STABLE NUMERICAL RESULTS TO A CLASS OF TIME-SPACE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA SPECTRAL METHOD

The aim of this work is to present numerical treatments to a complex order fractional nonlinear onedimensional problem of Burgers’ equations. A new parameter rt is presented in order to be consistent with the physical model problem. This parameter characterizes the existence of fractional structures[r]

Shandong University, Jinan 250061, Shandong, PR ChinaAuthors’ contributionsThe work presented here was carried out in collaboration between all authors.YZ carried out the design of the study, the statistical analysis and drafted the manuscript.SS and ZH conceived, instructed the design of<[r]

to the incompressible flow. Several authors h ave employ ed this method successfully incomputing unsteady problems. Mercle and Athavale [4] presented solution using thisapproach in tw o -dimensional generalized coordinates. P ark and Sankar [5] also presentsolutions for three-dimensiona[r]

gt, xt − ks, 1.2where x ∈ Rp, f ∈ CRp, Rp,andg ∈ CR × Rp, Rp,andr&gt;0,s &gt;0 are two given constants.2 Journal of Inequalities and ApplicationsFor the special case that n  1andp  1, various problems on the solutions of 1.1,such as the existence of periodi[r]

A study of non-commensurate fractional linear system is done in a parallel way to the commensurate case. A partial fraction decomposition is accomplished using a recursive procedure. Each partial fraction is inverted in two different ways. The decomposition integer/fractional is done also. Some exam[r]

Many dynamical processes contain both continuous and discrete elements simultaneously.Thus, traditional mathematical modeling techniques, such as differential equations ordifference equations, provide a limited understanding of these types of models. A simplee[r]

401–406].Although many authors have avoided repetitions by working on the abstract space Ꮾ,the delicacy of some investigations restricts their work on a class of concrete spaces thatverify many properties such as Cγwith γ&gt;0. In [1, 6–8], we have considered (1.1)wit[r]

2∆x(19.1.11)which can easily be rearranged to be a formula for un+1jin terms of the otherquantities. The FTCS scheme is illustrated in Figure 19.1.1. It’s a fine example ofan algorithm that is easy to derive, takes little storage, and executes quickly. Toobad it do[r]

∂x2(19.0.1)where v = constant is the velocity of wave propagation. The prototypicalparabolicequation is the diffusion equation∂u∂t=∂∂xD∂u∂x(19.0.2)where D is the diffusion coefficient. The prototypical elliptic equation is thePoisson equation∂2u∂x2+∂2u∂y2= ρ(x, y)(19.0.3)

6.2.3. Theorems about Differentiable Functions. L’Hospital Rule . . . . . . . . . . . . . . . . . 2546.2.4. Higher-Order Derivatives and Differentials. Taylor’s Formula . . . . . . . . . . . . . . . 2556.2.5. Extremal Points. Points of Inflection 2576.2.6. Qualitative Analysis of Functi[r]

matrix and SNR in advance but can achieve almost thesame performance with the conventional LMMSE channelestimation in terms of the normalized mean square error(NMSE) of channel estimation and bit error rate (BER).Secondly, the proposed method needs only fast Fouriertransform (FF[r]

In this paper, the Mawhin’s continuation theorem in the theory of coincidence degree has been used to investigate the existence of solutions for a class of nonlinear second-order differential systems of equations in ℝ

the necessary condition (15.14.1.2) results in an equation with x, y,andz. Treating it asan algebraic (transcendental) equation for z,wefind z = z(x, y). The direct substitution ofthe expression z = z(x, y) into both equations (15.14.1.1) gives an answer to the questionwhether it[r]

y = –2–23Figure 31.9◆EXAMPLE 31.17 An industrial plant produces radioactive material at a constant rate of 4 kilograms per year.The radioactive material decays at a rate proportional to the amount present and has a half-life of 20 years.(a) Write a[r]

speed. Unlike the rapid methods, however, the multigrid methods can solve generalelliptic equations with nonconstant coefficients with hardly any loss in efficiency.Even nonlinear equations can be solved with comparable speed.Unfortunately there is not a single multigrid algorithm[r]

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper c[r]

(New York:McGraw-Hill), Chapter 6. [5]Temperton, C. 1980,Journal of Computational Physics, vol. 34, pp. 314–329. [6]19.5 Relaxation Methods for Boundary ValueProblemsAs we mentioned in §19.0, relaxation methods involve splitting the sparsematrix that arises from finite differencing and then it[r]

The authors declare that they have no competing interests.Authors’ contributionsQZ carried out the theoretical proof and drafted the manuscript. XH participated in thedesign and coordination. Both of the two authors read and approved the final manuscript.References[1] Liapunov, AM: Probl`eme g[r]

scale of interest. This number of steps is usually prohibitive. We must thereforefind a stable way of taking timesteps comparable to, or perhaps — for accuracy —somewhat smaller than, the time scale of (19.2.7).This goal poses an immediate “phil[r]

AcknowledgmentsWe would like to thank professor Fang AiNong for many useful comments on the sub-ject, the project supported by the Natural Science Foundation of Hebei University, theNational Natural Science Foundation of China under Grant 10471039, and the ZhejiangProvince Natur[r]