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E-grāmata: Fractional Partial Differential Equations And Their Numerical Solutions

(South China Univ Of Technology, China), (Chongqing Univ, China), (Inst Of Applied Physics & Computational Mathematics, China)
  • Formāts: 348 pages
  • Izdošanas datums: 09-Mar-2015
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789814667067
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Fractional Partial Differential Equations And Their Numerical SolutionsPalielināt
  • Formāts: 348 pages
  • Izdošanas datums: 09-Mar-2015
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789814667067
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Guo, Pu, and Huang present students, academics, researchers, and practicing mathematicians with an examination of fractional calculus and related subjects. The authors have organized the main body of their text in six chapters devoted to fractional calculus and fractional differential equations, fractional partial differential equations, numerical approximations in fractional calculus, and a wide variety of other related subjects. Boling Guo is a faculty member of the Institute of Applied Physics and Computational Mathematics, China. Xueke Pu is a faculty member of Chongquing University, China. Fenghui Huang is a faculty member of South China University of Technology. Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)

Preface v
1 Physics Background
1(33)
1.1 Origin of the fractional derivative
1(4)
1.2 Anomalous diffusion and fractional advection-diffusion
5(11)
1.2.1 The random walk and fractional equations
6(4)
1.2.2 Fractional advection-diffusion equation
10(1)
1.2.3 Fractional Fokker-Planck equation
11(4)
1.2.4 Fractional Klein-Framers equation
15(1)
1.3 Fractional quasi-geostrophic equation
16(4)
1.4 Fractional nonlinear Schrodinger equation
20(3)
1.5 Fractional Ginzburg-Landau equation
23(4)
1.6 Fractional Landau-Lifshitz equation
27(2)
1.7 Some applications of fractional differential equations
29(5)
2 Fractional Calculus and Fractional Differential Equations
34(75)
2.1 Fractional integrals and derivatives
34(22)
2.1.1 Riemann-Liouville fractional integrals
34(8)
2.1.2 R-L fractional derivatives
42(6)
2.1.3 Laplace transforms of R-L fractional derivatives
48(2)
2.1.4 Caputo's definitions of fractional derivatives
50(2)
2.1.5 Weyl's definition for fractional derivatives
52(4)
2.2 Fractional Laplacian
56(26)
2.2.1 Definition and properties
56(7)
2.2.2 Pseudo-differential operator
63(6)
2.2.3 Riesz potential and Bessel potential
69(2)
2.2.4 Fractional Sobolev space
71(5)
2.2.5 Commutator estimates
76(6)
2.3 An existence theorem
82(7)
2.4 Distributed order differential equations
89(7)
2.4.1 Distributed order diffusion-wave equation
91(3)
2.4.2 Initial boundary value problem of distributed order
94(2)
2.5 Appendix A: the Fourier transform
96(8)
2.6 Appendix B: Laplace transform
104(2)
2.7 Appendix C: Mittag-Leffler function
106(3)
2.7.1 Gamma function and Beta function
106(1)
2.7.2 Mittag-Leffler function
107(2)
3 Fractional Partial Differential Equations
109(148)
3.1 Fractional diffusion equation
109(4)
3.2 Fractional nonlinear Schrodinger equation
113(25)
3.2.1 Space fractional nonlinear Schrodinger equation
113(12)
3.2.2 Time fractional nonlinear Schrodinger equation
125(4)
3.2.3 Global well-posedness of the one-dimensional fractional nonlinear Schrodinger equation
129(9)
3.3 Fractional Ginzburg-Landau equation
138(17)
3.3.1 Existence of weak solutions
138(5)
3.3.2 Global existence of strong solutions
143(7)
3.3.3 Existence of attractors
150(5)
3.4 Fractional Landau-Lifshitz equation
155(44)
3.4.1 Vanishing viscosity method
155(7)
3.4.2 Ginzburg-Landau approximation and asymptotic limit
162(8)
3.4.3 Higher dimensional case---Galerkin approximation
170(15)
3.4.4 Local well-posedness
185(14)
3.5 Fractional QG equations
199(30)
3.5.1 Existence and uniqueness of solutions
200(9)
3.5.2 Inviscid limit
209(4)
3.5.3 Decay and approximation
213(8)
3.5.4 Existence of attractors
221(8)
3.6 Fractional Boussinesq approximation
229(18)
3.7 Boundary value problems
247(10)
4 Numerical Approximations in Fractional Calculus
257(29)
4.1 Fundamentals of fractional calculus
258(3)
4.2 G-Algorithms for Riemann-Liouville fractional derivative
261(5)
4.3 D-Algorithm for Riemann-Liouville fractional derivative
266(3)
4.4 R-Algorithms for Riemann-Liouville fractional integral
269(3)
4.5 L-Algorithms for fractional derivative
272(2)
4.6 General form of fractional difference quotient approximations
274(2)
4.7 Extensions of integer-Order numerical differentiation and integration
276(7)
4.7.1 Extensions of backward and central difference quotient schemes
276(3)
4.7.2 Extension of interpolation-type integration quadrature formulas
279(1)
4.7.3 Extension of linear multi-step method: Lubich fractional linear multi-step method
280(3)
4.8 Applications of other approximation techniques
283(3)
4.8.1 Approximations of fractional integral and derivative of periodic function using fourier expansion
283(1)
4.8.2 Short memory principle
284(2)
5 Numerical Methods for the Fractional Ordinary Differential Equations
286(13)
5.1 Solution of fractional linear differential equation
286(1)
5.2 Solution of the general fractional differential equations
287(12)
5.2.1 Direct method
289(3)
5.2.2 Indirect method
292(7)
6 Numerical Methods for Fractional Partial Differential Equations
299(24)
6.1 Space fractional advection-diffusion equation
301(4)
6.2 Time fractional partial differential equation
305(5)
6.2.1 Finite difference schemes
306(1)
6.2.2 Stability analysis: Fourier-von Neumann method
307(1)
6.2.3 Error analysis
308(2)
6.3 Time-space fractional partial differential equation
310(8)
6.3.1 Finite difference schemes
310(2)
6.3.2 Stability and convergence analysis
312(6)
6.4 Numerical methods for non-linear fractional partial differential equations
318(5)
6.4.1 Adomina decomposition method
318(2)
6.4.2 Variational iteration method
320(3)
Bibliography 323
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