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The relationship between convexity and symmetry is widely recognized. In fuzzy theory, both concepts exhibit similar behavior. It is crucial to remember that real and interval-valued mappings are special instances of fuzzy-number-valued mappings ( F - N - V - M s ), as fuzzy theory relies on the unit interval, which is crucial to resolving problems with interval analysis and fuzzy number theory. In this paper, a new harmonic convexities class of fuzzy numbers has been introduced via up and down relation. We show several Hermite–Hadamard ( H ⋅ H ) and Fejér-type inequalities by the implementation of fuzzy Aumann integrals using the newly defined class of convexities. Some nontrivial examples are also presented to validate the main outcomes. [ABSTRACT FROM AUTHOR]

The symmetric function class interacts heavily with other types of functions. One of these is the convex function class, which is strongly related to symmetry theory. In this study, we define a novel class of convex mappings on planes using a fuzzy inclusion relation, known as coordinated up and down convex fuzzy-number-valued mapping. Several new definitions are introduced by placing some moderate restrictions on the notion of coordinated up and down convex fuzzy-number-valued mapping. Other uncommon examples are also described using these definitions, which can be viewed as applications of the new outcomes. Moreover, Hermite–Hadamard–Fejér inequalities are acquired via fuzzy double Aumann integrals, and the validation of these outcomes is discussed with the help of nontrivial examples and suitable choices of coordinated up and down convex fuzzy-number-valued mappings. [ABSTRACT FROM AUTHOR]

This article studies a biological population model in the context of a fractional Caputo-Fabrizio operator using double Laplace transform combined with the Adomian method. The conditions for the existence and uniqueness of solution of the problem under consideration is established with the use of the Banach principle and some theorems from fixed point theory. Furthermore, the convergence analysis is presented. For the accuracy and validation of the technique, some applications are presented. The numerical simulations present the obtained approximate solutions with a variety of fractional orders. From the numerical simulations, it is observed that when the fractional order is large, then the population density is also large; on the other hand, population density decreases with the decrease in the fractional order. The obtained results reveal that the considered technique is suitable and highly accurate in terms of the cost of computing, and can be used to analyze a wide range of complex non-linear fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the effect of torsion on the initiation of localized bulging in an inflated cylindrical tube that is made of an incompressible material. When the twisting moment M is fixed, the bifurcation condition for localized bulging is derived in a simple form. Moreover, it is shown that the value of pressure associated with the onset of the bulge increases as the twisting moment increases. Therefore, localized bulging may be delayed or prevented when a torsion is applied to the tube. Our results indicate that the torsion reduces the radial and axial expansions of the tube during the inflation. The effect of the tube thickness on the onset of localized bulging is examined, and we find that the torsion plays a very important role in thin tubes. [ABSTRACT FROM AUTHOR]

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of the modified BBM equation are obtained. Furthermore, the findings of the current study remain strong since Jacobi function solutions generate hyperbolic function solutions and trigonometric function solutions, as liming cases of interest. Some of the obtained solutions are illustrated graphically using appropriate values for the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we propose two scaled Dai–Yuan (DY) directions for solving constrained monotone nonlinear systems. The proposed directions satisfy the sufficient descent condition independent of the line search strategy. We also reasonably proposed two different relations for computing the scaling parameter at every iteration. The first relation is proposed by approaching the quasi-Newton direction, and the second one is by taking the advantage of the popular Barzilai–Borwein strategy. Moreover, we propose a robust projection-based algorithm for solving constrained monotone nonlinear equations with applications in signal restoration problems and reconstructing the blurred images. The global convergence of this algorithm is also provided, using some mild assumptions. Finally, a comprehensive numerical comparison with the relevant algorithms shows that the proposed algorithm is efficient. [ABSTRACT FROM AUTHOR]

This manuscript investigates a fractional piecewise dengue transmission model using singular and non-singular kernels. The existence results and uniqueness of the solution are established by using the approach of fixed point and in the framework of piecewise derivative and integral. To obtain the approximate solution of the considered models we apply a piecewise numerical iteration scheme which is based on Newton interpolation polynomials. Furthermore, the numerical scheme for piecewise derivatives encompasses singular and non-singular kernels. This study aims to enhance our understanding of dengue internal transmission dynamics by using a novel piecewise derivative approach that considers both singular and non-singular kernels. This work contributes to clarifying the concept of piecewise derivatives and their significance in understanding crossover dynamics. Moreover, a deep neural network approach is employed with high accuracy in training, testing, and validation of data to investigate the specified disease problem. This methodology is employed to thoroughly investigate the intricacies of the specified disease problem. [ABSTRACT FROM AUTHOR]

In this article, we investigate a tumor-immune and antigen-presenting cells population in the form of a mathematical model. To achieve greater accuracy in understanding the spread of tumor and immune cell populations, we apply the Caputo–Fabrizio (CF) fractional-order derivative. The fixed-point theorems are employed to analyze the uniqueness and existence of the model. The Laplace transform along with the Adomian decomposition approach is used to construct an algorithm for a semi-analytical solution under the CF fractional derivative. The chaotic dynamics of the cancer-immune model are confirmed by the Lyapunov exponents. The article examines how different vaccination protocols can affect tumor dormancy and recurrence. Furthermore, we offer a description for why adoptive immunotherapy techniques may potentially increase tumor growth rather than suppress it. [ABSTRACT FROM AUTHOR]

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings ( U D E T - convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for U D E T - convex   F N V M s. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems. [ABSTRACT FROM AUTHOR]

The scattering of x linearly polarized non-diffracting Lommel beam by a perfect electromagnetic conductor (PEMC) sphere is analyzed within the generalized Lorenz–Mie theory (GLMT) framework. The electromagnetic fields of the incident Lommel beams are expressed by employing spherical vector wave functions (SVWFs) and beam shape coefficients (BSCs). These BSCs are mainly considered in the expression of shaped beams in the GLMT with slight beam axicon angle. The scattered electromagnetic fields are constructed using unknown scattering field coefficients (SFCs) and SVWFs. These undetermined SFCs are calculated employing boundary conditions (BCs) at the PEMC surface. The expressions of the far-region/far-zone scattering intensity (FSI) are given and computed. The influences of Lommel beam configuration parameters like as beam order, beam half cone angle, beam asymmetry parameter, beam center coordinates positions, and PEMC sphere size parameter in addition to PEMC scalar admittance on the FSI are analyzed and discussed. The experience of this research work will be quite helpful in analyzing optical radiation force and torque, electromagnetic scattering, light manipulation, antenna engineering domain, optical tweezers, particle interaction, and trapping. Moreover, findings of the study can also be extended to study the interaction between the non-diffracting Lommel beam and the metamaterial structures. [ABSTRACT FROM AUTHOR]

In this work, we use symbolic computation and ansatz function schemes, to investigate the soliton solutions for the nonlinear Schrödinger equation (NLSE) along with Kudryashov's quintuple self-phase modulation system (KQSPMS) including dual-form of nonlocal nonlinearity (DFNLN). We initially determine the ordinary differential (OD) form for this model through a variable transformation. Then we introduce numerous new dynamical soliton types: the M-shaped rational soliton, the M-shaped interaction between one and two stripe solitons, the periodic cross-M-shaped rational (PCMR) soliton, the periodic cross-kink (PCK) soliton, multi-waves, and the homoclinic breather soliton. Secondly, we determine the partial differential (PD) form for this model through a variable transformation. Moreover, a lump soliton, a periodic wave, a rogue wave, a lump interaction with a periodic and kink wave, and three different types of breather soliton will obtain. We'll demonstrate these solutions' unique structure and extremely interesting interaction behavior. We'll also use graphs (3-D and contour plots) to discuss the dynamics of the results after setting the parameters to the proper values. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger–Poisson equation with gravitational potential field is examined, along with its many soliton interactions and travelling wave solutions. By applying Hirota Bilinear scheme, we will give a brief study on transformation of solitons, also by Sine–Gordon expansion scheme solitary travelling wave solutions will be obtained in terms of trigonometric and hyperbolic functions. The innovative Hirota bilinear method is a powerful and standard technique, play a crucial role in producing soliton solutions as well as lump solutions. With the assistance of this method one can transformed integrable equation into Hirota bilinear form under dependent variable transformation. Soliton solutions obtained by Hirota bilinear method are significant in mathematical physics, may be superimposed in fibers. These solitons are applicable in optical communications, which enable to produce faster, richer, more secure and more flexible communication systems. Also we will investigate the gravitational potential's behaviour on soliton solutions, such as parabolic, bright, dark, anti-dark, combination and S-shaped solitons. Contour plots and three-dimensional soliton solutions are provided. These solitons are important in nonlinear processes, magnetic devices, and lasers beams because of variations in gravitational field. [ABSTRACT FROM AUTHOR]

In this research, we study a variety of solutions like periodic, exponential, rational, hyperbolic and Jacobian elliptic function solutions of Rangwala–Rao equation (RRE) by employing complete discrimination system for polynomial (CDSP) technique. We deduce the relevant travelling wave system from the original equation using the travelling wave transformation and generate a conserved quantity, the Hamiltonian, from it. With the aid of qualitative theory of differential equation and bifurcation theory of planar dynamical systems a variety of phase portraits to the relevant travelling wave system of RRE are explored. The CDSP approach is not only useful for generating soliton solutions but also for carrying out qualitative analysis. Furthermore, 3D and 2D graphs are displayed for some of the soliton solutions of RRE. [ABSTRACT FROM AUTHOR]

In this paper, we derived a modified conjugate gradient (CG) parameter by adopting the Birgin and Mart i ´ nez strategy using the descent three-term CG direction and the Newton direction. The proposed CG parameter is applied and suggests a robust algorithm for solving constrained monotone equations with an application to image restoration problems. The global convergence of this algorithm is established using some proper assumptions. Lastly, the numerical comparison with some existing algorithms shows that the proposed algorithm is a robust approach for solving large-scale systems of monotone equations. Additionally, the proposed CG parameter can be used to solve the symmetric system of nonlinear equations as well as other relevant classes of nonlinear equations. [ABSTRACT FROM AUTHOR]

In this manuscript, various types of breather, periodic cross-kink and rational solitons, M-shaped rational solutions with one and two kink and multi-waves will be discussed for Bose–Einstien condensate (BEC) model. We will compute the Kuznetsov–Ma breather, Akhmediev breather and generalized breather along with discussion on the interaction between them. At the end we will present the graphical description for our newly obtained solutions. [ABSTRACT FROM AUTHOR]

Highlights • Bulging formation can be prevented in a bilayer tube of arbitrary thickness while retaining moderate extensibility. • Several critical parameters for avoiding localized bulging are determined. • The applicability of an explicit bifurcation condition to layered tubes is validated. Abstract This paper studies the bifurcation behavior in an inflated bilayer tube of arbitrary thickness under inflation and uni-axial extension. It is assumed that both layers are composed of the Gent material with each layer having its own J m , where J m is a material parameter in the Gent model that signifies the maximum extensibility. First, we determine several critical parametrical regions where localized bulging disappears for a single-layer tube. Then we investigate localized bulging in an inflated bilayer tube, where one layer (layer I) of the tube cannot bulge whereas the other part (layer II) can. Surprisingly, we find that such a composite tube is still susceptible to localized bulging and localized bulging can be prevented only if the proportion of layer I exceeds a critical value, no matter whether layer I occupies the inner side or the outer side. Even for a very thin bilayer tube, the same feature holds. The cases of fixed axial force and fixed axial stretch are both studied, and the critical geometrical parameters marking the transition between bulging and no bulging are determined. Moreover, we carry out a numerical analysis by use of the finite element method to verify the applicability of an explicit bifurcation condition and the predicted bifurcation behavior. This paper offers a possible way to avoid bulging formation in a cylindrical tube while retaining moderate extensibility. [ABSTRACT FROM AUTHOR]

Abstract The developed article considers SIR problems for the recent COVID-19 pandemic, in which each component is divided into two subgroups: young and adults. These subgroups are distributed among two classes in each compartment, and the effect of COVID-19 is observed in each class. The fractional problem is investigated using the non-singular operator of Atangana Baleanu in the Caputo sense (ABC). The existence and uniqueness of the solution are calculated using the fundamental theorems of fixed point theory. The stability development is also determined using the Ulam-Hyers stability technique. The approximate solution is evaluated using the fractional Adams-Bashforth technique, providing a wide range of choices for selecting fractional order parameters. The simulation is plotted against available data to verify the obtained scheme. Different fractional-order approximations are compared to integer-order curves of various orders. Therefore, this analysis represents the recent COVID-19 pandemic, differentiated by age at different fractional orders. The analysis reveals the impact of COVID-19 on young and adult populations. Adults, who typically have weaker immune systems, are more susceptible to infection compared to young people. Similarly, recovery from infection is higher among young infected individuals compared to infected cases in adults. [ABSTRACT FROM AUTHOR]

Thermal performance can be enhanced due to the mixing of nanoparticles in base fluid. This research discusses the involvement of ternary hybrid nanoparticles in the mixture of pseudo-plastic fluid model past over a two dimensional porous stretching sheet. Modelling of energy equation is carried out in the presence of external heat source or sink and viscous dissipation. The flow presenting equations and derived in Cartesian coordinate system under usual boundary layer theory in the form of complex coupled partial differential equations (PDEs). The derived PDEs have been converted into corresponding ordinary differential equations (ODEs) with the engagement of suitable transformation. The engineers, scientists and mathematicians have great interest in the solution of differential equations because to understand the real physics of the problem. Here, finite element scheme has been used to approximate the solution of the converted problem. The contribution of several emerging parameters on solution have been displayed through graphs and discussed. It is recommended that the finite element method can be engaged to approximate the solution of nonlinear problems arising in modelling the problem in mathematical physics. [ABSTRACT FROM AUTHOR]

Thermal performance can be enhanced due to the mixing of nanoparticles in base fluid. This research discusses the involvement of ternary hybrid nanoparticles in the mixture of pseudo-plastic fluid model past over a two dimensional porous stretching sheet. Modelling of energy equation is carried out in the presence of external heat source or sink and viscous dissipation. The flow presenting equations and derived in Cartesian coordinate system under usual boundary layer theory in the form of complex coupled partial differential equations (PDEs). The derived PDEs have been converted into corresponding ordinary differential equations (ODEs) with the engagement of suitable transformation. The engineers, scientists and mathematicians have great interest in the solution of differential equations because to understand the real physics of the problem. Here, finite element scheme has been used to approximate the solution of the converted problem. The contribution of several emerging parameters on solution have been displayed through graphs and discussed. It is recommended that the finite element method can be engaged to approximate the solution of nonlinear problems arising in modelling the problem in mathematical physics. [ABSTRACT FROM AUTHOR]

In the present study, we explore the time-dependent convectional flow of a rheological nanofluid over a turning cone with the consolidated impacts of warmth and mass exchange. It has been shown that if the angular velocity at the free stream and the cone's angular velocity differ inversely as a linear time function, a self-similar solution can be obtained. By applying sufficient approximation to the boundary layer, the managed conditions of movement, temperature, and nanoparticles are improved; afterward, the framework is changed to a non-dimensional framework utilizing proper comparability changes. A numerical solution for the obtained system of governing equations is achieved. The effect of different parameters on the velocity, temperature, and concentration profiles are discussed. Tangential velocity is observed to decrease with an increase in the Deborah number, whereas tangential velocity increases with increasing values of the angular velocity ratio, relaxation to the retardation time ratio, and buoyancy parameter. Expansion in the Prandtl number is noted to decrease the boundary layer temperature and thickness. The temperature is seen to decrease with an expansion in the parameters of lightness, thermophoresis parameter, and Brownian movement. It is discovered that the Nusselt number expands by expanding the lightness parameter and Prandtl number, whereas it increases by decreasing the Deborah number. We also noticed that the Sherwood number falls incrementally in Deborah and Prandtl numbers, but it upsurges with an increase in the buoyancy parameter. [ABSTRACT FROM AUTHOR]

In this paper, we study chirp-free solitons (dromions) for the higher-order nonlinear Schrödinger equation (HNLSE) having non-Kerr terms with the aid of sub-ODE scheme. We find bright and singular dromion with domain walls, rational and periodic solitons, Weierstrass elliptic functions (WEFs), Jacobi elliptic solutions (JESs) and other solitary wave solutions for our governing model. The Sub-ODE scheme is very useful scheme, providing so many new solitary wave solutions especially JES and WEF solutions. We will also construct graphs of our results in different dimensions along with constraint conditions. [ABSTRACT FROM AUTHOR]

In the present work, the propagation of (non)linear electrostatic waves is reported in a normal (electron–ion) magnetoplasma. The inertialess electrons follow a non-extensive q-distribution, while the positive inertial ions are assumed to be warm mobile. In the linear regime, the dispersion relation for both the fast and slow modes is derived, whose properties are analyzed parametrically, focusing on the effect of nonextensive parameter, component of parallel anisotropic ion pressure, component of perpendicular anisotropic ion pressure, and magnetic field strength. The reductive perturbation technique is employed for reducing the fluid equation of the present plasma model to a Zakharov–Kuznetsov (ZK) equation. The parametric role of physical parameters on the characteristics of the symmetry planar structures such solitary waves is investigated. Furthermore, the stability of the pulse soliton solution of the ZK equation against the oblique perturbations is investigated. Furthermore, the dependence of the instability growth rate on the related physical parameters is examined. The present investigation could be useful in space and astrophysical plasma systems. [ABSTRACT FROM AUTHOR]

This study focuses on the Ablowitz–Kaup–Newell–Segur (AKNS) water waves equation. Painleve test (P-test) will be implemented to check the integrability of AKNS equation and an extended modified auxiliary equation mapping (EMAEM) architectonic is implemented to get a new set of traveling wave solutions like periodic and doubly periodic, bell type, kink, singular kink, anti-kink, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions. Furthermore, it is analyzed that the implemented algorithm is efficient and accurate for solving nonlinear evolution equations (NLEEs). Finally, graphical simulations (2D, 3D and contours) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on the Davey–Stewartson (DS)-II equation, and the extended modified auxiliary equation mapping (EMAEM) architectonic is used to develop a new set of solutions such as kink, singular kink, rational, combined soliton-like solutions, bell-type solutions, trigonometric and hyperbolic solutions. Furthermore, this study reveals that the used technique is efficient for solving other nonlinear evolution equations (NLEEs). The Painleve test (P -test) will also be used to examine the integrability of the DS-II equation. Finally, graphical simulations are designed to show the exact behavior of solutions as well as the efficacy of the suggested strategy. [ABSTRACT FROM AUTHOR]

In this work, we utilize the extended simplest equation method to obtain exact solutions for the higher-order nonlinear Schrödinger's equation with fourth-order dispersion and cubic-quintic nonlinearity that describe the propagation of extremely short pulses in optical fibers. With the aid of this method, we get many exact solutions like singular-bright combo soliton solutions, bright, dark and singular soliton solutions, periodic solutions and other solutions. Moreover, for the physical illustration of the obtained solutions, 3D and 2D graphs are presented. [ABSTRACT FROM AUTHOR]

In this paper, we will compute the conservation laws (CLs) of Chen–Lee–Liu equation (CLLE) with the help of scaling invariance technique. We will use Euler and Homotopy operators for the evaluation of conserved densities and fluxes. We will also obtain optical dromions with the help of Unified method (UM). By using this method, we will get domain walls, solitary wave and elliptic wave solutions. Moreover, we will investigate the stability as well as the integrability of the governing model by using linear stability technique and Painlevé analysis respectively. [ABSTRACT FROM AUTHOR]

Correction to: I Scientific Reports i https://doi.org/10.1038/s41598-022-12857-3, published online 02 June 2022 In the original version of this Article Muhammad Sohail was incorrectly affiliated with 'Department of Mechanical Engineering, Sejong University, Seoul, 05006, Korea'. The original article can be found online at https://doi.org/10.1038/s41598-022-12857-3. [Extracted from the article]

[Display omitted] • Jet flow of ternary nanofluid is considered. • Modified Buongiorno's model is used for the bioconvective flow of gyrotactic microorganisms. • Chemical Reaction is included. • Three different types of nanoparticles rGO , F e 3 O 4 , T i O 2 are suspended into H 2 O. • Thermophoresis and Brownian motion are discussed along with the nanoparticle volume fraction. A few typographical errors have been identified in our paper titled "Implementation of modified Buongiorno's model for the investigation of chemically reacting rGO - F e 3 O 4 - T i O 2 - H 2 O ternary nanofluid jet flow in the presence of bio-active mixers". This corrigendum addresses those errors however, these errors have no impact on the obtained results. [ABSTRACT FROM AUTHOR]

This article examined the Silver based Di‑hydrogen carbon nanotubes flow model (SDH-CNTFM) between two stretchable coaxially disks by utilizing the Method of Levenberg Marquardt with Back-propagated Neural Networks (MLM-BPNN). Here the base liquid is silver (Ag) and the nanoparticles are SWCNTs and MWCNTs (single and multiwall carbon nanotubes). The governing PDEs for SDH-CNTFM are transformed into ODEs by utilizing similarity transformation. Energy equations are developed through heat generation and viscous dissipation joule heating. Also calculated the total entropy optimization. Flow parameters velocity, entropy optimization, temperature, Nusselt number and Bejan number are discussed for both single and multi-walls carbon nanotubes (SWCNTs and MWCNTs) graphically and in Tabular form. The reference dataset is calculated through implementation of Optimal Homotopy Analysis method (OHAM) for variants of SDH-CNTFM. For the variation of different parameters this reference dataset is utilized in MATLAB to clarify the solution and error analysis plots. Moreover, the approximated solution is assessed through adopting training/testing/validation procedure and comparing it with standard solution which is endorsed by performance study based on MSE convergence, error histogram and regression studies. Heat transfer rate and surface drag force are discussed for both SWCNTs and MWCNTs numerically by using different flow parameters. From obtained outcomes, it is observed that entropy rate boosts up for higher approximation of nanoparticles of volume friction and Brickman number (Br) which is controlled due to the minimization of Brickman number. [ABSTRACT FROM AUTHOR]

[Display omitted] The analysis of enhancing the heat transfer of a traditional fluid by adding nanoparticles was effectively studied by many researchers across the globe. In later stages, these nanofluids were made chemically stable by suspending an additional inert nanoparticle thus forming a hybrid nanofluid. The heat transfer characteristics of hybrid nanofluids are discussed in various aspects. Considering these studies, the heat and mass transfer characteristics of ternary nanofluid formed by suspending three different nanoparticles is analysed in this article. The self-propelled microorganisms move within the nanofluid due to the density gradient and it ensures proper mixing of nanoparticles. In order to achieve proper bioconvection caused by microorganisms, the nanoparticle concentration is assumed to be dilute and the fluid with these characteristics is assumed to flow as a jet past a stretching sheet. The mathematical model to analyse such a characteristic flow is framed using the modified Buongiorno's model that describes the impact of volume fraction, thermophoresis and Brownian motion. The mathematical model obtained will be further converted into non-linear differential equations that are solved through the RKF - 45 method. The results obtained through this method are interpreted graphically and the impact of fluid flow parameters on the heat and mass transfer rates are tabulated. It is perceived that the mixed convection parameter enhances the velocity profile. Similarly, the increase in the Brownian motion and thermophoresis enhances the thermal profile. Meanwhile, the increase in the nanoparticle volume fraction helps in enhancing the thermal conductivity and thus the temperature is found to be increasing. [ABSTRACT FROM AUTHOR]

In this paper, we consider the time-fractional two-mode coupled Burgers equation with the Caputo fractional derivative. A modified homotopy perturbation method coupled with Laplace transform (He-Laplace method) is applied to find its approximate analytical solution. The method is to decompose the equation into a series of linear equations, which can be effectively and easily solved by the Laplace transform. The solution process is illustrated step by step, and the results show that the present method is extremely powerful for fractional differential equations. [ABSTRACT FROM AUTHOR]