Advances in Mathematical Physics The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions Wed, 22 Feb 2017 13:37:29 +0000 A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method. Bashar Zogheib, Emran Tohidi, and Stanford Shateyi Copyright © 2017 Bashar Zogheib et al. All rights reserved. Third-Order Approximate Solution of Chemical Reaction-Diffusion Brusselator System Using Optimal Homotopy Asymptotic Method Wed, 22 Feb 2017 00:00:00 +0000 The objective of this paper is to investigate the effectiveness and performance of optimal homotopy asymptotic method in solving a system of nonlinear partial differential equations. Since mathematical modeling of certain chemical reaction-diffusion experiments leads to Brusselator equations, it is worth demanding a new technique to solve such a system. We construct a new efficient recurrent relation to solve nonlinear Brusselator system of equations. It is observed that the method is easy to implement and quite valuable for handling nonlinear system of partial differential equations and yielding excellent results at minimum computational cost. Analytical solutions of Brusselator system are presented to demonstrate the viability and practical usefulness of the method. The results reveal that the method is explicit, effective, and easy to use. Salem Alkhalaf Copyright © 2017 Salem Alkhalaf. All rights reserved. Hyers-Ulam Stability and Existence of Solutions for Nigmatullin’s Fractional Diffusion Equation Tue, 21 Feb 2017 00:00:00 +0000 We discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability of time-fractional order heat conduction equations via fractional Green function involving Wright function. In addition, an interesting existence result for solution is given. Zhuoyan Gao and JinRong Wang Copyright © 2017 Zhuoyan Gao and JinRong Wang. All rights reserved. The Cauchy Problem for Space-Time Monopole Equations in Temporal and Spatial Gauge Mon, 20 Feb 2017 06:39:57 +0000 We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition and the temporal gauge condition . Hyungjin Huh and Jihyun Yim Copyright © 2017 Hyungjin Huh and Jihyun Yim. All rights reserved. Turing Bifurcation and Pattern Formation of Stochastic Reaction-Diffusion System Sun, 12 Feb 2017 09:22:50 +0000 Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction-Diffusion System (RDS), Stochastic Reaction-Diffusion System (SRDS) is more complex and it is very difficult to deal with the noise function. In this paper, we have presented a method to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation. Qianiqian Zheng, Zhijie Wang, Jianwei Shen, and Hussain Muhammad Ather Iqbal Copyright © 2017 Qianiqian Zheng et al. All rights reserved. Third-Order Conditional Lie–Bäcklund Symmetries of Nonlinear Reaction-Diffusion Equations Thu, 09 Feb 2017 13:19:02 +0000 The third-order conditional Lie–Bäcklund symmetries of nonlinear reaction-diffusion equations are constructed due to the method of linear determining equations. As a consequence, the exact solutions of the resulting equations are derived due to the compatibility of the governing equations and the admitted differential constraints, which are resting on the characteristic of the admitted conditional Lie–Bäcklund symmetries to be zero. Keqin Su and Jie Cao Copyright © 2017 Keqin Su and Jie Cao. All rights reserved. The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses Thu, 09 Feb 2017 06:12:58 +0000 Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result. Xianzhen Zhang, Zuohua Liu, Hui Peng, Xianmin Zhang, and Shiyong Yang Copyright © 2017 Xianzhen Zhang et al. All rights reserved. Unsteady Helical Flows of a Size-Dependent Couple-Stress Fluid Wed, 08 Feb 2017 12:46:06 +0000 The helical flows of couple-stress fluids in a straight circular cylinder are studied in the framework of the newly developed, fully determinate linear couple-stress theory. The fluid flow is generated by the helical motion of the cylinder with time-dependent velocity. Also, the couple-stress vector is given on the cylindrical surface and the nonslip condition is considered. Using the integral transform method, analytical solutions to the axial velocity, azimuthal velocity, nonsymmetric force-stress tensor, and couple-stress vector are obtained. The obtained solutions incorporate the characteristic material length scale, which is essential to understand the fluid behavior at microscales. If characteristic length of the couple-stress fluid is zero, the results to the classical fluid are recovered. The influence of the scale parameter on the fluid velocity, axial flow rate, force-stress tensor, and couple-stress vector is analyzed by numerical calculus and graphical illustrations. It is found that the small values of the scale parameter have a significant influence on the flow parameters. Qammar Rubbab, Itrat Abbas Mirza, Imran Siddique, and Saadia Irshad Copyright © 2017 Qammar Rubbab et al. All rights reserved. Dirichlet Forms Constructed from Annihilation Operators on Bernoulli Functionals Wed, 08 Feb 2017 00:00:00 +0000 The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anticommutation relation (CAR) in equal-time. As a mathematical structure, Dirichlet forms play an important role in many fields in mathematical physics. In this paper, we apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Let be a nonnegative function on . By using the Bernoulli annihilators, we first define in a dense subspace of -space of Bernoulli functionals a positive, symmetric, bilinear form associated with . And then we prove that is closed and has the contraction property; hence, it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with on -space of Bernoulli functionals, which we call the -Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, we show that the -Ornstein-Uhlenbeck semigroup is a Markov semigroup. Caishi Wang and Beiping Wang Copyright © 2017 Caishi Wang and Beiping Wang. All rights reserved. A Non-Convex Partition of Unity and Stress Analysis of a Cracked Elastic Medium Tue, 07 Feb 2017 10:00:31 +0000 A stress analysis using a mesh-free method on a cracked elastic medium needs a partition of unity for a non-convex domain whether it is defined explicitly or implicitly. Constructing such partition of unity is a nontrivial task when we choose to create a partition of unity explicitly. We further extend the idea of the almost everywhere partition of unity and apply it to linear elasticity problem. We use a special mapping to build a partition of unity on a non-convex domain. The partition of unity that we use has a unique feature: the mapped partition of unity has a curved shape in the physical coordinate system. This novel feature is especially useful when the enrichment function has polar form, , because we can partition the physical domain in radial and angular directions to perform a highly accurate numerical integration to deal with edge-cracked singularity. The numerical test shows that we obtain a highly accurate result without refining the background mesh. Won-Tak Hong Copyright © 2017 Won-Tak Hong. All rights reserved. Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator Thu, 02 Feb 2017 06:23:35 +0000 Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered. The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution. The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at are also considered. Augusto Beléndez, Enrique Arribas, Tarsicio Beléndez, Carolina Pascual, Encarnación Gimeno, and Mariela L. Álvarez Copyright © 2017 Augusto Beléndez et al. All rights reserved. Stability Monitoring of Batch Processes with Iterative Learning Control Tue, 31 Jan 2017 10:04:13 +0000 In recent years, the iterative learning control (ILC) is widely used in batch processes to improve the quality of the products. Stability is a preoccupation of batch processes when the ILC is applied. Focusing on the stability monitoring of batch processes with ILC, a method based on innerwise matrix with considering the uncertainty of the model and disturbance was proposed. First, the batch process with ILC was derived as a two-dimensional autoregressive and moving average (2D-ARMA) model. Then two kinds of stability indices are constructed based on the innerwise matrix through the identification of the 2D-ARMA. Finally, the statistical process control (SPC) chart was adopted to monitor those stability indices. Numerical results are presented to demonstrate the effectiveness of the proposed method. Yan Wang, Junwei Sun, Taishan Lou, and Lexiang Wang Copyright © 2017 Yan Wang et al. All rights reserved. The Analytical Form of the Dispersion Equation of Elastic Waves in Periodically Inhomogeneous Medium of Different Classes of Crystals Sun, 29 Jan 2017 13:57:18 +0000 The investigation of thermoelastic wave propagation in elastic media is bound to have much influence in the fields of material science, geophysics, seismology, and so on. The heat conduction equations and bound equations of motions differ by the difficulty level and presence of many physical and mechanical parameters in them. Therefore thermoelasticity is being extensively studied and developed. In this paper by using analytical matrizant method set of equation of motions in elastic media are reduced to equivalent set of first-order differential equations. Moreover, for given set of equations, the structure of fundamental solutions for the general case has been derived and also dispersion relations are obtained. Nurlybek A. Ispulov, Abdul Qadir, Marat K. Zhukenov, Talgat S. Dossanov, and Tanat G. Kissikov Copyright © 2017 Nurlybek A. Ispulov et al. All rights reserved. Asymptotic Stability and Asymptotic Synchronization of Memristive Regulatory-Type Networks Thu, 26 Jan 2017 14:36:30 +0000 Memristive regulatory-type networks are recently emerging as a potential successor to traditional complementary resistive switch models. Qualitative analysis is useful in designing and synthesizing memristive regulatory-type networks. In this paper, we propose several succinct criteria to ensure global asymptotic stability and global asymptotic synchronization for a general class of memristive regulatory-type networks. The experimental simulations also show the performance of theoretical results. Jin-E Zhang Copyright © 2017 Jin-E Zhang. All rights reserved. Calculations on Lie Algebra of the Group of Affine Symplectomorphisms Mon, 23 Jan 2017 00:00:00 +0000 We find the image of the affine symplectic Lie algebra from the Leibniz homology to the Lie algebra homology . The result shows that the image is the exterior algebra generated by the forms . Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such a map is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology is isomorphic to . Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations. Zuhier Altawallbeh Copyright © 2017 Zuhier Altawallbeh. All rights reserved. Symmetries and Properties of the Energy-Casimir Mapping in the Ball-Plate Problem Wed, 18 Jan 2017 10:11:55 +0000 In this paper a system derived by an optimal control problem for the ball-plate dynamics is considered. Symplectic and Lagrangian realizations are given and some symmetries are studied. The image of the energy-Casimir mapping is described and some connections with the dynamics of the considered system are presented. Cristian Lăzureanu and Tudor Bînzar Copyright © 2017 Cristian Lăzureanu and Tudor Bînzar. All rights reserved. Principal Component Analysis in the Nonlinear Dynamics of Beams: Purification of the Signal from Noise Induced by the Nonlinearity of Beam Vibrations Mon, 16 Jan 2017 08:55:58 +0000 The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA). A. V. Krysko, Jan Awrejcewicz, Irina V. Papkova, Olga Szymanowska, and V. A. Krysko Copyright © 2017 A. V. Krysko et al. All rights reserved. New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations Mon, 16 Jan 2017 05:59:15 +0000 It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function . This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval . This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution . A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods. Jian Rong Loh, Chang Phang, and Abdulnasir Isah Copyright © 2017 Jian Rong Loh et al. All rights reserved. Analysis of Waterman’s Method in the Case of Layered Scatterers Mon, 16 Jan 2017 00:00:00 +0000 The method suggested by Waterman has been widely used in the last years to solve various light scattering problems. We analyze the mathematical foundations of this method when it is applied to layered nonspherical (axisymmetric) particles in the electrostatic case. We formulate the conditions under which Waterman’s method is applicable, that is, when it gives an infinite system of linear algebraic equations relative to the unknown coefficients of the field expansions which is solvable (i.e., the inverse matrix exists) and solutions of the truncated systems used in calculations converge to the solution of the infinite system. The conditions obtained are shown to agree with results of numerical computations. Keeping in mind the strong similarity of the electrostatic and light scattering cases and the agreement of our conclusions with the numerical calculations available for homogeneous and layered scatterers, we suggest that our results are valid for light scattering as well. Victor Farafonov, Vladimir Il’in, Vladimir Ustimov, and Evgeny Volkov Copyright © 2017 Victor Farafonov et al. All rights reserved. On the Symmetries and Conservation Laws of the Multidimensional Nonlinear Damped Wave Equations Thu, 12 Jan 2017 00:00:00 +0000 We carry out a classification of Lie symmetries for the ()-dimensional nonlinear damped wave equation with variable damping. Similarity reductions of the equation are performed using the admitted Lie symmetries of the equation and some interesting solutions are presented. Employing the multiplier approach, admitted conservation laws of the equation are constructed for some new, interesting cases. Usamah S. Al-Ali, Ashfaque H. Bokhari, A. H. Kara, and F. D. Zaman Copyright © 2017 Usamah S. Al-Ali et al. All rights reserved. The Neumann Problem for a Degenerate Elliptic System Near Resonance Wed, 11 Jan 2017 12:44:35 +0000 This paper studies the following system of degenerate equations , , , , , Here is a bounded domain, and is the exterior normal vector on . The coefficient function may vanish in , with . We show that the eigenvalues of the operator are discrete. Secondly, when the linear part is near resonance, we prove the existence of at least two different solutions for the above degenerate system, under suitable conditions on , and . Yu-Cheng An and Hong-Min Suo Copyright © 2017 Yu-Cheng An and Hong-Min Suo. All rights reserved. Dynamics of a Computer Virus Propagation Model with Delays and Graded Infection Rate Wed, 04 Jan 2017 13:46:51 +0000 A four-compartment computer virus propagation model with two delays and graded infection rate is investigated in this paper. The critical values where a Hopf bifurcation occurs are obtained by analyzing the distribution of eigenvalues of the corresponding characteristic equation. In succession, direction and stability of the Hopf bifurcation when the two delays are not equal are determined by using normal form theory and center manifold theorem. Finally, some numerical simulations are also carried out to justify the obtained theoretical results. Zizhen Zhang and Limin Song Copyright © 2017 Zizhen Zhang and Limin Song. All rights reserved. Some Discussions about the Error Functions on SO(3) and SE(3) for the Guidance of a UAV Using the Screw Algebra Theory Wed, 04 Jan 2017 12:03:16 +0000 In this paper a new error function designed on 3-dimensional special Euclidean group SE(3) is proposed for the guidance of a UAV (Unmanned Aerial Vehicle). In the beginning, a detailed 6-DOF (Degree of Freedom) aircraft model is formulated including 12 nonlinear differential equations. Secondly the definitions of the adjoint representations are presented to establish the relationships of the Lie groups SO(3) and SE(3) and their Lie algebras so(3) and se(3). After that the general situation of the differential equations with matrices belonging to SO(3) and SE(3) is presented. According to these equations the features of the error function on SO(3) are discussed. Then an error function on SE(3) is devised which creates a new way of error functions constructing. In the simulation a trajectory tracking example is given with a target trajectory being a curve of elliptic cylinder helix. The result shows that a better tracking performance is obtained with the new devised error function. Yi Zhu, Xin Chen, and Chuntao Li Copyright © 2017 Yi Zhu et al. All rights reserved. A New Unconditionally Stable Method for Telegraph Equation Based on Associated Hermite Orthogonal Functions Thu, 29 Dec 2016 07:09:39 +0000 The present paper proposes a new unconditionally stable method to solve telegraph equation by using associated Hermite (AH) orthogonal functions. Unlike other numerical approaches, the time variables in the given equation can be handled analytically by AH basis functions. By using the Galerkin’s method, one can eliminate the time variables from calculations, which results in a series of implicit equations. And the coefficients of results for all orders can then be obtained by the expanded equations and the numerical results can be reconstructed during the computing process. The precision and stability of the proposed method are proved by some examples, which show the numerical solution acquired is acceptable when compared with some existing methods. Di Zhang, Fusheng Peng, and Xiaoping Miao Copyright © 2016 Di Zhang et al. All rights reserved. The Stochastic Resonance Behaviors of a Generalized Harmonic Oscillator Subject to Multiplicative and Periodically Modulated Noises Wed, 28 Dec 2016 09:03:30 +0000 The stochastic resonance (SR) characteristics of a generalized Langevin linear system driven by a multiplicative noise and a periodically modulated noise are studied (the two noises are correlated). In this paper, we consider a generalized Langevin equation (GLE) driven by an internal noise with long-memory and long-range dependence, such as fractional Gaussian noise (fGn) and Mittag-Leffler noise (M-Ln). Such a model is appropriate to characterize the chemical and biological solutions as well as to some nanotechnological devices. An exact analytic expression of the output amplitude is obtained. Based on it, some characteristic features of stochastic resonance phenomenon are revealed. On the other hand, by the use of the exact expression, we obtain the phase diagram for the resonant behaviors of the output amplitude versus noise intensity under different values of system parameters. These useful results presented in this paper can give the theoretical basis for practical use and control of the SR phenomenon of this mathematical model in future works. Suchuan Zhong, Kun Wei, Lu Zhang, Hong Ma, and Maokang Luo Copyright © 2016 Suchuan Zhong et al. All rights reserved. A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos Wed, 21 Dec 2016 14:12:55 +0000 A new no-equilibrium chaotic system is reported in this paper. Numerical simulation techniques, including phase portraits and Lyapunov exponents, are used to investigate its basic dynamical behavior. To confirm the chaotic behavior of this system, the existence of topological horseshoe is proven via the Poincaré map and topological horseshoe theory. Chunmei Wang, Chunhua Hu, Jingwei Han, and Shijian Cang Copyright © 2016 Chunmei Wang et al. All rights reserved. Stored Coulomb Self-Energy of a Uniformly Charged Rectangular Plate Sun, 18 Dec 2016 12:22:55 +0000 A large number of electronic devices contain charged, flat plates (electrodes) as their components. The approximation of considering such components as infinitely large plates is not satisfactory for the current status of consumer electronics where size is now extremely small. In particular, the nanotechnology revolution has made the fabrication of truly finite systems with arbitrary shape and characteristic lengths that measure in nanometers possible. As a result the only accurate approach for such situations is to consider the system realistically as one with a finite size extent. In this work we calculate the amount of electrostatic energy that is stored in a charged finite size electrode that is modelled as a uniformly charged rectangular plate with arbitrary length and width. Nontrivial mathematical transformations allow us to derive a closed form exact expression for the Coulomb self-energy of such a system as a function of its length and width (therefore, shape, too). The exact result derived can be useful to understand the storage process of electrostatic energy as a function of size/shape in uniformly charged plate systems. The result also applies to calculations that deal with the properties of a finite two-dimensional electron gas within the jellium model where the finite jellium domain can have an arbitrary rectangular shape. Orion Ciftja Copyright © 2016 Orion Ciftja. All rights reserved. The Rational Solutions and Quasi-Periodic Wave Solutions as well as Interactions of -Soliton Solutions for 3 + 1 Dimensional Jimbo-Miwa Equation Thu, 15 Dec 2016 10:28:01 +0000 The exact rational solutions, quasi-periodic wave solutions, and -soliton solutions of 3 + 1 dimensional Jimbo-Miwa equation are acquired, respectively, by using the Hirota method, whereafter the rational solutions are also called algebraic solitary waves solutions and used to describe the squall lines phenomenon and explained possible formation mechanism of the rainstorm formation which occur in the atmosphere, so the study on the rational solutions of soliton equations has potential application value in the atmosphere field; the soliton fission and fusion are described based on the resonant solution which is a special form of the -soliton solutions. At last, the interactions of the solitons are shown with the aid of -soliton solutions. Hongwei Yang, Yong Zhang, Xiaoen Zhang, Xin Chen, and Zhenhua Xu Copyright © 2016 Hongwei Yang et al. All rights reserved. Chebyshev Collocation Method for Parabolic Partial Integrodifferential Equations Thu, 15 Dec 2016 09:38:31 +0000 An efficient technique for solving parabolic partial integrodifferential equation is presented. This technique is based on Chebyshev polynomials and finite difference method. A priori error estimate for the proposed technique is deduced. Some examples are presented to illustrate the validity and efficiency of the presented method. M. Sameeh and A. Elsaid Copyright © 2016 M. Sameeh and A. Elsaid. All rights reserved. Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations Wed, 14 Dec 2016 13:25:22 +0000 This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method. Weishi Yin, Fei Xu, Weipeng Zhang, and Yixian Gao Copyright © 2016 Weishi Yin et al. All rights reserved.