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In this paper, a local fractional Riccati differential equation method is applied. A new travelling wave solution to the nonlinear local fractional (2 + 1) -dimensional dispersive long water wave system is investigated. After travelling wave transformation, the governing model studied is converted into nonlinear ordinary differential equation. Some properties with the strain conditions are also reported. [ABSTRACT FROM AUTHOR]

This study aims to address the difficulties in solving coupled generalized non-linear Burger equations using local fractional calculus as a framework. The methodology used in this work, particularly in the area of local fractional calculus, combines the Elzaki transform with the Adomian decomposition method. This combination has proven to be a highly effective strategy for addressing non-linear partial differential equations within the local fractional context, which finds numerous practical applications. The proposed method offers a systematic and easily understandable procedure for tackling both linear and non-linear partial differential equations (PDEs). It provides an easy-to-follow path to solve these problems. We offer a real-world example that exhibits the method's successful use in resolving issues to corroborate its efficacy. The obtained solution is visually represented to illustrate the practical utility of this approach. [ABSTRACT FROM AUTHOR]

This mathematical model is based on a system of two non-linear delay differential equations representing GDP (Gross Domestic Product) growth and national debt respectively. Growth of GDP is directly proportional to national debt. In the absence of national debt, there could be no or very slow GDP growth. The national debt is heavily dependent on external loans and external investments. These investments act as a catalyst for accelerating the rate of GDP. It is assumed that the external debt is never paid off fully. The availability of external investments is not immediate as per demand but takes some time for the maturation of the deal. The time delay in actual arrival of the foreign investment and its effect on GDP-National debt dynamics is the main focus of this study. This effect is studied using a delay parameter τ. The non-zero equilibrium of the system is calculated, and the stability analysis is performed on it. Hopf bifurcation is observed for a critical value of the delay parameter. Numerical simulation is performed using MATLAB code. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear Schrödinger equation in the sense of conformable derivative. Bifurcations and phase portraits are first proposed by using bifurcation theory, which investigates the dynamical behavior of this equation. This bifurcation theory classifies the plausible solutions to infinite periodic wave solutions, periodic wave solutions, two kink (anti-kink) wave solutions, and two families of breaking wave solutions. A generalized double auxiliary equation approach that generates three families of exact exact traveling wave solutions is then proposed using the conformable operator under various parameter conditions. The 3D behavior of various solutions with absolute real and imaginary parts is displayed. The obtained results show that the proposed methodology is efficient and applicable to a broad class of conformable nonlinear partial differential equations in mathematical physics. [ABSTRACT FROM AUTHOR]

The time-fractional generalized biological population model and the (2, 2, 2) Zakharov–Kuznetsov (ZK) equation are significant modeling equations to analyse biological population, ion-acoustic waves in plasma, electromagnetic waves, viscoelasticity waves, material science, probability and statistics, signal processing, etc. The new generalized G ′ / G -expansion method is consistent, computer algebra friendly, worthwhile through yielding closed-form general soliton solutions in terms of trigonometric, rational and hyperbolic functions associated to subjective parameters. For the definite values of the parameters, some well-established and advanced solutions are accessible from the general solution. The solutions have been analysed by means of diagrams to understand the intricate internal structures. It can be asserted that the method can be used to compute solitary wave solutions to other fractional nonlinear differential equations by means of fractional complex transformation. [ABSTRACT FROM AUTHOR]

In this paper, the fractional natural decomposition method (FNDM) is employed to find the solution for the KunduEckhaus equation and coupled fractional differential equations describing the massive Thirring model. The massive Thirring model consists of a system of two nonlinear complex differential equations, and it plays a dynamic role in quantum field theory. The fractional derivative is considered in the Caputo sense, and the projected algorithm is a graceful mixture of Adomian decomposition scheme with natural transform technique. In order to illustrate and validate the efficiency of the future technique, we analyzed projected phenomena in terms of fractional order. Moreover, the behaviour of the obtained solution has been captured for diverse fractional order. The obtained results elucidate that the projected technique is easy to implement and very effective to analyze the behaviour of complex nonlinear differential equations of fractional order arising in the connected areas of science and engineering [ABSTRACT FROM AUTHOR]

Bernoulli sub‐equation function method is applied to obtain exact solutions of Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation. As a result of this, exact traveling‐wave and some new oscillating solutions to CDGSK are obtained. It may be observed that Bernoulli sub‐equation function method employed here is very effective and reliable to get explicit solutions for this nonlinear partial differential equation. Profiles of all constructed solutions are graphically illustrated entirely as well. [ABSTRACT FROM AUTHOR]

This paper presents a newly developed method, namely, the rational sine-Gordon expansion method to find novel exact solutions to nonlinear differential equations. This method is based on the sine-Gordon expansion method. To generalize the approach, we utilize the ansatz, which is a rational function as different to a polynomial function. In this way, we have more general wave solutions for nonlinear dynamic systems. We apply this method to the (2 + 1)-dimensional conformable Zakharov–Kuznetsov modified equal width (ZK-MEW) equation and the modified regularized long wave (MRLW) equation. Some new solutions are reported. Consequently, we submit the new soliton solutions to the literature. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the behaviour of solution for the system exemplifying model of tumour invasion and metastasis by the help of q-homotopy analysis transform method (q-HATM) with the fractional operator. The analyzed model consists of a system of three nonlinear differential equations elucidating the activation and the migratory response of the degradation of the matrix, tumour cells and production of degradative enzymes by the tumour cells. The considered method is graceful amalgamations of q-homotopy analysis techniquewith Laplace transform(LT), and Caputo-Fabrizio (CF) fractional operator is hired in the present study. By using the fixed point theory, existence and uniqueness are demonstrated. To validate and present the effectiveness of the considered algorithm, we analyzed the considered system in terms of fractional order with time and space. The error analysis of the considered scheme is illustrated. The variations with small change time with respect to achieved results are effectively captured in plots. The obtained results confirm that the considered method is very efficient and highly methodical to analyze the behaviors of the system of fractional order differential equations. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present work is to find the numerical solution for fractional Benney–Lin equation by using two efficient methods, called q‐homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three‐dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology. [ABSTRACT FROM AUTHOR]

In this paper, we study on the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity. The proposed technique is graceful amalgamations of Laplace transform technique with q -homotopy analysis scheme and fractional derivative defined with Atangana–Baleanu (AB) operator. The fixed-point hypothesis is considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to illustrate and validate the efficiency of the future technique, we consider three different cases and analyzed the projected model in terms of fractional order. Moreover, the physical behavior of the obtained solution has been captured in terms of plots for diverse fractional order, and the numerical simulation is demonstrated to ensure the exactness. The obtained results elucidate that the proposed scheme is easy to implement, highly methodical as well as accurate to analyze the behavior of coupled nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we present the sine-Gordon expansion method to prepare the mixed dark bright wave patterns to the nonlinear partial differential equations arising in mathematical physics. Then, we apply the proposed method for a credible recourse of two nonlinear physical models: the modified Vakhnenko-Parkes and modified α -equation. These exact solutions comprise the hyperbolic, trigonometric, rational and exponential function with few licentious parameter. The analytical solutions have different physical structures and they are graphically analyzed in order to show their dynamical behavior by means of 2D, 3D and contour plots. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present work is to find the solution for fractional Drinfeld–Sokolov–Wilson equation using q - homotopy analysis transform method (q -HATM). The proposed technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to validate and illustrate the efficiency of the future technique, we analysed the projected model in terms of fractional order. Meanwhile, the physical behaviour of the q -HATM solutions have been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The achieved results illuminate that, the future algorithm is easy to implement, highly methodical as well as effective and very accurate to analyse the behaviour of coupled nonlinear differential equations of fractional order arisen in the connected areas of science and engineering. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present work is to obtain an approximated analytical solution for the fractional smoking epidemic model with the aid of a novel technique called q-homotopy analysis transform method (q-HATM). The considered nonlinear mathematical model has been effectively employed to elucidate the evolution of smoking in a population and its impact on public health in a community. We find some new approximate solutions in a series form, which converges rapidly, and the proposed algorithm provides auxiliary parameters, which are very reliable and feasible in controlling the convergence of obtained approximate solutions. Further, we present novel simulations for all cases of results to validate the applicability and effectiveness of proposed scheme. The outcomes of the study reveal that the q-HATM is computationally very effective to analyse nonlinear fractional differential equations arises in daily life problems. [ABSTRACT FROM AUTHOR]

In the present work, an efficient numerical technique, called q-homotopy analysis transform method (briefly, q -HATM), is applied to nonlinear Fisher's equation of fractional order. The homotopy polynomials are employed, in order to handle the nonlinear terms. Numerical examples are illustrated to examine the efficiency of the proposed technique. The suggested algorithm provides the auxiliary parameters ħ and n , which help us to control and adjust the convergence region of the series solution. The outcomes of the study reveal that the q -HATM is computationally very effective and accurate to analyse nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

The current study provides the solutions of the mathematical model based on the coronavirus including the effects of vaccination and quarantine. The numerical stochastic process relying on Levenberg-Marquardt backpropagation technique (L-MB) neural networks (NN), i.e., L-MBNNs, is presented to solve the model. The entire dynamics of the proposed model depends upon the human population, which is represented by N and is further divided into multiple subgroups. The detail of these subgroups is presented in the form of susceptible population (S), exposed population (E), and infected people (I). Likewise, Q represents the quarantined and R shows the recovered or deceased individuals. Those who have been immunized are symbolized by V. All these categories make the model SEIQRV, that is based on a system of nonlinear differential equations. The statistics that is used to provide the numerical solutions of the SEIQRV model is 76% for training, 10% for testing and 14% for authorization. The correctness of the L-MBNNs is tested by using the comparison of the proposed and reference solutions (Adam method). The statistical representations are provided in order to check the reliability, competence and validity of L-MBNNs using the procedures of error histograms (EH), state transitions (ST), regression and correlation. [ABSTRACT FROM AUTHOR]

The non dissipative double dispersive micro strain wave in the micro structured solid is solved via the F -expansion method and obtained the new exact doubly periodic Jacobi elliptic function solutions. The F -expansion method converts the nonlinear partial differential equation into ordinary differential equation, then the initially assumed solution converts the ordinary differential equation into systems of algebraic equations. By solving these algebraic equations, the six families of the periodic wave solutions are classified of which three are real valued and three are complex valued. Next by treating the Jacobi elliptic function modulus to 0 (or) 1, some of the new exact solitons and their classifications are obtained. The two dimensional and three dimensional graphs are plotted to show the traversing of the obtained wave trains. • Non dissipative DDMSW is treated with F -expansion method. • Six families of unique doubly periodic solutions are obtained. • Existence condition for the doubly periodic wave solutions are given. • Non-topological, new singular and compound solitons are computed. • 2D and 3D plots are simulated. [ABSTRACT FROM AUTHOR]