Lie Symmetry Analysis of Fractional Differential Equations

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Authors
1. Lie symmetry analysis of integer order differential equations
	1.1 Classical Lie symmetry analysis
		1.1.1 Lie symmetries of the Fornberg-Whitham equation
			1.1.1.1 Similarity reductions and exact solutions
		1.1.2 Lie symmetries of the modified generalized Vakhnenko equation
		1.1.3 Lie symmetries of the Magneto-electro-elastic circular rod equation
		1.1.4 Lie symmetries of the couple stress fluid-filled thin elastic tubes
		1.1.5 Lie symmetries of the generalized Kadomtsev-Petviashvili-modi ed equal width equation
		1.1.6 Lie symmetries of the mKdV-KP equation
	1.2 Nonclassical Lie symmetry analysis
		1.2.1 Nonclassical symmetries for a class of reaction-diffusion equations
			1.2.1.1 Heir-equations and nonclassical symmetries
			1.2.1.2 R (u,x) =−½x2 u3+3u2+−c2u.
			1.2.1.3 R (u,x) =−− ecxu3+c24u+ecx2.
		1.2.2 Nonclassical symmetries of the Black-Scholes equation
	1.3 Self-adjointness and conservation laws
		1.3.1 Conservation laws of the Black-Scholes equation
		1.3.2 Conservation laws of the couple stress fluid-filled thin elastic tubes
		1.3.3 Conservation laws of the Fornberg-Whitham equation
		1.3.4 Conservation laws of the mKdV-KP equation
2. Group analysis and exact solutions of fractional partial differential equations
	2.1 Basic theory of fractional differential equations
	2.2 Group analysis of fractional differential equations
	2.3 Group analysis of time-fractional Fokker-Planck equation
		2.3.1 Exact solutions of time-fractional Fokker-Planck equation by invariant subspace method
	2.4 Lie symmetries of time-fractional Fisher equation
	2.5 Lie symmetries of time-fractional K(m, n) equation
	2.6 Lie symmetries of time-fractional gas dynamics equation
	2.7 Lie symmetries of time-fractional diffusion-absorption equation
		2.7.1 Exact solutions of time-fractional diffusion-absorption by invariant subspace method
	2.8 Lie symmetries of time-fractional Clannish Random Walker's parabolic equation
		2.8.1 Exact solutions of time-fractional Clannish Random Walker's equation by invariant subspace method
	2.9 Lie symmetries of the time-fractional Kompaneets equation
	2.10 Lie symmetry analysis of the time-fractional variant Boussinesq and coupled Boussinesq-Burger's equations
		2.10.1 Exact solutions of time-fractional VB and BB equations by invariant subspace method
3. Analytical Lie group approach for solving the fractional integro-differential equations
	3.1 Lie groups of transformations for FIDEs
	3.2 The invariance criterion for FIDEs
	3.3 Symmetry group of FIDEs
	3.4 Kernel function, free term and related symmetry group of the FIDEs
		3.4.1 General conditions for K and f
		3.4.2 Some special cases
4. Nonclassical Lie symmetry analysis to fractional differential equations
	4.1 General solutions extracted from invariant surfaces to fractional differential equations
		4.1.1 Fractional diffusion equation
		4.1.2 Fractional Burger's equation
		4.1.3 Fractional Airy's equation
		4.1.4 Fractional KdV equation
		4.1.5 Fractional gas dynamic equation
	4.2 Lie symmetries of space fractional diffusion equations
		4.2.1 Nonclassical method
	4.3 Lie symmetries of time-fractional diffusion equation
		4.3.1 Nonclassical method
	4.4 General solutions to fractional diffusion equations by invariant surfaces
5. Conservation laws of the fractional differential equations
	5.1 Description of approach
		5.1.1 Time-fractional diffusion equations
		5.1.2 Conservation laws and nonlinear self-adjointness
		5.1.3 Fractional Noether operators
		5.1.4 Nonlinear self-adjointness of linear TFDE
		5.1.5 Conservation laws for TFDE with the Riemann-fractional derivative
		5.1.6 Conservation laws for TFDE with the Caputo fractional derivative
		5.1.7 Symmetries and nonlinear self-adjointness of nonlinear TFDE
		5.1.8 Conservation laws for nonlinear TFDE with the Riemann-fractional derivative
		5.1.9 Conservation laws for nonlinear TFDE with the Caputo fractional derivative
	5.2 Conservation laws of fractional diffusion-absorption equation
	5.3 Nonlinear self-adjointness of the Kompaneets equations
		5.3.1 Conservation laws for approximations of the Eq. (2.82)
		5.3.2 Conservation laws for approximations of the Eq. (2.84)
		5.3.3 Conservation laws for approximations of the Eq. (2.85)
		5.3.4 Conservation laws for approximations of the Eq. (2.86)
		5.3.5 Noninvariant particular solutions
	5.4 Conservation laws of the time-fractional CRW equation
	5.5 Conservation laws of the time-fractional VB equation and time-fractional BB equation
		5.5.1 Construction of conservation laws for Eq. (5.91)
Bibliography
Index