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New fractional operators, the COVID-19 model has been studied in this paper. By using different numerical techniques and the time fractional parameters, the mechanical characteristics of the fractional order model are identified. The uniqueness and existence have been established. The model's Ulam-Hyers stability analysis has been found. In order to justify the theoretical results, numerical simulations are carried out for the presented method in the range of fractional order to show the implications of fractional and fractal orders. We applied very effective numerical techniques to obtain the solutions of the model and simulations. Also, we present conditions of existence for a solution to the proposed epidemic model and to calculate the reproduction number in certain state conditions of the analyzed dynamic system. COVID-19 fractional order model for the case of Wuhan, China, is offered for analysis with simulations in order to determine the possible efficacy of Coronavirus disease transmission in the Community. For this reason, we employed the COVID-19 fractal fractional derivative model in the example of Wuhan, China, with the given beginning conditions. In conclusion, again the mathematical models with fractional operators can facilitate the improvement of decision-making for measures to be taken in the management of an epidemic situation. [ABSTRACT FROM AUTHOR]

This article explores a wide range of envelope soliton pulses and their behavior during propagation through a birefringent optical fiber. The propagation of light in such fibers is governed by two coupled non-linear Schrödinger equations (CNLSEs), which account for both coherent and incoherent non-linear couplings. To investigate these solitons, an improved modified extended tanh function scheme (IMETFS) is employed. The study focuses on elliptical core optical fibers and examines the significance of various optical solitons in the presence of group-velocity dispersion and third-order nonlinearity. The findings illustrate that birefringent optical fibers can generate a diverse array of nonlinear periodic waves and localized pulses. The different types of soliton solutions identified include singular, bright, rational, exponential, Jacobi elliptic functions (JEFs), Weierstrass elliptic doubly periodic solutions, and singular periodic solutions. Comparisons with existing literature highlight the novelty and substantive contributions of the discovered wave solutions to current research in nonlinear optics. The success of this approach suggests its potential applicability in addressing nonlinear challenges across different fields, particularly in soliton theory, where similar models are frequently encountered. Additionally, the article presents visual representations of these solutions through 3D and 2D plots, aiding in the understanding of their behaviors. [ABSTRACT FROM AUTHOR]

This study examines the analytic wave solutions of a highly dispersive perturbed complex Ginzburg–Landau equation (CGLE) with conformable fractional derivative and polynomial law of nonlinearity using the improved modified extended tanh-function method. The results show a wide range of solutions including (bright, dark, singular) solitons, Jacobi elliptic solutions, exponential solutions, and Weierstrass elliptic solutions. The obtained soliton solutions showcase diverse dynamics, encompassing different solitary waves and localized structures. The polynomial nonlinearity adds complexity to the dynamics, resulting in the emergence of new solitons with distinct characteristics. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. [ABSTRACT FROM AUTHOR]

This study investigates optical solitons and other traveling wave solutions in fiber Bragg gratings for coupled nonlinear Schrödinger equations with cubic quadratic nonlinearity using the improved modified extended tanh-function method (IMETFM). We discover a highly dispersive solitons in fiber Bragg such as bright solitons, dark solitons, combo bright–dark solitons and singular solitons. Moreover, we obtain hyperbolic solutions, combo hyperbolic solutions, combo periodic solutions, singular periodic solutions, rational solutions, exponential solutions, and Jacobi elliptic solutions. By comparing our obtained travelling wave solutions with existing literature, we demonstrate their novelty and substantive contribution to current research. The success of our approach suggests its potential applicability to address various nonlinear challenges in different fields, particularly within soliton theory, since the analyzed model appears in a multitude of applications. These findings are of great significance for the further development of optical communication systems. Additionally, the physical meaning of the geometrical structures for some solutions is discussed for different choices of the free parameters. It is important to mention that our proposed method is highly effective, consistent, and impactful and can be utilized to solve different physical models. [ABSTRACT FROM AUTHOR]

This paper introduces the Nucci reduction method, a novel and efficient approach for deriving exact solutions to the perturbed Gerdjikov–Ivanov equation, offering a significant advancement in the field. The suggested technique involves transforming the equation into real and imaginary components prior to application. We successfully obtained four distinct exact and explicit solutions, along with the corresponding first integrals. Explanations and presentations of solutions are given in a logical manner. We derive an analytical expression for the instability gain and examine its key features using linear stability analysis. Finally, we compare the correctness of the analytical and numerical solutions. We demonstrate the robustness and stability of solitary waves through numerical simulations. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is a fundamental equation in the field of nonlinear optics and plays an important role in the study of many physical phenomena. The present study introduces a new model that demonstrates the novelty of the paper and provides the advancement of knowledge in the area of nonlinear optics by solving a challenging problem known as the extended (3 + 1)-dimensional nonlinear conformable Kudryashov's equation (CKE) with generalized anti-cubic nonlinearity, which is a generalization of the NLSE to three spatial dimension and one temporal dimension for the first time. This work is significant because it advances our understanding of nonlinear optics and its applications to solve complex equations in physics and related disciplines. The extended hyperbolic function method (EHFM) and Nucci's reduction method are applied to the extended (3 + 1)-dimensional nonlinear CKE with generalized anti-cubic nonlinearity. The equation is solved by using the concept of conformable derivative, a recently developed operator in fractional calculus, which has advantages over other fractional derivatives in terms of accuracy and flexibility. The attained solutions include periodic singular, dark 1-soliton, singular 1-soliton, and bright 1-soliton which are visualized using 3D and contour plots. This study highlights the potential of using conformable derivative and the applied techniques to solve complex nonlinear differential equations in various fields. The obtained solutions and analysis will be useful in the design and analysis of optical communication systems and other related fields. Overall, this study contributes for the understanding of the dynamics of the extended (3+1)-dimensional nonlinear CKE and offers new insights into the use of mathematical techniques to tackle complex problems in physics and related fields. [ABSTRACT FROM AUTHOR]

This article extracts analytical solutions of the Fokas-Lenells equation describing pulse propagation in optical fibers and presents a comprehensive analysis of the analytical solutions. This paper focuses on deriving analytical, non-perturbative solutions for the Fokas-Lenells equation by reducing it to a system of ordinary differential equations using the Lie symmetry method. Three generators for the Lie algebra are obtained, and three reductions are explored by combining vector generators. These algebraic structures provide a powerful framework for simplifying complex mathematical and physical problems by revealing underlying symmetries and patterns. The graphical representations of obtained solutions are rendered. Besides, the influence of parameters within the equation on the behavior of the solutions has been demonstrated in the figures. These results contribute to a deeper understanding of the Fokas-Lenells equation and have implications in various fields of physics and engineering. [ABSTRACT FROM AUTHOR]

This article investigates the dynamics of optical solitons in a magneto-optic waveguide with the nonlinear perturbed Schrödinger equation. The model incorporates two generalized nonlocal laws, a Kudryashov's sextic-power law nonlinear structure, and high dispersion up to the sixth-order dispersion. The proposed innovative model ensures that all the obtained solutions are original and have not been reported elsewhere. The impetus for this work is the recognition of the crucial significance of studying nonlinear magneto-optical interactions in optical communication systems. The nonlinear Schrödinger equation and its derivatives play crucial roles in various scientific areas, particularly in the field of nonlinear optics and optical fibers. In order to investigate the new structure of our model, we utilize the Z ′ Z -expansion technique and the enhanced Kudrtashov's method. The solitons obtained are of several types, including bright, singular, and combinations thereof as well as kink-type solution. The strategies utilized are efficacious and distinctive tools in their methodologies and results. To further elucidate the dynamics of the acquired solitons, we offer a discussion section where we present plots of specific solitons. [ABSTRACT FROM AUTHOR]

In this paper, (1 + 1) -dimensional Schrödinger–Hirota equation with Kerr law having inter-modal dispersion is considered. This model designates the propagation of pulses in optical fibers, so it has a significant impact on the model and optimization of optical fiber communication systems. Utilizing the Lie symmetry technique, three generators are produced and three sub-algebras are derived by the combination of these generators. Lie symmetry analysis is the basis of the reduction of the presented nonlinear partial differential equation to the nonlinear ordinary differential equation system. We implement the unified Riccati equation expansion method so that we acquire the analytical soliton solutions. Thus, periodic singular, dark, and singular solitons are retrieved; besides, their 3-dimensional portraits are illustrated by selecting appropriate parameter values. Moreover, the influence of the parameters of the presented model is investigated via various 2-dimensional graphical visualizations. This study reports that all produced solutions satisfy the presented model. [ABSTRACT FROM AUTHOR]

This study centers on examining the characteristics of the recently extended (3 + 1) -dimensional nonlinear equation proposed by Kudryashov. The extended F-expansion technique is employed to derive solitons and other exact wave solutions for this model. Applying this technique results in a diverse set of solutions, encompassing bright solitons, dark solitons, singular solitons, hyperbolic solutions, periodic solutions, singular periodic solutions, exponential solutions, rational solutions, and solutions involving Jacobi elliptic functions (JEF). This method proves to be a dependable and efficient approach for obtaining exact solutions for various nonlinear partial differential equations (NPDE). Visual representations through graphical illustrations are provided to depict the dynamics of selected solutions, and the influence of the parameter n on the obtained solutions is scrutinized and presented in pertinent figures. These discoveries not only advance our comprehension of nonlinear wave phenomena but also hold practical significance for fields related to wave propagation, nonlinear optics, and optical systems. [ABSTRACT FROM AUTHOR]

In this study, the space-time fractional generalised reaction duffing model is investigated analytically, which is a generalization for a collection of prominent fractional models describing various key phenomenon in science and engineering. The governing equation is converted to a nonlinear ODE by the compatible travelling wave transform. The investigation established that for analysing nonlinear evolution equations of fractional order, the recommended approach is more effective and realistic. The findings are given extensively in rational forms of trigonometric function series or clearly in powers of specific trigonometric functions. A collection of bright, dark, periodic, and optical solitons is discovered. Mathematica is used to flourish the presence of some obtained solutions in 3D graphs with different fractional orders. The results show that the recommended methods are more practical and effective ways to illustrate the dynamics of several complex wave structures in modern science and technology. [ABSTRACT FROM AUTHOR]

In this work, we investigate the optical solitons and other waves through magneto-optic waveguides with Kudryashov’s law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation factors using the modified extended mapping approach. Many classifications of solutions are established like bright solitons, dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions, rational wave, solutions, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic function solutions. Some of the extracted solutions are described graphically to provide their physical understanding of the acquired solutions. [ABSTRACT FROM AUTHOR]

New fractional operators, the COVID-19 model has been studied in this paper. By using different numerical techniques and the time fractional parameters, the mechanical characteristics of the fractional order model are identified. The uniqueness and existence have been established. The model’s Ulam-Hyers stability analysis has been found. In order to justify the theoretical results, numerical simulations are carried out for the presented method in the range of fractional order to show the implications of fractional and fractal orders. We applied very effective numerical techniques to obtain the solutions of the model and simulations. Also, we present conditions of existence for a solution to the proposed epidemic model and to calculate the reproduction number in certain state conditions of the analyzed dynamic system. COVID-19 fractional order model for the case of Wuhan, China, is offered for analysis with simulations in order to determine the possible efficacy of Coronavirus disease transmission in the Community. For this reason, we employed the COVID-19 fractal fractional derivative model in the example of Wuhan, China, with the given beginning conditions. In conclusion, again the mathematical models with fractional operators can facilitate the improvement of decision-making for measures to be taken in the management of an epidemic situation. [ABSTRACT FROM AUTHOR]

In this article, Heir‐equations method has been used to investigate nonclassical symmetries and new exact solutions of the generalized Fitzhugh–Nagumo equation with variable coefficients. Different types of variable coefficients with corresponding exact solutions have been considered. [ABSTRACT FROM AUTHOR]

In this work, we investigate the optical solitons and other waves through magneto-optic waveguides with Kudryashov's law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation factors using the modified extended mapping approach. Many classifications of solutions are established like bright solitons, dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions, rational wave, solutions, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic function solutions. Some of the extracted solutions are described graphically to provide their physical understanding of the acquired solutions. [ABSTRACT FROM AUTHOR]

In the present paper, a model of the gravitational waves of water known as the (2 + 1) -dimensional generalized time-fractional (2DGTF) Zakharov Kuznetsov Benjamin Bona Mahony (ZK–BBM) equation is studied. To this end, through employing the Lie symmetry method, exact solutions and reductions are derived for the governing equation. Additionally, classical and nonclassical Lie symmetry generators are acquired, and based on such Lie symmetry generators, conservation laws for the classical and nonclassical vector fields related to the 2DGTF ZK–BBM equation are constructed. Some case studies are given to analyze the impact of the order of the time-fractional derivative on the dynamics of the exact solutions. [ABSTRACT FROM AUTHOR]

In this study, we investigated Schrödinger–Hirota equation Kerr law in the presence of spatio-temporal dispersion which describes pulse propagation in optical fibers provoked by nonlinear effect and reduced to multi-plane dispersion. We used the Lie symmetry method to reduce the model into the nonlinear ordinary differential equations. Moreover, the unified Riccati equation expansion method was used to obtain optical soliton solutions and other soliton solutions of the model. We obtained dark, kink, singular, and periodic-singular soliton solutions. Obtained results were shown via 3D and contour graphs. Thus, the model was analyzed for the first time with the Lie symmetry method and UREEM. [ABSTRACT FROM AUTHOR]

The current research investigates the behavior of femtosecond solitary waves in an inhomogeneous optical fiber using the generalized derivative nonlinear Schrödinger equation with quintic nonlinearity. The extended F-expansion technique is utilized to obtain various exact solutions such as bright soliton solutions, dark soliton solutions, combo bright–dark soliton solutions, singular soliton solutions, periodic solutions, Jacobi elliptic functions solutions, rational solutions, Weierstrass elliptic solutions, exponential solutions. The obtained solutions are presented in three-dimensional and contour graphics by selecting appropriate parameters. Ibragimov's conservation technique is also applied to obtain conservation laws for the given model. These findings are crucial for comprehending various of scientific and physical applications. [ABSTRACT FROM AUTHOR]

In this study, the Nucci's reduction approach and the method of generalized projective Riccati equations (GPREs) were utilized to derive novel analytical solutions for the (1+1)-dimensional classical Boussinesq equations, the generalized reaction Duffing model, and the nonlinear Pochhammer-Chree equation. The nonlinear systems mentioned earlier have been solved using analytical methods, which impose certain limitations on the interaction parameters and the coefficients of the guess solutions. However, in the case of the double sub-equation guess solution, analytic solutions were allowed. The soliton solutions that were obtained through this method display real positive values for the wave phase transformation, which is a novel result in the application of the generalized projective Riccati method. In previous applications of this method, the real positive properties of the solutions were not thoroughly investigated. [ABSTRACT FROM AUTHOR]

This paper focuses on the use of the Sardar sub-equation method to obtain optical solitons for a nonlinear model that incorporates Kudryashov's quintuple power law and dual-form nonlocal nonlinearity. The model studies the propagation of pulses in optical fibers and the proposed technique is used to investigate various types of solitons including bright, dark, singular, periodic, combined dark-bright, and combined dark-singular solitons. The proposed technique is highly efficient and can also be utilized to solve higher-order nonlinear partial differential equations in fields such as fluid mechanics, plasma physics, and fiber optics. The paper includes a comparative analysis of the model and presents graphical representations for some of the obtained solutions. [ABSTRACT FROM AUTHOR]

This paper explores the interactions of capillary–gravity waves by presenting novel exact solutions to a coupled Schrödinger–KdV equation. The authors introduce a reduction method based on Nucci's approach to derive these exact solutions. The results provide a new approach to solving the Schrödinger–KdV equation, offering potential applications in various fields such as mathematical physics, hydrodynamics, and soliton theory. Effectiveness and versatility of utilized reduction technique make it a valuable tool for researchers in various fields, including mathematical physics, fluid mechanics, and nonlinear dynamics. Overall, this research contributes to the understanding of the complex behavior of capillary–gravity waves and provides a useful tool for modeling and analyzing such phenomena. [ABSTRACT FROM AUTHOR]

In mathematics, physics, and engineering, establishing numerical or analytical solutions for fractional mathematical models for specific phenomena and developing fractional mathematical models for specific phenomena are important topics. In this work the (G ′ G) - expansion method with the generalized fractional derivative has been used to obtained the explicit solutions of nonlinear fractional Tzitz e ´ ica type nonlinear evolution equations. These models are tested to illustrate the pertinent features of the proposed method. [ABSTRACT FROM AUTHOR]

In this study, we dealt with the new conservation theorem and the auxiliary method. We have applied the theorem and method to (3 + 1)-dimensional modified Wazwaz–KdV equations. When we applied a new conservation theorem to given equations, the obtained conservation laws did not satisfy the divergence condition. So, we modified the obtained conservation laws. These conservation laws contain extra terms. Finally, we applied the auxiliary method to given equations. We obtained two solution families with six exact solutions. All the obtained solutions are different from each other. For a suitable value of the solutions, the 3D and 2D surfaces have been plotted by Maple. [ABSTRACT FROM AUTHOR]

In this paper, we provide the Noether symmetries with gauge fields of the area-minimizing hypersurface Lagrangian according to some vacuum classes of pp-waves. Also we show that they are elements of the Killing algebra of vacuum classes of pp-waves. Finally, we construct the conserved fields for the area-minimizing Lagrangian according to Noether's theorem. [ABSTRACT FROM AUTHOR]

Fractional derivatives are significant mathematical instruments that have been practiced to model real phenomena in different areas of science. Through the current investigation study, we develop the Human Liver and Hearing Loss models by employing three fractional operators called Atangana–Baleanu–Caputo, Caputo and Caputo–Fabrizio derivatives. The numerical techniques regarding the basic theories of fractional analysis considered to obtain the approximate solutions of the offered systems. Indeed, approximate results are presented for showing the efficacy of the methods during graphs of solutions. To see the performance of the suggested models, different fractional orders are tested. [ABSTRACT FROM AUTHOR]

The purpose of this work is to seek various innovative exact solutions using the new Kudryashov approach to the nonlinear partial differential equations (NLPDEs). This technique obtains novel exact solutions of soliton types. Moreover, several 3D and 2D plots of the higher dimensional Klein-Gordon, Kadomtsev-Petviashvili, and Boussinesq equations are demonstrated by considering the relevant values of the aforementioned parameters to exhibit the nonlinear wave structures more adequately. The new Kudryashov technique is an effective, and simple technique that provides new generalized solitonic wave profiles. It is anticipated that these novel solutions will enable a thorough understanding of the development and dynamic nature of such models. [ABSTRACT FROM AUTHOR]

The current work is based on performing the non‐classical and classical Lie group analysis for a system of nonlinear time‐fractional Heisenberg equation. Here, we apply analysis of the Lie group symmetry as one of the powerful tools dealing with the large class of fractional order differential equations in the Riemann–Liouville (RL) sense. Indeed, classical and non‐classical Lie symmetries group is used to similarity reduction of nonlinear time‐fractional Heisenberg equation. Especially, based on the obtained Lie symmetry generators, we construct conservation laws for the classical and non‐classical vector fields related to the Heisenberg time‐fraction equation. [ABSTRACT FROM AUTHOR]

In this work, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equation in the Riemann-Liouville concept. We employ the classical Lie symmetries to obtain similarity reductions of nonlinear time-fractional Benjamin-Ono equation and then, we find the related exact solutions for the derived generators. Finally, according to the Lie symmetry generators obtained, we construct conservation laws for related classical vector fields of time-fractional Benjamin-Ono equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to use the Nucci's reduction method to obtain some novel exact solutions to the s-dimensional generalized nonlinear dispersive mK(m,n) equation. To the best of the authors' knowledge, this paper is the first work on the study of differential equations with local derivatives using the reduction technique. This higher-dimensional equation is considered with three types of local derivatives in the temporal sense. Different types of exact solutions in five cases are reported. Furthermore, with the help of the Maple package, the solutions found in this study are verified. Finally, several interesting 3D, 2D and density plots are demonstrated to visualize the nonlinear wave structures more efficiently. [ABSTRACT FROM AUTHOR]

In this work, a generalized (G ′ / G) -expansion method has been used for solving the nonlinear Chen–Lee–Liu equation. This method is a more common, general, and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations (NPDEs), where G = G (τ) follows the Jacobi elliptic equation [ G ′ (τ) ] 2 = H (G) , and we let H (G) be a fourth-order polynomial. Many new exact solutions such as the hyperbolic, rational, and trigonometric solutions with different parameters in terms of the Jacobi elliptic functions are obtained. The distinct solutions obtained in this paper clearly explain the importance of some physical structures in the field of nonlinear phenomena. Also, this method deals very well with higher-order nonlinear equations in the field of science. The numerical results described in the plots were obtained by using Maple. [ABSTRACT FROM AUTHOR]

This work provides an analytical investigation of the time-fractional Kundu–Eckhaus (KE) equation. We use the symmetry of the Lie group as an appropriate tool which deals with the wide class of fractional-order differential equations in Riemann–Liouville sense. In the current work, firstly, we employ classical Lie symmetries to obtain similarity reductions of nonlinear generalised time-fractional KE equation. At the final step, we find relevant exact solutions for the extracted generators. [ABSTRACT FROM AUTHOR]

This paper shows out optical soliton solutions of the twin-core couplers with parabolic law nonlinearity with optical metamaterials parameters. It is used the improved new sub-ODE method (SOM) to found dark optical soliton, trigonometric function solutions, bright optical soliton and Jacobian elliptic function (JEF) solutions. Besides, physical explanation of the obtained bright and dark optical soliton solutions have been done and the influence of some parameters of the model have been set out. As the two-core coupler is favorable for modulation instability (MI), it has been studied the steady state of the results. More importantly, we have checked the effects of the optical metamaterials parameters on the formation of the Modulation bands in normal and anomalous dispersive regime. The results obtained in this paper set out Weierstrass Elliptic Function solutions compared to Mirzazadeh et al. (J Nonlinear Opt Phys Mater 24:1550017, 2015), Vega-Guzman et al. (Acta Phys Pol A 133:167–178, 2018) and will be helpful in the feature in optical birefringent fibers. [ABSTRACT FROM AUTHOR]

We investigate the nonlinear boundary value problems by reproducing kernel Hilbert space technique in this paper. We construct some reproducing kernel Hilbert spaces. We define a bounded linear operator to obtain the solutions of the problems. We demonstrate our numerical results by some tables. We compare our numerical results with some results exist in the literature to present the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the squared remainder minimization method for solving multi-term fractional differential equations. We restrict our attention to linear equations. Approximate solutions of these equations are considered in terms of linearly independent functions. We change our problem into a minimization problem. Finally, the Lagrange-multiplier method is used to minimize the resultant problem. The convergence of this approach is discussed and theoretically investigated. Some relevant examples are investigated to illustrate the accuracy of the method, and obtained results are compared with other methods to show the power of applied method. [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]

We apply reproducing kernel method to the density-dependent diffusion Nagumo equation. Powerful method has been applied by reproducing kernel functions. The approximations to the exact solution are obtained. In particular, series solutions are obtained. These solutions demonstrate the certainty of the method. The results acquired in this work conceive many attracted behaviors that assure further work on the Nagumo equation. [ABSTRACT FROM AUTHOR]

We apply the reproducing kernel method and group preserving scheme for investigating the Lane–Emden equation. The reproducing kernel method is implemented by the useful reproducing kernel functions and the numerical approximations are given. These approximations demonstrate the preciseness of the investigated techniques. [ABSTRACT FROM AUTHOR]

In this paper, we study the magneto-electro-elastic (MEE) circular rod by the aid of Lie group symmetry method. Corresponding symmetry reductions of MEE and its some invariant solutions using the Nucci’s method are completely considered too. Subsequently, the soliton solutions are obtained using the first integral method. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a numerical method to approximate the solutions of time fractional diffusion equation which is in the class of Lie group integrators. Our utilized method, namely fictitious time integration method transforms the unknown dependent variable to a new variable with one dimension more. Then the group preserving scheme is used to integrate the new fractional partial differential equations in the augmented space R3+1. Effectiveness and validity of method demonstrated using two examples. [ABSTRACT FROM AUTHOR]