Search

This paper investigates the optical soliton solutions of the Lakshmanan–Porsezian–Daniel equation, a nonlinear evolution equation modeling the propagation of optical soliton waves in materials influenced by parabolic law nonlinearity and spatio-temporal dispersion. Employing both the Kudryashov and new Kudryashov methods, the study captures various soliton waves, deriving both dark and bright soliton solutions. The research delves into the effects of the model parameters on these solutions, presenting the findings through detailed 3D and 2D simulation images. Practical and rapid results achieved using both Kudryashov methods are highlighted, positioning this work as a valuable reference for peers. The study's distinct combination of the Lakshmanan–Porsezian–Daniel equation and the chosen analytical techniques not only introduces novelty but also emphasizes practicality and efficiency. This innovative approach, coupled with its significant contribution to the knowledge base, underscores the research's relevance in the field. [ABSTRACT FROM AUTHOR]

Recent decades have witnessed the emergence of interesting models of fractional partial differential equations. In the current work, a class of parabolic equations with regularized Hyper-Bessel derivative and the exponential source is investigated. More specifically, we examine the existence and uniqueness of mild solutions in Hilbert scale-spaces which are constructed by a uniformly elliptic symmetry operator on a smooth bounded domain. Our main argument is based on the Banach principle argument. In order to achieve the necessary and sufficient requirements of this argument, we have smoothly combined the application of the Fourier series supportively represented by Mittag-Leffler functions, with Hilbert spaces and Sobolev embeddings. Because of the presence of the fractional operator, we face many challenges in handling proper integrals which appear in the representation of mild solutions. Besides, the source term of an exponential type also causes trouble for us when deriving the desired results. Therefore, powerful embeddings are used to limit the growth of nonlinearity. [ABSTRACT FROM AUTHOR]

The main motive of this article is to study the recently developed Atangana‐Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag‐Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two‐dimensional cable equation in which time‐fractional derivative is of Mittag‐Leffler kernel type. First, we derive an approximation formula of the fractional‐order ABC derivative of a function tk using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two‐dimensional model of the time‐fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two‐dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag‐Leffler fractional derivative. [ABSTRACT FROM AUTHOR]

In this study, we study an inverse source problem for the time-fractional diffusion equation, where the final data t = T are given. We show that our problem is ill-posed in the sense of Hadamard. Applying a truncation method, we give the regularized solution. Finally, convergence estimates under a priori and a posteriori parameter choice rules are proved. [ABSTRACT FROM AUTHOR]

In this paper we consider the singular conformable sequential equation with distributional potentials. We present Weyl's theory in the frame of conformable derivatives. Moreover we give two theorems on limit-point case. [ABSTRACT FROM AUTHOR]

In this article, we investigate the generalized fractional operator Caputo type (ABC) with kernels of Mittag–Lefller in three parameters Eα,μγ(λt) and its fractional integrals with arbitrary order for solving the time fractional parabolic nonlinear equation. The generalized definition generates infinitely many problems for a fixed fractional derivative α. We utilize this operator with homotopy analysis method for constructing the new scheme for generating successive approximations. This procedure is used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the ℏ‐curves as well as by calculating the residual error and the results were accurate. Based on the experiment, we verify the existence of the solution for the new parameters. Depending on these results, this treatment can be used to find approximate solutions to many fractional differential equations. [ABSTRACT FROM AUTHOR]

This work deals with the conecept of a Caputo-Fabrizio type non-integer variable order differential opertor on time scales that involves a non-singular kernel. A measure theoretic discussion on the limit cases for the order of differentiation is presented. Then, corresponding to the fractional derivative, we discuss on an integral for constant and variable orders. Beside the obtaining solutions to some dynamic problems on time scales involving the proposed derivative, a fractional folrmulation for the viscoelastic oscillation problem is studied and its conversion into a third order dynamic equation is presented. [ABSTRACT FROM AUTHOR]

In this paper, we aim to generalize a fractional integro-differential operator in the open unit disk utilizing Jackson calculus (quantum calculus or q-calculus). Next, by consuming the generalized operator to define a formula of normalized analytic functions, we present a set of integral inequalities using the concepts of subordination and superordination. In addition, as an application, we determine the maximum and minimum solutions of the extended fractional 2D-shallow water equation in a complex domain. [ABSTRACT FROM AUTHOR]

The fractional generalization of Nambu mechanics is constructed by using the differential forms and exterior derivatives of fractional orders. The generalized Pfaffian equations are obtained and one example is investigated in details. [ABSTRACT FROM AUTHOR]

• A highly accurate numerical scheme is proposed for solving a class of variable-order fractional advection-reaction-subdiffusion equations. • The stability and convergence analysis of the present scheme are addressed. • Some extensive numerical experiments are performed to confirm the theoretical analysis. • Comparisons are made with the available schemes in the literature. This paper is concerned with a highly accurate numerical scheme for a class of one- and two-dimensional time-fractional advection-reaction-subdiffusion equations of variable-order α (x , t) ∈ (0 , 1). For the spatial and temporal discretization of the equation, a fourth-order compact finite difference operator and a third-order weighted-shifted Grünwald formula are applied, respectively. The stability and convergence of the present scheme are addressed. Some extensive numerical experiments are performed to confirm the theoretical analysis and high-accuracy of this novel scheme. Comparisons are also made with the available schemes in the literature. [ABSTRACT FROM AUTHOR]

Nonlinear dynamical problems, characterized by unpredictable and chaotic changes among variables over time, pose unique challenges in understanding. This paper explores the coupled nonlinear (2+1)-dimensional complex modified Korteweg-de-Vries (cmKdV) equation-a fundamental equation in applied magnetism and nanophysics. The study focuses on dynamic behaviors, specifically examining bifurcations and equilibrium points leading to chaotic phenomena by introducing an external term to the system. Employing chaos theory, we showcase the chaotic tendencies of the perturbed dynamical system. Additionally, a sensitivity analysis using the Runge-Kutta method reveals the solution's stability under slight variations in initial conditions. Innovatively, the paper utilizes the planar dynamical system technique to construct various solitons within the governing model. This research provides novel insights into the behavior of the (2+1)-dimensional cmKdV equation and its applications in applied magnetism and nanophysics. [ABSTRACT FROM AUTHOR]

The solution of the Lax tensor equations in the case L[sub αβγ]= -L[sub βαγ] was analyzed. The Lax tensors on the dual metrics were investigated. We classified all two-dimensional metrics having the symmetric Lax tensor L[sub αβγ]. The Lax tensors of the flat space, Rindler system and its dual were found. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions for an increasing variables singular m-dimensional system of fractional q-differential equations on a time scale. In this singular system, the first equation has two variables and the number of variables increases permanently. By using some fixed point results, we study the singular system under some different conditions. Also, we provide two examples involving practical algorithms, numerical tables, and some figures to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of finding a source term of a Rayleigh–Stokes equation. Our problem is not well-posed in the sense of Hadamard. The sought solution does not depend continuously on the given data. Using the truncation method and some new techniques on trigonometric estimators, we give the regularized solution. Moreover, the mean square error and convergence rates are established. [ABSTRACT FROM AUTHOR]

A numerical method for solving inhomogeneous nonlocal time fractional convection-diffusion-reaction equations with variable coefficients has been developed. The fractional time operator is taken in the sense of the modified operator with the Mittag-Leffler kernel. The numerical method is based on the modified Gauss elimination with Taylor's expansion. Through rigorous analysis, it has been proved that the given method is unconditionally stable and second-order convergent in space and time. The numerical results for three test problems illustrate the efficiency and validity of the theoretical estimates. [ABSTRACT FROM AUTHOR]

In this paper, for the first time the inverse problem of reconstructing the time-dependent potential (TDP) and displacement distribution in the hyperbolic problem with periodic boundary conditions (BCs) and nonlocal initial supplemented by over-determination measurement is numerically investigated. Though the inverse problem under consideration is ill-posed by being unstable to noise in the input data, it has a unique solution. The Crank–Nicolson-finite difference method (CN-FDM) along with the Tikhonov regularization (TR) is applied for calculating an accurate and stable numerical solution. The programming language MATLAB built-in lsqnonlin is used to solve the obtained nonlinear minimization problem. The simulated noisy input data can be inverted by both analytical and numerically simulated. The obtained results show that they are accurate and stable. The stability analysis is performed by using Fourier series. [ABSTRACT FROM AUTHOR]

This study consider Gerdjikov-Ivanov equation where the perturbation terms appear with full non-linearity. The Jacobi elliptic function ansatz method is implemented to obtain exact solutions of this equation that models pulse dynamics in optical fibers. It is retrieved some bright, dark optical and singular solitons profile in the limiting cases of the Jacobi elliptic functions. The constraint conditions depending on the parameters for the existence of solitons are also presented. [ABSTRACT FROM AUTHOR]

In this article, we have considered Wick-type stochastic Korteweg de Vries (KdV) equation with conformable derivatives. By the help of white noise analysis, Hermit transform and extended G'/G- expansion method, we have obtained exact travelling wave solutions of KdV equation with conformable derivatives. We have applied the inverse Hermit transform for stochastic soliton solutions and then we have shown how stochastic solutions can be presented as Brownian motion functional solutions by an application example. [ABSTRACT FROM AUTHOR]

In this paper, the fractional reduced differential transform method (FRDTM) is used to obtain the series solution of time-fractional seventh-order Sawada–Kotera (SSK) and Lax's KdV (LKdV) equations under initial conditions (ICs). Here, the fractional derivatives are considered in the Caputo sense. The results obtained are contrasted with other previous techniques for a specific case, α = 1 revealing that the presented solutions agree with the existing solutions. Further, convergence analysis of the present results with an increasing number of terms of the solution and absolute error has also been studied. The behavior of the FRDTM solution and the effects on different values α are illustrated graphically. Also, CPU-time taken to obtain the solutions of the title problems using FRDTM has been demonstrated. [ABSTRACT FROM AUTHOR]

This article uses the extension of the Lie symmetry analysis (LSA) and conservation laws (Cls) (Singla et al. in Nonlinear Dyn. 89(1):321-331, 2017; Singla et al. in J. Math. Phys. 58:051503, 2017) for the space-time fractional partial differential equations (STFPDEs) to analyze the space-time fractional Rosenou-Haynam equation (STFRHE) with Riemann-Liouville (RL) derivative. We transform the space-time fractional RHE to a nonlinear ordinary differential equation (ODE) of fractional order using its Lie point symmetries. The reduced equation’s derivative is in Erdelyi-Kober (EK) sense. We use the power series (PS) technique to derive explicit solutions for the reduced ODE for the first time. The Cls for the governing equation are constructed using a new conservation theorem. [ABSTRACT FROM AUTHOR]

A class of tempered fractional differential equations with terminal value problems are investigated in this paper. Discretized collocation methods on piecewise polynomials spaces are proposed for solving these equations. Regularity results are constructed on weighted spaces and convergence order is studied. Several examples are supported the theoretical parts and compared with other methods. [ABSTRACT FROM AUTHOR]

• An accurate method is developed to solve one- and two-dimensional Fredholm integral equations of the second kind. • The numerical method involves the use of a new shifted Cosine-trigonometric basis functions to approximate the solution of the problem. • The error analysis of the proposed method is derived. • Comparative results show that this method is accurate than the other existing methods in the literature. In this paper, an accurate numerical method based on the cosine-trigonometric basis functions is developed to solve the Fredholm integral equations of the second kind. By using the proposed method, the presented equation is converted into a system of algebraic equations. The convergence analysis of the proposed method is also investigated. To demonstrate the efficiency of the proposed method, the numerical simulations of various types of one- and two-dimensional examples are prepared. Comparative results show that this method is accurate than the other existing methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the approximated analytical solution for fractional Swift‐Hohenberg (S‐H) equation is found with the aid of novel technique called q‐homotopy analysis transform method (q‐HATM). To ensure the applicability and efficiency of the proposed algorithm, we consider non‐linear arbitrary‐order S‐H equation in presence and absence of dispersive term. The convergence analysis for the projected problem is presented, and the numerical simulations have been conducted to verify the future scheme is reliable and accurate. Further, the effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are presented through plots. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyse the complex problems that arose in science and technology. [ABSTRACT FROM AUTHOR]

The present article deals with a fractional extension of the vibration equation for very large membranes with distinct special cases. The fractional derivative is considered in Atangana‐Baleanu sense. A numerical algorithm based on homotopic technique is employed to examine the fractional vibration equation. The stability analysis is conducted for the suggested scheme. The maple software package is utilized for numerical simulation. In order to illustrate the effects of space, time, and order of Atangana‐Baleanu derivative on the displacement, the outcomes of this study are demonstrated graphically. The results revel that the Atangana‐Baleanu fractional derivative is very efficient in describing vibrations in large membranes. [ABSTRACT FROM AUTHOR]

In this article, an efficient analytical technique, called Laplace–Adomian decomposition method, is used to obtain the solution of fractional Zakharov– Kuznetsov equations. The fractional derivatives are described in terms of Caputo sense. The solution of the suggested technique is represented in a series form of Adomian components, which is convergent to the exact solution of the given problems. Furthermore, the results of the present method have shown close relations with the exact approaches of the investigated problems. Illustrative examples are discussed, showing the validity of the current method. The attractive and straightforward procedure of the present method suggests that this method can easily be extended for the solutions of other nonlinear fractional-order partial differential equations. [ABSTRACT FROM AUTHOR]

Recently a new kind of derivatives, namely the conformable derivative is introduced which have not many drawbacks of other fractional derivatives. Two types of KdV equations with conformable derivative are investigated in this paper. Existence and uniqueness of two different equations of KdV class with conformable derivatives are investigated. It is also shown that the invariant subspace method can be extended to find the exact solutions of these equations. [ABSTRACT FROM AUTHOR]

In this work a poweful approach is presented to solve the time-fractional gas dynamics equation. In fact, we use a fictitious time variable y to convert the dependent variable w(x, t) into a new one with one more dimension. Then by taking a initial guess and implementing the group preserving scheme we solve the problem. Finally four examples are solved to illustrate the power of the offered method. [ABSTRACT FROM AUTHOR]

The main goal of this manuscript is to investigate a fractional optimal control problem subject to a dynamical system involving Hadamard fractional derivatives. Necessary conditions for the optimality of the considered problem are derived in terms of the corresponding Euler–Lagrange equations. An iterative method is also proposed to numerically solve the obtained equations from the necessary optimality conditions. Two illustrative examples are considered and simulated in order to show the applicability and efficiency of the proposed method. Numerical simulations show that the used method presents some satisfying results regarding the absolute error values. [ABSTRACT FROM AUTHOR]

In this paper, new analytical obliquely propagating wave solutions for the time fractional extended Zakharov–Kuzetsov (FEZK) equation of conformable derivative are investigated. By using the main properties of the conformable derivative, the FEZK equation is transformed into integer-order differential equations, and the reduced equations are solved via the generalized exponential rational function method (GERFM). The shape and features for the resulting solutions are illustrated through three-dimensional (3D) plots and corresponding contour plots for various values of the free parameters. [ABSTRACT FROM AUTHOR]

In this work, a new modified fractional form of the Nagumo equation has been presented and deeply analyzed. Using the Caputo–Fabrizio and Atangana–Baleanu time-fractional derivatives, classical Nagumo model is transformed to a new fractional version. The modified equation has been solved by using the homotopy analysis transform method. The convergence analysis has been also examined with the help of the so-called h -curves and average residual error. Comparing the obtained approximate solution with the exact solution leaves no doubt believing that the proposed technique is very efficient and converges toward the exact solution very rapidly. [ABSTRACT FROM AUTHOR]

The idea of this work is to provide a pseudo‐operational collocation scheme to deal with the solution of the variable‐order time‐space fractional KdV–Burgers–Kuramoto equation (VOSTFKBKE). Such the fractional partial differential equation (FPDE) has three characteristics of dissipation, dispersion, and instability, which make this equation is used to model many phenomena in diverse fields of physics. Numerical solutions are sought in a linear combination of two‐dimensional Jacobi polynomials as basis functions. In order to approximate unknown functions in terms of the basis vector, pseudo‐operational matrices are constructed to avoid integration. An error bound of the residual function is estimated in a Jacobi‐weighted space in the L2$$ {L}^2 $$ norms. Numerical results are compared with exact ones and those reported by other researchers to demonstrate the effectiveness of the recommended method. [ABSTRACT FROM AUTHOR]

This paper is devoted to identifying an unknown source for a time-fractional diffusion equation in a general bounded domain. First, we prove the problem is non-well posed and the stability of the source function. Second, by using the Modified Fractional Landweber method, we present regularization solutions and show the convergence rate between regularization solutions and sought solution are given under a priori and a posteriori choice rules of the regularization parameter, respectively. Finally, we present an illustrative numerical example to test the results of our theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the q -Shehu transform. Further, we define the generalized hyper-Bessel function. Also, we state the q -Shehu transform for some elementary functions. The present aim in this paper is to obtain the solutions of the q -fractional kinetic equations in terms of the established generalized hyper-Bessel function by applying the established q -Shehu transform. Also, we give some special cases of our main results. At the end of this paper, we give the numerical values and the graphical representations of these solutions by using the software MATLAB. [ABSTRACT FROM AUTHOR]

This paper presents the modulation instability (MI) analysis and the different types of optical soliton solutions of the Triki–Biswas model equation. The aforesaid model equation is the generalization of the derivative nonlinear Schrödinger equation which describes the ultrashort pulse propagation with non-Kerr dispersion. The study is carried out by means of a novel efficient integration scheme. During this work, a sequence of optical solitons is produced that may have an important in optical fiber systems. The results show that the studied model hypothetically has incredibly rich optical soliton solutions. The constraint conditions for valid soliton solutions are also reported. The gained results show that the applied method is efficient, powerful and can be applied to various complex models with the help of representative calculations. [ABSTRACT FROM AUTHOR]

In this article, we construct multi-wave, homoclinic breather, M-shaped rational and periodic cross kink wave solutions for coupled-Higgs equation (CHE) by using distinct transformations. We obtain these solutions with the aid of logarithmic transformations and symbolic computations. Moreover, we also derive some soliton wave solutions for CHE in polynomial forms. These solutions include solitary wave, soliton wave and Jacobi elliptic function solutions and are found by implementing unified technique. We will also discuss the graphical structure of our newly achieved results. [ABSTRACT FROM AUTHOR]

In this paper, the Homotopy perturbation Laplace method is implemented to investigate the solution of fractional-order Whitham–Broer–Kaup equations. The derivative of fractional-order is described in Caputo's sense. To show the reliability of the suggested method, the solution of certain illustrative examples are presented. The results of the suggested method are shown and explained with the help of its graphical representation. The solutions of fractional-order problems as well as integer-order problems are determined by using the present technique. It has been observed that the obtained solutions are in significant agreement with the actual solutions to the targeted problems. Computationally, it has been analyzed that the solutions at different fractional-orders have a higher rate of convergence to the solution at integer-order of the derivative. Due to the analytical analysis of the problems, this study can further modify the solution of other fractional-order problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the spectral collocation method with help of Chebyshev polynomials. We consider the space fractional Korteweg-de Vries and the space fractional Korteweg-de Vries-Burgers equations based on the Caputo-Fabrizio fractional derivative. The proposed method reduces the models under study to a set of ordinary differential equations and then solves the system via the finite difference method. To the best our knowledge this is the first work which studies the Caputo-Fabrizio space fractional derivative for the proposed equations. The results were validated in the case of the classic differential equations in comparison with the exact solution and the calculation of the absolute error, and in the case of fractional differential equations, the results were verified by calculating the residual error function. In both cases, the results are very accurate and effective. The presented method is easy and accurate, and can be applied to many fractional systems. [ABSTRACT FROM AUTHOR]

We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed L ∞ , L 2 , and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique. [ABSTRACT FROM AUTHOR]

The dynamical attitude of the transmission for the nerve impulses of a nervous system, which is mathematically formulated by the Atangana–Baleanu (AB) time-fractional FitzHugh–Nagumo (FN) equation, is computationally and numerically investigated via two distinct schemes. These schemes are the improved Riccati expansion method and B-spline schemes. Additionally, the stability behavior of the analytical evaluated solutions is illustrated based on the characteristics of the Hamiltonian to explain the applicability of them in the model's applications. Also, the physical and dynamical behaviors of the gained solutions are clarified by sketching them in three different types of plots. The practical side and power of applied methods are shown to explain their ability to use on many other nonlinear evaluation equations. [ABSTRACT FROM AUTHOR]

This paper adopts the Adomian decomposition method and the Padé approximation techniques to derive the approximate solutions of a conformable Rosenau‐Hyman equation by considering the new definition of the Adomian polynomials. The Padé approximate solutions are derived along with interesting figures showing both the analytic and approximate solutions. [ABSTRACT FROM AUTHOR]

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are compared with each other as well as with the exact results of the problems. The comparison shows the best agreement of solutions with each other and with the exact solution as well. Moreover, the proposed methods are found to be accurate, effective, and straightforward while dealing with the fractional-order system of partial differential equations and therefore can be generalized to other fractional order complex problems from engineering and science. [ABSTRACT FROM AUTHOR]

This paper proposes a new effective pseudo-spectral approximation to solve the Sylvester and Lyapunov matrix differential equations. The properties of the Chebyshev basis operational matrix of derivative are applied to convert the main equation to the matrix equations. Afterwards, an iterative algorithm is examined for solving the obtained equations. Also, the error analysis of the propounded method is presented, which reveals the spectral rate of convergence. To illustrate the effectiveness of the proposed framework, several numerical examples are given. [ABSTRACT FROM AUTHOR]

We present a collocation approach based on redefined cubic B-spline (RCBS) functions and finite difference formulation to study the approximate solution of time fractional Allen–Cahn equation (ACE). We discretize the time fractional derivative of order α ∈ (0 , 1 ] by using finite forward difference formula and bring RCBS functions into action for spatial discretization. We find that the numerical scheme is of order O (h 2 + Δ t 2 − α) and unconditionally stable. We test the computational efficiency of the proposed method through some numerical examples subject to homogeneous/nonhomogeneous boundary constraints. The simulation results show a superior agreement with the exact solution as compared to those found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce generalized exponential rational function method (GERFM) to obtain an exact solutions for the Biswas-Milovic (BM) equation with quadratic-cubic and parabolic nonlinearities. A wide range of closed solutions are acquired. The most important feature of the new method is that it is very effective and simple. The main merits of the proposed is that it gives more general solutions with some free parameters and can be applied to other types of nonlinear partial differential equations. Some interesting Figures for the physical features of some of the obtained solutions are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell‐Whitehead‐Segel–type equations which are very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement. [ABSTRACT FROM AUTHOR]

In this study, we implement three efficient integration algorithms to retrieve the solutions of optical soliton space-time fractional nonlinear equation for the dynamics of microtubules MTs, which considered as one of the most important part in cellular processes biology. In this work we used three integration methods, firstly, the method of exp(-Ω)-expansion equation function, secondly the Kudryashov equation method in the general case and the method of extended Bernoulli sub-equation function, with the help of the fractional complex transformation and conformable derivatives which including the solution of complex function, rational function, hyperbolic function and exponential function. Finally, our results give good solution and understanding of the properties of the non-linear waves in fractional medium. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present a technique for the numerical solution of Caputo time fractional superdiffusion equation. The central difference approximation is used to discretize the time derivative, while non-polynomial quintic spline is employed as an interpolating function in the spatial direction. The proposed method is shown to be unconditionally stable and O (h 4 + Δ t 2) accurate. In order to check the feasibility of the proposed technique, some test examples have been considered and the simulation results are compared with those available in the existing literature. [ABSTRACT FROM AUTHOR]