numerical assessment of multiple vaccinations to mitigate the transmission of COVID-19 via a new epidemiological modeling approach

Vaccines play a crucial role in mitigating the transmission of COVID-19. The primary objective of COVID-19 vaccination campaigns is to induce immunity in individuals, thereby reducing the risk of infection, severe illness, and subsequent transmission within communities. The present study mainly focuses on examining the impact of multiple vaccinations on the transmission dynamics of COVID-19 using a new epidemic modeling approach. The vaccinated individuals are divided into three subclasses according to their vaccination status. The primary, secondary and booster shots of COVID-19 vaccines are considered in the construction of the model. Initially, the model is constructed using classical integer order differential equations and basic necessary properties are carried out. Furthermore, utilizing a fractional and fractal–fractional epidemiological approach with an exponential kernel, the model is extended with these operators in order to provide insights into the potential benefits of this vaccination strategy in controlling the spread of the disease. The uniqueness and existence criteria of the fractional and fractal–fractional models are presented and an efficient numerical solution is investigated using the modified Adams–Bashforth method. The impact of both fractional (memory index) and fractal parameters is demonstrated graphically for R0<1 and R0>1 showing the converges of solution trajectories to the disease-free and endemic steady states respectively. Moreover, we examine the impact of various vaccination rates (first and second dose, and booster short) on the reduction and mitigation of the disease incidence.

Mathematical assessment of monkeypox disease with the impact of vaccination using a fractional epidemiological modeling approach

This present paper aims to examine various epidemiological aspects of the monkeypox viral infection using a fractional-order mathematical model. Initially, the model is formulated using integer-order nonlinear differential equations. The imperfect vaccination is considered for human population in the model formulation. The proposed model is then reformulated using a fractional order derivative with power law to gain a deeper understanding of disease dynamics. The values of the model parameters are determined from the cumulative reported monkeypox cases in the United States during the period from May 10th to October 10th, 2022. Besides this, some of the demographic parameters are evaluated from the population of the literature. We establish sufficient conditions to ensure the existence and uniqueness of the model’s solution in the fractional case. Furthermore, the stability of the endemic equilibrium of the fractional monkeypox model is presented. The Lyapunov function approach is used to demonstrate the global stability of the model equilibria. Moreover, the fractional order model is numerically solved using an efficient numerical technique known as the fractional Adams-Bashforth-Moulton method. The numerical simulations are conducted using estimated parameters, considering various values of the fractional order of the Caputo derivative. The finding of this study reveals the impact of various model parameters and fractional order values on the dynamics and control of monkeypox.