Differential Equations Of Integer Order Research Articles

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.

Applications of the Laplace variational iteration method to fractional heat like equations

The importance of differential equations of integer order and fractional order can be seen in many areas of engineering and applied sciences. The present work involves fractional order heat equations that arise in numerous applications of engineering and aims to find series solutions by the Laplace variational iteration method (LVIM). The method combines the Laplace transform and the variational iteration method. To show the efficiency and validity of LVIM, we have exemplarily considered 1-D, 2-D, and 3-D fractional heat equations and solve them by LVIM. Exact solutions are gained in expressions of the Mittag-Leffler function. The results are also explored through graphs and charts.

A Comparative Study of the Fractional Partial Differential Equations via Novel Transform

In comparison to fractional-order differential equations, integer-order differential equations generally fail to properly explain a variety of phenomena in numerous branches of science and engineering. This article implements efficient analytical techniques within the Caputo operator to investigate the solutions of some fractional partial differential equations. The Adomian decomposition method, homotopy perturbation method, and Elzaki transformation are used to calculate the results for the specified issues. In the current procedures, we first used the Elzaki transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. For each targeted problem, the generalized schemes of the suggested methods are derived under the influence of each fractional derivative operator. The current approaches give a series-form solution with easily computable components and a higher rate of convergence to the precise solution of the targeted problems. It is observed that the derived solutions have a strong connection to the actual solutions of each problem as the number of terms in the series solution of the problems increases. Graphs in two and three dimensions are used to plot the solution of the proposed fractional models. The methods used currently are simple and efficient for dealing with fractional-order problems. The primary benefit of the suggested methods is less computational time. The results of the current study will be regarded as a helpful tool for dealing with the solution of fractional partial differential equations.

An inventory model for partial backlogging items with memory effect

Inventory control is considered one of the most widely documented topics in the reality. Fractional derivatives and integration is the part of fractional calculus. Fractional calculus is the generalized part of ordinary calculus. The memory of physical phenomena is a highly concerning topic but it is neglected with describing in terms of integer-order differential equation. To discuss the memory of the inventory model, fractional derivative tools are considered. A fractional derivative at any point gives the previous marginal output and current point output for any current point input. In this model, a shortage is considered, and during the shortage period, demand is partially backlogged. Depending on the low partial backlogging rate and high partial backlogging rate, the result of the memory effect varies on the total average cost and optimal ordering interval. Moreover, in order to show the relationship between fractional models and ordinary classical model, two types of memory indices have been considered: (i) differential memory index and (ii) integral memory index. For a certain memory effect, minimized total average cost is the same for low partial backlogging rate and high partial backlogging rate.

An investigation of a closed-form solution for non-linear variable-order fractional evolution equations via the fractional Caputo derivative

Determining the non-linear traveling or soliton wave solutions for variable-order fractional evolution equations (VO-FEEs) is very challenging and important tasks in recent research fields. This study aims to discuss the non-linear space–time variable-order fractional shallow water wave equation that represents non-linear dispersive waves in the shallow water channel by using the Khater method in the Caputo fractional derivative (CFD) sense. The transformation equation can be used to get the non-linear integer-order ordinary differential equation (ODE) from the proposed equation. Also, new exact solutions as kink- and periodic-type solutions for non-linear space–time variable-order fractional shallow water wave equations were constructed. This confirms that the non-linear fractional variable-order evolution equations are natural and very attractive in mathematical physics.

Traveling Wave Solutions for Time-Fractional mKdV-ZK Equation of Weakly Nonlinear Ion-Acoustic Waves in Magnetized Electron–Positron Plasma

In this paper, we discuss the time-fractional mKdV-ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on the properties of certain employed truncated M-fractional derivatives, we reduce the time-fractional mKdV-ZK equation to an integer-order ordinary differential equation utilizing an adequate traveling wave transformation. Further, we derive a dynamical system to present bifurcation of the equation equilibria and show existence of solitary and kink singular wave solutions for the time-fractional mKdV-ZK equation. Furthermore, we establish symmetric solitary, kink, and singular wave solutions for the governing model by using the ansatz method. Moreover, we depict desired results at different physical parameter values to provide physical interpolations for the aforementioned equation. Finally, we introduce applications of the governing model in detail.

A New COVID-19 Pandemic Model including the Compartment of Vaccinated Individuals: Global Stability of the Disease-Free Fixed Point

Owing to the COVID-19 pandemic, which broke out in December 2019 and is still disrupting human life across the world, attention has been recently focused on the study of epidemic mathematical models able to describe the spread of the disease. The number of people who have received vaccinations is a new state variable in the COVID-19 model that this paper introduces to further the discussion of the subject. The study demonstrates that the proposed compartment model, which is described by differential equations of integer order, has two fixed points, a disease-free fixed point and an endemic fixed point. The global stability of the disease-free fixed point is guaranteed by a new theorem that is proven. This implies the disappearance of the pandemic, provided that an inequality involving the vaccination rate is satisfied. Finally, simulation results are carried out, with the aim of highlighting the usefulness of the conceived COVID-19 compartment model.

NOVEL BRIGHT AND KINK OPTICAL SOLITON SOLUTIONS OF FRACTIONAL LAKSHMANAN–PORSEZIAN–DANIEL EQUATION WITH KERR LAW NONLINEARITY IN CONFORMABLE SENSE

The fractional Lakshmanan–Porsezian–Daniel equation (LPD) is a significant complex model for the fractional Schrödinger family which arises in quantum physics. This paper explores new bright and kink soliton solutions of the space-time fractional LPD equation with the Kerr law of nonlinearity. By considering the conformable derivatives, the governing model is translated into integer-order differential equations with the aid of an appropriate complex traveling wave transformation. Dynamic behavior and phase portrait of traveling wave solutions are investigated. Further, various types of bright and kinked soliton solutions under definite parametric settings are discussed. Moreover, graphical representations of the obtained solution of the diverse fractional order are depicted to naturally illustrate the constructed solution.

Stieltjes integral boundary value problem involving a nonlinear multi-term Caputo-type sequential fractional integro-differential equation

&lt;abstract&gt;&lt;p&gt;In this article, we analyze the existence and uniqueness of mild solution to the Stieltjes integral boundary value problem involving a nonlinear multi-term, Caputo-type sequential fractional integro-differential equation. Krasnoselskii's fixed-point theorem and the Banach contraction principle are utilized to obtain the existence and uniqueness of the mild solution of the aforementioned problem. Furthermore, the Hyers-Ulam stability is obtained with the help of established methods. Our proposed model contains both the integer order and fractional order derivatives. As a result, the exponential function appears in the solution of the model, which is a fundamental and naturally important function for integer order differential equations and its many properties. Finally, two examples are provided to illustrate the key findings.&lt;/p&gt;&lt;/abstract&gt;

A stability analysis for multi-term fractional delay differential equations with higher order

As a widely used tool modeling some processes and systems in a variety of fields, fractional delay differential equations (FDDEs) with higher order have attracted much attention of the scientific community for years. Motivated by Yao et al. (2022), in which a single term has been done, we are much more interested in the stability analysis for multi-term FDDEs. In addition to the widely used Laplace transform method and decoupling technique for the characteristic equation, a region embedding technique is first introduced to handle the multiple fractional exponents. The existing results are generalized to multi-term FDDEs with higher order and the damping term of the classical integer-order delay differential equation is extended to fractional calculus. Numerical simulations for FDDEs and time-fractional telegraph equations with time delay are presented to illustrate the efficiency and validity of our results.