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In this work, we develop multi-wave, homoclinic breathers, M-shaped rational, 1-kink interactions with M-shaped, periodic-cross rational and kink-cross rational solutions for the fifth-order Sawada-Kotera equation, which represents the motion of long waves under gravity in shallow water, using several ansatz transformations. A three-wave approach is used to identify multi-wave solitons. Additionally, novel forms of exact solutions are constructed using the homoclinic breathers technique. The kink-cross rational and periodic-cross rational solitons are investigated using appropriate transformations. We develop M-shaped solitons and demonstrate their behavior by selecting suitable parameter values. Furthermore, the interactions between kink waves and M-shaped solitons are also studied. The obtained solutions are determined in 3D, 2D, and contour profiles, where the free parameters involved in the solutions are assigned specific values. Their physical relevance is explored to highlight the inner context of tangible incidents in the natural domain. Since these recently discovered solutions contain a few arbitrary constants, they can be used to explain the variation in qualitative characteristics of wave phenomena.

One of the fastest growing fields of study in nonlinear optics is optical solitons. This article investigates the optical soliton solutions of the coupled Kaup-Newell equation in birefringent fibers without four wave mixing. This model helps explain how modified structures, particularly Alfven waves, propagate in optical fiber and plasma physics. The results presented in this research clearly show how well the new mapping method handle nonlinear Kaup-Newell equation in coupled vector form, in birefringent fiber without four-wave mixing, to extract the analytical soliton solutions. Four-wave mixing expresses a nonlinear optical effect in which four waves interact each other as a result of the third order nonlinearity. As a valuable outcome of this article, a variety of solitons are discovered including bright, dark, periodic, singular bell-shaped, gap, flat-kink, dark-singular, and w-shaped. The existence criteria for these solitons are also obtained that are presented as constraint conditions. The constraint relations guarantee the existence of such solitons. Furthermore, graphs in 2D, 3D and contour are created to show the physical characteristics of the solutions that are produced. The new mapping method is clear and effective, and the solutions provide a wealth of avenues for additional research. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. The outcomes presented in this work are new versions which have never been observed before in existing literature.

Abstract The present research investigates the double-chain deoxyribonucleic acid model, which is important for the transfer and retention of genetic material in biological domains. This model is composed of two lengthy uniformly elastic filaments, that stand in for a pair of polynucleotide chains of the deoxyribonucleic acid molecule joined by hydrogen bonds among the bottom combination, demonstrating the hydrogen bonds formed within the chain’s base pairs. The modified extended Fan sub equation method effectively used to explain the exact travelling wave solutions for the double-chain deoxyribonucleic acid model. Compared to the earlier, now in use methods, the previously described modified extended Fan sub equation method provide more innovative, comprehensive solutions and are relatively straightforward to implement. This method transforms a non-linear partial differential equation into an ODE by using a travelling wave transformation. Additionally, the study yields both single and mixed non-degenerate Jacobi elliptic function type solutions. The complexiton, kink wave, dark or anti-bell, V, anti-Z and singular wave shapes soliton solutions are a few of the creative solutions that have been constructed utilizing modified extended Fan sub equation method that can offer details on the transversal and longitudinal moves inside the DNA helix by freely chosen parameters. Solitons propagate at a consistent rate and retain their original shape. They are widely used in nonlinear models and can be found everywhere in nature. To help in understanding the physical significance of the double-chain deoxyribonucleic acid model, several solutions are shown with graphics in the form of contour, 2D and 3D graphs using computer software Mathematica 13.2. All of the requisite constraint factors that are required for the completed solutions to exist appear to be met. Therefore, our method of strengthening symbolic computations offers a powerful and effective mathematical tool for resolving various moderate nonlinear wave problems. The findings demonstrate the system’s potentially very rich precise wave forms with biological significance. The fundamentals of double-chain deoxyribonucleic acid model diffusion and processing are demonstrated by this work, which marks a substantial development in our knowledge of double-chain deoxyribonucleic acid model movements.

The thin-film ferroelectric materials equation is a fundamental mathematical model to explain the behavior of polarized electric fields in ferroelectric materials possessing unique properties such as piezoelectricity, pyroelectricity, electrostriction, hysteresis, spontaneous polarization, etc. These significant properties of ferroelectric materials are important in the progress of capacitors, non-volatile memories, electrical sensors, as well as optical and biomedical devices, among other applications. The aim of the present study is to explore applicable soliton solutions to the fractional-order model to the aforementioned equation by addressing the improved F-expansion method. Through analysis and rigorous identification, various radical solutions, including trigonometric, hyperbolic, and rational forms of the model, are established, and significant solitons are restored by setting appropriate parameter values in the analytical solutions obtained, which are demonstrated through 3D, 2D, and contour plots. We also explore the impact of fractional derivative as well as the parameters, like pressure, temperature, and others included in the model to determine the polarization effect in ferroelectric materials. This study uncovers several novel solutions along with their states in response to external or internal field parameters, which might be helpful in understanding the nature of ferroelectric materials.

The response of neurons is highly sensitive to the stimulus. The stimulus can be associated with a direct injection in vitro experimentation (e.g., time dependent and independent inputs); or post-synaptic potentials resulting from the interaction of many neurons. A typical incoming stimulus resembles a noise which in principle can be described as a random variable. In computational neuroscience, the noise has been extensively studied for different setups. In this study, we investigate the effect of noisy inputs in a minimal network of two identical leaky integrate-and-fire (LIF) neurons interacting with finite pulses. In particular, we consider a Gaussian white noise as a standard function for stochastic modelling of neurons, while taking into account the pulse width as an elementary component for the signal transmission. By exploring the role of noise and finite pulses, the two neurons show a synchronous spiking behaviour characterized by fluctuations in the interspike intervals. Above some critical values the synchronous regime collapses onto asynchronous dynamics. The abrupt change in such dynamics is accompanied by a hysteresis, i.e., the coexistence of synchronous and asynchronous firing behaviour.

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots.

The time fractional Schrödinger equation contributes to our understanding of complex quantum systems, anomalous diffusion processes, and the application of fractional calculus in physics and cubic B-spline is a versatile tool in numerical analysis and computer graphics. This paper introduces a numerical method for solving the time fractional Schrödinger equation using B-spline functions and the Atangana-Baleanu fractional derivative. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while a θ-weighted scheme is used to discretize the space directions. The efficiency of the method is demonstrated through numerical results, and error norms are examined at various values of the non-integer parameter, temporal directions, and spatial directions.

The B-spline function is made up of a set of smooth piecewise polynomials that are controlled by a set of control points. A linear combination of B-spline basis of a particular degree can be used to express any spline function of that degree. The spline functions and their derivatives are continuous and depending on the multiplication of knots. In this study, an extended cubic B-spline (ECBS) approximation is used for the numerical solution of time-fractional advection diffusion equation (TFADE) involving Atangana-Baleanu fractional derivative (ABFD). Initially, the non-singular kernel ABFD is discretized using finite difference method (FDM). The spatial derivatives are discretized using ECBS functions. Convergence and stability of the proposed scheme are studied. The results tabulated in tables that show how they accurate and desirable the outcomes are by comparing the obtained approximate results with the available exact solutions. To the authors’ knowledge, this work is the first time to use the ABFD for numerical solution of time-fractional advection diffusion equation using ECBS.

The time-fractional regularized long wave equation is a significant model in the study of ocean engineering, plasma physics, and fluid dynamics. The fractional derivative is contemplated in the beta derivative sense, which is consistent with all the principles of derivative. In this article, the reliable and viable two-variable (G'/G,1/G)-expansion approaches is instigated to generate a variety of novel and generic wave outcomes. Consequently, the presented research provides abundant robust soliton solutions, including periodic soliton, bell-shaped soliton, W-shaped soliton, and some other type solitons. The effect of the fractional-order derivative has also been shown, and it is noticed that the fractional-order derivative has a significant impact on the wave profile. The physical conduct of the solitons has been demonstrated thoroughly via three- and two-dimensional and contour graphs. It can be noted that the new solutions obtained will provide greater convenience in the study of the phenomena described by the regular long-wave equation.

The nonlinear perturbed Radhakrishnan-Kundu-Lakshmanan (RKL) equation is a significant model in the engineering field for analyzing data transmission through nonlinear optical fibers. The two-variable (ψ′/ψ,1/ψ)-expansion method is a reliable scheme to compute self-controlled solitary wave solutions. In this study, we have constructed a wide range of geometric shapes and further inclusive soliton solutions comprising rational, trigonometric, and hyperbolic functions and their integration to the stated equation, devising the acknowledged method. The physical details of the ensued solitary wave solutions and the effects of fractional parameters are explained by sketching three- and two-dimensional graphs. For the appropriate values of the parameters, the periodic, V-shaped, bell-shaped, compacton, singular periodic, flat bell-shaped, plane-shaped, and some other types of solitons are formulated. The effect of the fractional derivative has been demonstrated by plotting the two-dimensional graphs for different values of the fractional order. It is important to note that changes in physical and auxiliary parameters cause variations in the wave profile. The formerly described physical phenomena may be better understood using the standard and new types of solitons. The results obtained exhibit the effectiveness, compatibility, and reliability of the two-variable (ψ′/ψ,1/ψ)-expansion approach for extracting soliton solutions of fractional-order nonlinear evolution equations in science, technology, and engineering.

The analytical soliton solutions place a lot of value on birefringent fibres. The major goal of this study is to generate novel forms of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation, which depicts unstable optical solitons that arise from optical propagations using birefringent fibres. The (presumably new) extended direct algebraic (EDA) technique is used here to extract a large number of solutions for RKLE. It gives soliton solutions up to thirty-seven, which essentially correspond to all soliton families. This method's ability to determine many sorts of solutions through a single process is one of its key advantages. Additionally, it is simple to infer that the technique employed in this study is really straightforward yet one of the quite effective approaches to solving nonlinear partial differential equations so, this novel extended direct algebraic (EDA) technique may be regarded as a comprehensive procedure. The resulting solutions are found to be hyperbolic, periodic, trigonometric, bright and dark, combined bright-dark, and W-shaped soliton, and these solutions are visually represented by means of 2D, 3D, and density plots. The present study can be extended to investigate several other nonlinear systems to understand the physical insights of the optical propagations through birefringent fibre.

In this research work, we construct an epidemic model to understand COVID-19 transmission vaccination and therapy considerations. The model's equilibria were examined, and the reproduction parameter was calculated via a next-generation matrix method, symbolized by $ \mathcal{R}_0 $. We have shown that the infection-free steady state of our system is locally asymptotically stable for $ \mathcal{R}_0 < 1 $. Also, the local asymptotic stability of the endemic steady state has been established for $ \mathcal{R}_0 > 1 $. We have used a partial rank correlation coefficient method for sensitivity analysis of the threshold parameter $ \mathcal{R}_0 $. The contribution of vaccination to the threshold parameter is explored through graphical results. In addition to this, the uniqueness and existence of the solution to the postulated model of COVID-19 infection is shown. We ran various simulations of the proposed COVID-19 dynamics with varied input parameters to scrutinize the complex dynamics of COVID-19 infection. We illustrated the variation in the dynamical behavior of the system with different values of the input parameters. The key factors of the system are visualized for the public health officials for the control of the infection.

Non-local fractional derivatives are generally more effective in mimicking real-world phenomena and offer more precise representations of physical entities, such as the oscillation of earthquakes and the behavior of polymers. This study aims to solve the 2D fractional-order diffusion-wave equation using the Riemann–Liouville time-fractional derivative. The fractional-order diffusion-wave equation is solved using the modified implicit approach based on the Riemann–Liouville integral sense. The theoretical analysis is investigated for the suggested scheme, such as stability, consistency, and convergence, by using Fourier series analysis. The scheme is shown to be unconditionally stable, and the approximate solution is consistent and convergent to the exact result. A numerical example is provided to demonstrate that the technique is more workable and feasible.

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second order derivative approximation to obtain the numerical solution of the weakly singular kernel (SK) non-linear fractional partial integro-differential equation (FPIDE). The spatial and temporal fractional derivatives are discretized by ExCuBS and the Caputo finite difference scheme, respectively. The present study found that it is stable and convergent. The validity of the current approach is examined on a few test problems, and the obtained outcomes are compared with those that have previously been reported in the literature.

In this paper, a fractional-order Hantavirus infection model incorporating harvesting is formulated and investigated. The populations are divided into susceptible mice, infected mice and alien species. Mathematical analysis and numerical simulations are performed to clarify the characteristics of the proposed fractional-order Hantavirus infection model. The existence, uniqueness, non-negativity and boundedness of the solutions are examined. The local stability of the equilibrium points of the fractional-order model is studied. The mathematical proof of the existence of transcritical bifurcation is given by using Sotomayor’s theorem. The theoretical findings are illustrated by numerical simulations. The impact of fractional-order, competitive effect of alien species on mice, competitive effect of mice on alien species, carrying capacity and harvesting efforts on the stability of the Hantavirus infection model are studied. The basin of attraction regions is also illustrated.

This study, considers the fractional order cable model (FCM) in the sense of Riemann–Liouville fractional derivatives (R-LFD). We use a modified implicit finite difference approximation to solve the FCM numerically. The Fourier series approach is used to examine the proposed scheme’s theoretical analysis, including stability and convergence. The scheme is shown to be unconditionally stable, and the approximate solution converges to the exact solution. To demonstrate the application and feasibility of the proposed approach, a numerical example is provided.

In this article, developed the compact implicit difference method based Grünwald Letnikov formula (GLF) to compute the solution of the time-fractional diffusion-wave equation (TFDWE) describing wave propagation phenomenan having order α (1

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.

For uni-directional wave transmission in the smooth bottom of shallow sea water and the superconductivity of nonlinear media with dispersion systems, the (1 + 1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations are of particular interest in research to the academics. Analytical wave solutions to the stated models have been successfully constructed in this study, which might have considerable implications in describing the nonlinear dynamical behavior associated with the phenomena. The models we aim to uncover have been put into the form of differential equations with one characteristic variable expending the wave transformation coordinate and, thereupon, the rational (G′/G)-expansion technique is executed. Using the considered technique, diverse soliton solutions in suitable forms arrayed to trigonometric, rational, and hyperbolic functions have been determined. The achieved solutions are figured out in 3D profiles, assigning the free parameters involved in solutions to particular values and discussed their physical significance to bring out the inner context of the tangible incidents in the natural domain. The rational (G′/G)-expansion approach is efficient, concise, and capable of finding analytical solutions to other nonlinear models that can be considered in subsequent studies.

Nonlinear fractional-order evolution equations are fundamental strategies for simulating nonlinear phenomena on a large scale in technology, science, and engineering. This article considers the nonlinear space–time fractional perturbed nonlinear Schrödinger (NLS) equation defined in the sense of the beta-derivative, a widely used model to simulate nonlinear optics, ultra-short pulse lasers, optical communication systems, plasmas, and other domains. The model is converted into a nonlinear equation defined by fractional wave transformation. We construct diverse sorts of soliton solutions in the integration of rational, trigonometric, exponential, and hyperbolic functions with open parameters using the typical generalized exponential rational function (GERF) and the improved Bernoulli sub-equation function (IBSEF) schemes. Some standard shapes of waveforms, including parabolic soliton, kink, flat kink, bright-dark soliton, periodic soliton, breather, breathing periodic, singular periodic, breathing-like soliton, flat parabolic, and several other types of solitons, are determined. Periodic solitons under perturbations are required to understand ultra-short pulse lasers and long-distance optical communication systems. To validate the physical characteristics of the established solitons, we sketch 3D, 2D, and contour plots of the solutions using consistent values of the parameters. The techniques used in this study to extract inclusive and standard solutions are approachable, efficient, and speedier to compute the soliton solutions for nonlinear models in communication engineering and optics.

The B-spline or the basis spline function, is a piecewise polynomial function made up of polynomials, its domain is subdivided by knots, and basis functions are non-zero on the entire domain. In this study, a new cubic B-spline (NCBS) approximation together with the θ-weighted scheme is formed to approximate the numerical solution of the time fractional advection diffusion equation (TFADE) involving the Atangana–Baleanu time fractional derivative (ABTFD). The finite difference scheme (FDS) is utilized to discretize the ABTFD with a non-singular kernel. The spatial derivative is discretized by using NCBS functions. The stability analysis of the proposed technique is studied. Convergence analysis of the current technique is also analyzed. The proposed technique is examined on a variety of problems, and the numerical outcomes are contrasted with the previously published technique’s results to ensure the correctness and accuracy of the current technique.

The Sharma-Tasso-Olver and Klein–Gordon equations are significant models to interpret plasma physics, relativistic physics, quantum mechanics, nonlinear optics, etc. In the present article, an advanced generalized approach, namely the generalized (G′/G)-expansion approach is scheduled to unfold certain wave solutions to the nonlinear evolution equations earlier described. The soliton solutions exposed are estimated in the form of hyperbolic, rational and trigonometric functions. The physical implications of the ascertained solutions are summed up by setting specific values of the integrated constraints and delineating the profiles to decode the physical phenomena. The numerical simulation of the solutions extracted is standard kink, rogue wave, bright-dark soliton, periodic soliton, breather type soliton, compaction, singular kink soliton, etc. This study affirms that the introduced scheme is robust, efficient in searching for nonlinear evolution equations, computer algebra compatible, and capable of finding further inclusive wave solutions.

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

The nonlinear Fokas and breaking soliton equations are important modeling equations for coastal and ocean waves, such as long water waves and tsunami waves. Moreover, these models have effective applications for simulating surface and internal waves in oceans and rivers, wave packets in finite depth water, and Riemann wave propagation interactions. By exploiting the enhanced modified simple equation approach, in this article, we have established further generic, radical, and standard solitons to the erstwhile nonlinear framework with some parameters. The reputed schematic solitons, such as bell-shaped, periodic, cuspon, compacton, plane-shaped, singular soliton, kink, and other solitons, are devised for varied scores of parameters. The accomplished results might assist in better understanding the dynamics of coastal waves modulated by the formerly acknowledged couple of nonlinear models. We have studied the impacts of physical parameters and propagation rate on wave structure and affirmed their role in changing the waveform. Three- and two-dimensional diagrams are sketched to interpret the context profiles of the obtained solitary wave solutions. The established solutions demonstrate the applicability and competence of the approach to investigating nonlinear models.

Quantum field theory and solid-state physics phenomena are interpreted using the Tzitzeica-Dodd-Bullough model and nonlinear optics and electromagnetic waves are studied through the Dodd-Bullough-Mikhailov equation. The nonlinear Cahn-Allen model is important for describing the evolution of non-conserved order fields during anti-period sphere coarsening and the phase separation reaction–diffusion processes in multicomponent alloy structures. In this article, we have assessed the advanced and broad-spectrum solitary wave solutions to the stated models and studied the effects of wave speed as well as physical coordinates on the wave profiles and affirmed that waveform varies with the change of the free parameters associated to them. The soliton solutions are extracted by balancing the exponents of the highest-order linear and nonlinear terms. To thoroughly analyze the wave characteristic of the closed-form solutions, different standard wave profiles including kink, bell-shaped soliton, anti-peakon, periodic, quasi-periodic, complex singular, and several general solitons are depicted. Both Painlevé and traveling wave transformation play important roles in converting the equations into nonlinear differential equations. The solutions formulated in this probe are further generic, wide-spectral and some are distinctive which have not been established in the former studies.

The current paper uses the cubic B-spline functions and θ-weighted scheme to achieve numerical solutions of the time fractional Burgers’ equation with Atangana–Baleanu derivative. A non-singular kernel is involved in the Atangana–Baleanu fractional derivative. For discretization along temporal and spatial grids, the proposed numerical technique employs the finite difference approach and cubic B-spline functions, respectively. This scheme is unconditionally stable and second order convergent in spatial and temporal directions. The presented approach is endorsed by some numerical examples, which show that it is applicable and accurate.

The purpose of this investigation is to examine the phenomenon of a slip flow of upper-convected Maxwell (UCM) nanofluid generated by an inclined stretching sheet through a porous medium and a magnetic field. The diffusion model for mass and heat transfer introduced by Cattaneo and Christov is incorporated in the modeling process. The relaxation framework of visco-elastic system is formulated for UCM nanofluid to determine both heat and mass transfer by the Cattaneo-Christov model. Thermophoresis, Brownian motion and heat generation effects in the presence of chemical reaction have also been incorporated. Similarity transformations are used to achieve the dimensionless form of governing equations. These dimensionless equations have been solved numerically with the shooting technique. The present analysis primarily focuses on the research of important parameters appearing in the governing equations and their impacts on − θ′(0), − φ′(0), − f′′(0), velocity, concentration and temperature distributions. The velocity inside the boundary layer is decreased for the augmented Maxwell parameter while the temperature is upturned. Moreover, it is noted that a reduction in the non-dimensional velocity in a porous medium for larger magnetic parameter is more significant as compared with a non-porous medium. The heat transfer rate increases significantly as the Prandtl number increases whereas it declines for the Maxwell and thermal relaxation parameters.

Abstract A fractional-order eco-epidemiological model with disease in the prey population is formulated and analyzed. Mathematical analysis and numerical simulations are performed to clarify the characteristics of the proposed fractional-order model. The existence, uniqueness, non-negativity and boundedness of the solutions are proved. The local and global asymptotic stability of all equilibrium points are investigated. Finally, numerical simulations are conducted to illustrate the analytical results. The occurrence of Hopf bifurcations and transcritical bifurcations for the fractional-order eco-epidemiological model are demonstrated. It is observed that the fractional order has a stabilization effect and it may help to control the coexistence between susceptible prey, infected prey and predator populations.

Tuberculosis (TB), caused by Mycobacterium tuberculosis is one of the treacherous infectious diseases of global concern. In this paper, we consider a deterministic model of TB infection with the public health education and hospital treatment impact. The effective reproductive number, Rph, that measures the potential spread of TB is presented by employing the next generation matrix approach. We investigate local and global stability of the TB-free equilibrium point, endemic equilibrium point, and sensitivity analysis. The analyses of the proposed model show that the model undergoes the phenomenon of backward bifurcation when the effective reproduction number (Rph) is less than one, where two stable equilibria, namely, the DFE and an EEP coexist. Further, we compute the sensitivity of the impact of each parameter on the effective reproductive number of the model by employing a normalized sensitivity index formula. Numerical simulation of the proposed model was conducted using Maple 2016 and MatLab R2020b software and compared with the theoretical results for illustration purposes. The investigation results can be useful in providing information to policy makers and public health authorities in mitigating the spread of TB infection by public health education and hospital treatment.

Fractional differential equations describe nature adequately because of the symmetry properties that describe physical and biological processes. In this paper, a new approximation is found for the variable-order (VO) Riemann–Liouville fractional derivative (RLFD) operator; on that basis, an efficient numerical approach is formulated for VO time-fractional modified subdiffusion equations (TFMSDE). Complete theoretical analysis is performed, such as stability by the Fourier series, consistency, and convergence, and the feasibility of the proposed approach is also discussed. A numerical example illustrates that the proposed scheme demonstrates high accuracy, and that the obtained results are more feasible and accurate.

Tuberculosis (TB) is a serious global health threat that is caused by the bacterium Mycobacterium tuberculosis, is extremely infectious, and has a high mortality rate. In this paper, we developed a model of TB infection to predict the impact of saturated recovery. The existence of equilibrium and its stability has been investigated based on the effective reproduction number RC. The model displays interesting dynamics, including backward bifurcation and Hopf bifurcation, which further results in the stable disease-free and stable endemic equilibria to be coexisting. Bifurcation analysis demonstrates that the saturation parameter is accountable for the phenomenon of backward bifurcation. We derive a condition that guarantees that the model is globally asymptotically stable using the Lyapunov function approach to global stability. The numerical simulation also reveals that the extent of saturation of TB infection is the mechanism that is fuelling TB disease in the population.

Abstract In this paper, we solve two-dimensional modified anomalous fractional sub-diffusion equation using modified implicit finite difference approximation. The stability and convergence of the proposed scheme are analyzed by the Fourier series method. We show that the scheme is unconditionally stable and the approximate solution converges to the exact solution. A numerical example is given to show the application and feasibility of the proposed scheme.

This study extends a deterministic mathematical model for the dynamics of tuberculosis transmission to examine the impact of an imperfect vaccine and other exogenous factors, such as re-infection among treated individuals and exogenous re-infection. The qualitative behaviors of the model are investigated, covering many distinct aspects of the transmission of the disease. The proposed model is observed to show a backward bifurcation, even when Rv<1. As such, we assume that diminishing Rv to less than unity is not effective for the elimination of tuberculosis. Furthermore, the results reveal that an imperfect tuberculosis vaccine is always effective at reducing the spread of infectious diseases within the population, though the general effect increases with the increase in effectiveness and coverage. In particular, it is shown that a limited portion of people being vaccinated at steady-state and vaccine efficacy assume a equivalent role in decreasing disease burden. From the numerical simulation, it is shown that using an imperfect vaccine lead to effective control of tuberculosis in a population, provided that the efficacy of the vaccine and its coverage are reasonably high.

The variable-order (VO) fractional calculus can be seen as a natural extension of the constant-order, which can be utilized in physical and biological applications. In this study, we derive a new numerical approximation for the VO fractional Riemann–Liouville integral formula and developed an implicit difference scheme (IDS) for the variable-order fractional sub-diffusion equation (VO-FSDE). The derived approximation used in the VO time fractional derivative with the central difference approximation for the space derivative. Investigated the unconditional stability by the van Neumann method, consistency, and convergence analysis of the proposed scheme. Finally, a numerical example is presented to verify the theoretical analysis and effectiveness of the proposed scheme.

Fractional differential equations describe nature adequately because of the symmetry properties which describe physical and biological processes. In this article, a fourth-order new implicit difference scheme is formulated and applied to solve the two-dimensional time-fractional modified sub-diffusion equation involving two times Riemann–Liouville fractional derivatives. The stability of the fourth-order implicit difference scheme is investigated using the von Neumann technique. The proposed scheme is shown to be unconditionally stable. Numerical examples are given to illustrate the feasibility of the proposed scheme.

In this article, new extension of the generalized and improved (G′/G)-expansion method is proposed for constructing more general and a rich class of new exact traveling wave solutions of nonlinear evolution equations. To demonstrate the novelty and motivation of the proposed method, we implement it to the Korteweg-de Vries (KdV) equation. The new method is oriented toward the ease of utilize and capability of computer algebraic system and provides a more systematic, convenient handling of the solution process of nonlinear equations. Further, obtained solutions disclose a wider range of applicability for handling a large variety of nonlinear partial differential equations.

The generalized and improved (G'/G)-expansion method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. In this article, we investigate the higher dimensional nonlinear evolution equation, namely, the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation via this powerful method. The solutions are found in hyperbolic, trigonometric and rational function form involving more parameters and some of our constructed solutions are identical with results obtained by other authors if certain parameters take special values and some are new. The numerical results described in the figures were obtained with the aid of commercial software Maple.

In this article, we investigate the compound KdV-Burgers equation involving parameters by applying the improved (G′/G)-expansion method for constructing some new exact traveling wave solutions including solitons and periodic solutions. The second order linear ordinary differential equation with constant coefficients is used, in this method. The obtained solutions are presented through the hyperbolic, the trigonometric and the rational functions. Further, it is significant to point out that some of our solutions are in good agreement for special cases with the existing results which validates our other solutions. Moreover, some of the obtained solutions are described in the figures.

The generalized Riccati equation mapping is extended with the basic (G′/G)-expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equation G′(η)=w+uG(η)+vG2(η) is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.

we apply the improved (G′/G)-expansion method for constructing abundant new exact traveling wave solutions of the (2+1)-dimensional Modified Zakharov-Kuznetsov equation. In addition, G″+λG′+μG=0 together with b(α)=∑q=−wwpq(G′/G)q is employed in this method, where pq (q=0,±1,±2,…,±w), λ and μ are constants. Moreover, the obtained solutions including solitons and periodic solutions are described by three different families. Also, it is noteworthy to mention out that, some of our solutions are coincided with already published results, if parameters taken particular values. Furthermore, the graphical presentations are demonstrated for some of newly obtained solutions.

We construct new analytical solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.

In this article, the generalized Riccati equation mapping together with the basic (G′/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G′(ϕ) = h + f G(ϕ) + g G2(ϕ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.

In this article, new (G′/G)-expansion method and new generalized (G′/G)-expansion method is proposed to generate more general and abundant new exact traveling wave solutions of nonlinear evolution equations. The novelty and advantages of these methods is exemplified by its implementation to the KdV equation. The results emphasize the power of proposed methods in providing distinct solutions of different physical structures in nonlinear science. Moreover, these methods could be more effectively used to deal with higher dimensional and higher order nonlinear evolution equations which frequently arise in many scientific real time application fields.