Jul 17
adminPh.D. Thesis anisotropic, Darcy-Stokes problem, finite element, mixed element, Stokes problem
The theme of this paper is the analysis of anisotropic Hermite finite elements and the nonconforming finite elements approximations for some mixed problems.The classical interpolation theory of finite elements is well known.To make the constant in the interpolation error independent of element size,the ratio of the diameter of the element and the diameter of the biggest ball contained in the element,is uniformly bounded.This condition is called regular or nondegenerate condition,which restrict the application of the finite element.In fact,on the one hand,the solution of some practicable problems have anisotropic behvious in the boundary layer or in the corner of the domain, which means that the solution varies significantly only in certain directions.For these problems,the accuracy of the standard finite element methods may decrease.On the other hand,for some problems(for example the complex material problem),the computation amount is very large if we still use the regular mesh.So it is necessary to analyze the anisotropic finite element.Rectantly,many researchers are interested in the anisotropic finite element.In the second chapter,we define the anisotropic C~(N-1)-continuous Hermite interpolation finite elements on rectangle and cuboid,which are applied in elliptic problems of any 2N order.At last,we carry out the numerical experiments,which support our theoretical analysis.Darcy-Stokes equation models porous media flow coupled with open fluid flow in a single form equation.In some sub-domain,the equation is reduced to the Darcy equation, in some sub-domain,the equation is reduced to the Stokes equation.In a single domain, when v(x)=ε~2,α(x)=1,the equation becomes a singular perturbed equation.Because the coeffients in the equation are discontinuous,many standard finite elements are not uniformly stable.In the chapter 3,we construct two new nonconforming rectangle and cuboid elements.We prove the elements are stable for the Darcy-Stokes equation and singular perturbed equation in one domain.For the Stokes equations which goven viscous fluid flow,the natural Galerkin approximation is a standard mixed problem.We should sovle the velocity and pressure at the same time.The key leading to the success of a mixed method is the finite element must satisfy the discrete inf-sup condition or Babuska-Brezzi condition.We define several triangular,rectangular and tetrahedron elements,which are stable for the Stokes problem. The discrete B-B condition holds for the elements,and error estimates in the energy norm for the velocity and L~2-norm for the pressure are O(h~2).
Jul 17
adminPh.D. Thesis analytic solution, discrete SG, integrable symplectic map, Lax-Moser matrix
In the present paper,some integrable partial differential equations(PDEs),including the continuous and the discrete sine-Gordon equations(SG),together with the modified Korteweg-de Vries equations(mKdV),are investigated.Algebraic curve methods,combined with the nonlinearization of the eigenvalue problems,are applied to calculate the exact analytic solutions of these integrable models.They are solved through three steps: decomposition,straightening out and inversion.Firstly,from two spectral problems associated with SG and mKdV,resorting to the powerful tools of the fundamental identities and the Lenard analysis,three families of integrable models(soliton equations),the mKdV,the algebraic SG and the(2+1)-dimensional SG hierarchies,together with their zero-curvature representations,are derived.Two finite-dimensional Liouville-integrable systems are obtained from the nonlinearization procedures of the two spectral problems under two Bargmann constraints.It is interesting that they share the same Lax-Moser matrix and the same N-set of integrals on motions,involutive with each other and functionally independent in an open set of the phase space.The soliton equations are decomposed into these Liouville integrable ODEs(ordinary differential equations),whose compatible solutions yield special solutions of the integrable PDEs.The Hamiltonian flows of these ODEs are straightened out in the Jacobi variety of an algebraic curve,determined by the Lax-Moser matrix.Thus they are integrated by quadrature and the final exact analytic solution for these PDEs are obtained through the Jacobi inversion,expressed by means of the multi-variable theta functions.Two discrete equations related to Toda lattice are studied with Darboux transformation(DT), in addition,a discrete spectral problem is obtained.The discrete SG equation is derived as a compatible condition of the Lax pair,composed of a continuous part and a discrete one.The discrete part is obtained from the DT of mKdV-SG hierarchy.It is succeeded in finding the associated Bargmann\’s constraint,and in nonlinearizing the DT into an integrable symplectic map,which is also straightened out in the same Jacobi variety.Exact analytic solution for the discrete SG is calculated in a similar procedure.
Jul 17
adminPh.D. Thesis Anisotropic Finite Element, Global Superconvergence, Maxwell’s Equations, Navier-Stokes Equations, Nonconforming Finite Element, Optimal Error Estimates, the Consistency Error
Several new nonconforming finite elements are constructed,the convergence analysis of these elements are discussed and their application are presented in this thesis systematically.These new nonconforming elements include:the Quasi-Carey element,the Quasi-Wilson element,the higher order Wilson element and the second order nonconforming mixed finite element.Compared with the conforming finite element methods,the finite element methods of nonconforming have many advantages.Generally speaking,nonconforming elements have fewer degrees of freedom for its simpler structure and good convergence properties,such as the Morley element and the Wilson element.In addition, the nonconforming mixed finite element methods are usually much easier to be constructed to satisfy the discrete inf-sup condition than the conforming ones. Therefore,nonconforming finite element methods have drawn increasing attention from scientists and engineers.As we know,according to the second Strang lemma, the error of every nonconforming element consists of two parts,one arises from the interpolation error and the other is the consistency error due to nonconformity of the element.In most cases,the order of the consistency error is lower than or equal to that of the interpolation error.But,in this paper,one can see that for the second order elliptic problems the consistency error of the Quasi-Carey element is of order O(h~2),one order higher than that of its interpolation error O(h).We proved that the consistency error of the traditional Quasi-Wilson element is of order O(h~3),two order higher than that of its interpolation error.At the same time,a new QuasiWilson element for arbitrary quadrilateral meshes possessing consistency error with order O(h~3) is presented.After a careful analysis,we first show that the interpolation error of the higher order Wilson element is of order O(h~3) on anisotropic meshes,one order higher than that of its consistency error.As application,in Chapter 2,we investigated the approximation of higher accuracy of the anisotropic nonconforming Quasi-Carey element for the Sobolev type equations.The superclose and global superconvergence with order O(h~2) are obtained. Moveover,by virtue of the extrapolation,we improved the approximate accuracy of the related approximate solution and derive a posteriori error estimate of higher accuracy of order O(h~4).In Chapter 3,based on the special convergence of the Quasi-Wilson element,we applied it to convection-diffusion equations and obtained the optimal convergence order O(h~(3/2)) as the bilinear element and the p_1~(mod) element.In Chapter 4,after analysing the error estimates of the higher order Wilson element on anisotropic meshes with a numerical test,the superclose properties of this element are proved.Then the interpolation postprocessing technique is used to obtain the global superconvergence and the posterior error estimate of higher accuracy.In Chapter 5,we applied the Quasi-Carey element and the modification higher order Wilson element to Maxwell\’s equations on the finite element scheme, the optimal convergence results are obtained.But the similar optimal convergence results can not be obtained for the nonconforming linear triangular Crouzeix-Raviart element,the Carey element and the higher order Wilson element.In Chapter 6,the new nonconforming mixed finite element schemes with second order convergence behavior are proposed for the stationary Navier-Stokes equations,the convergence analysis is presented and the error estimates of both H~1-norm of order O(h~2) and L~2-norm of order O(h~3) with respect to velocity as well as the L~2-norm of order O(h~2) for the pressure are derived.At the same time,the numerical results are presented to illustrate the error analysis.
Jul 17
adminPh.D. Thesis blowup of solution, Cauchy problem, decay behavior of global solution, energy perturbation method, generalized IMBq equation, global solution, initial boundary value problem
In this paper, we study the existence, the uniqueness of the local generalizedsolutions, the global generalized solutions, the global classical solutions, the decayrate of the global solutions for the initial boundary value problems and the Cauchyproblems to some nonlinear evolution equations, and give the sufficient conditionsof blowup of the solutions for some problems above. The main results include thefollowing four parts:In Chapter 2, by virtue of the Green function of the ordinary differential equation, the contraction mapping principle and the extension theroem of solutions, weprove the existence, the uniqueness and the regularities of the global generalizedsolution and the global classical solutions for the initial boundary value problemsfor a class of generalized IMBq equations with damping termAnd we give the sufficient conditions of blowup of the solution for the problem(1)-(3). The main results are as follows:Theorem 1 Suppose that u_0(x),u_1(x)∈C~3[0,1],u_0(0) = u_0(1) = u_1(0) =u_1(1) =0, f∈C~3(R), g∈C~1(R), |f\’(s)|,|g\’(s)|≤C_1, then the problem (1)-(3) hasa unique global classical solution u∈C~2([0, T]; C~2[0, 1]), (?)T > 0.Theorem 2 Suppose that u_0(x),u_1(x)∈C~3[0,1],u_0(0) = u_0(1) = u_1(0) =u_1(1) = 0, g = f∈C~3(R), and where F(s) = (?), A,B > 0 are all constants. Then the problem (1)-(3) has aunique global classical solution u∈C~2([0, T]; C~2[0, 1]), (?)T > 0.Theorem 3 Assume that u_0(x),u_1(x)∈C~3[0,1],u_0(0) = u_0(1) = u_1(0) =u_1(1) = 0, g = 0,αβ+γ= 0, f∈C~3(R), (4) holds andwhere C, D > 0 are constants. Then the problem (1)-(3) has a unique global classicalsolution u∈C~2([0, T]; C~2[0,1]), (?)T > 0.Theorem 4 Assume that u(x, t) is the classical solution of the problem (1)-(3),if the following conditions hold: . . .whereλ=π~2, (3) (?) grows fast enough, such thatconverges whenαβ-λγ≤0. Then whenαβ-λγ> 0,for some finite time t_0≤T_2 = (?); whenαβ-λγ≤0 ,for some finite time t_0≤T_1. In Chapter 3, by using of the fundamental solution of the ordinary differentialequation, the contraction mapping principle and the extension theorem of solutions,we prove the existence, the uniqueness and the regularities of the global generalizedsolutions and the global classical solutions in W~(k,p)(R) for the Cauchy problems fora class of generalized IMBq equation with damping termMoreover , we show the decay rate of the solutions for the problem (5), (6), andgive the sufficient conditions of blowup of the solution for the problem in finite time.The main results are stated as followsTheorem 5 If u_0, u_1∈W~(k+2,p)(R),f,g∈C~(k+3)(R), where k > 1/p, and(?)_s∈R, |f\’(s)|, |g\’(s)|≤A_0. Then the problem (5), (6) admits a unique global clas-sical solution u∈C~3([0,∞); W~(k+2,p)(R)), that is, u∈C~3([0,∞);C~2(R)∩L~∞(R)).Theorem 6 If u_0, u_1∈W~(k+2,p)(R),g = f∈C~(k+3)(R), where k > 1/p, F(s)≤0or f\'(s)≤A_0. Then the problem (5), (6) has a unique global classical solutionu∈C~3([0,∞);W~(k+2,p)(R)), that is, u∈C~3([0,∞);C~2(R)∩L~∞(R)).Theorem 7 Suppose that u_0, u_1∈W~(k+2,p)(R)∩L~2(R),∧~(-1)u_1∈L~2(R), g = 0,β= 0, f∈C~(k+3)(R), where k>1/p is nonnegative integral number, and F(u_0)∈L~1(R), f\'(s)≤A_0 or F(s)≤0 satisfyingwhere A, B > 0 are constants,∧~(-r)Ψ=F~(-1) (|x|~(-r)FΨ(x)),F,F~(-1) denote the Fouriertransform and the inverse transform. Then the problem (6), (7) admits a uniqueglobal classical solution u(x, t)∈C~3([0,∞); W~(k+2,p)(R)∩L~2(R)), that is,u∈C~3([O,∞);C~2(R)∩L~2(R)∩L~∞(R)).Theorem 8 Suppose thatβ= 0,γ≤0,g(s) = 0, f(s)∈C(R),F(s) = (?).u_0, u_1∈L~2(R),∧~(-1)u_0,∧~(-1)u_1∈L~2(R),F(u_0)∈L~1(R), there existδ_1,δ_2> 0, such thatThen the solution of the problem (5), (6) blows up in finite time if one of the followin9conditions holdswhereTheorem 9 Suppose thatβ= 0,γ≤0,g(s) = 0,f(s)∈C(R),F(s) = (?),u_0, u_1∈L~2(R),∧~(-1)u_0,∧~(-1)u_1∈L~2(R), F(u_0)∈L~1(R) and there existδ_1,δ_2 > 0,such thatThen when (?)(0)≤0, andthe solution of the problem (5), (6) blows up in .finite time.In Chapter 4, by using of the Fourier transform , the contraction mappingprinciple and the extension theorem of solutions, we study the existence and the uniqueness of the global generalized solutions and the global classical solutions inH~s(R) for the Cauchy problem (5), (6) are studied. The main results are stated asfollowsTheorem 10 If u_0, u_1∈H~s(R), g = f∈C~([s]+1)(R),s > 5/2, F(s)≤0, orf\’(s)≤A_0. Then the problem (5), (6) admits a unique global classical solutionu(x,t)∈([0,∞);C~2(R)).Theorem 11 If u_0, u_1∈H~s(R),g = 0, f∈C~([s]+1)(R), and s > 5/2.∧~(-1)u_1∈L~2(R), and F(u_0)∈L~1(R), f\’(s)≤A_0 or F(s)≤0 satisfyingwhere A, B > 0 are all constants. Then the problem (5), (6) has a unique globalclassical solution u∈C~2([0,∞); C_B~1(R)).In Chapter 5, we study the energy decay of the following problem of viscoelasticity equation with nonlinear damping on the boundaryBy virtue of a so-called energy perturbation method and a comparison inequality, we prove that the solution of the problem (8)-(10) and f(x, t) have the sameexponential decay or algebraic decay for the case F(s) = s + |s|~(α-2)(α≥2) andF(s) = |s|~(α-2)(α>2). The main results are as follows:Theorem 12 (1) Suppose thatwhere M_1≥0,λ_1 > 1, then there exists C_4 > 0, such that (2)Assume thatwhere M_2≥0,λ_2>0, then there exists C_5 > 0, such thatwhere (?) = min(λ_2, C_1).Theorem 13 If there exist M_3≥0,λ_4 > 1, such thatthen (?)t∈[0,∞), there exist C_6, C_7 > 0, such that
Jul 17
adminPh.D. Thesis anisotropic, double set parameter, finite element, mixed element, nonconforming plate element, planar elasticity, Stokes problem, superconvergence
In this paper, We focus on the plate problem, Stokes problem and the planar elasticity problem with pure displacement, we propose two anisotropic parallelogram nonconforming finite element methods for Stokes problem. Additionally, a lot of numerical experiments are carried out, which can be verified the theatrical results of this paper.The classical finite element convergence analysis relies on the following regular condition, that is, there exists a const C independent of the element K and the mesh, such that (?)≤C, where h_k andρ_k are diameters of K and the biggest ballcontained in K, respectively. Some works show that is unnecessary in some finite space, therefore, there appear some study of finite element method on anisotropic m(?)shes, i.e, under what weak conditions, the convergence of FEM is independent of (?). These new results mainly include the classical results of Ciarlet and the new results of Apel. We introduces a general theorem of anisotropic interpolation.By this theorem a new criterion is presented, which improves Apel\’s result and is easier to use.For the planar elasticity problem with pure displacement conditions, we present high order element by restricting the operator div in P_2 , show how to construct the element, and prove the optical error.Stokes problem is a typical mixed variational form, which contains pressure and velocity together. In this paper, we propose two anisotropic parallelogram nonconformingfinite elements.The anisotropic estimates of interpolation error, consistency error and LBB condition are obtained, which show that the convergence the method is independent of the regular and quasi-uniform assumptions on the meshes.Plate problem is one of main models in FEM. There are many results and practicing schemes, the regularity of subdivision is inevitable in computation.In this paper, we give a 8-12-2 nonconforming rectangular element by the double set parameter method, and prove the optimal anisotropic interpolation. error estimate and consistency error estimate.
Jul 17
adminPh.D. Thesis Birkhoff normalform, Lagrange equations, Lyapunov stability, Moser twist theorem, planar Hamiltonian systems
We are mainly interested in the Lyapunov stability of periodic solutions of La-grange equations and planar nonlinear Hamiltonian systems. This paper is divided intofour parts.In Chapter 1, we introduce the historical background and some recent results ob-tained in the literature. We also state some important basic results.In Chapter 2, we study the Lyapunov stability of elliptic periodic solutions of La-grange equations. First we give some reasonable estimates of the periodic solutions ofErmakov-Pinney equations when the linearized equation is in the first stability zone.These estimates can also give the estimates of the rotation numbers of the Hill equa-tions. The results concerning the lower bounds of the rotation numbers are completelynew in the literature. By using these estimates, we prove that two classes of nonlinear,scalar, time-periodic, Lagrange equations will have twist periodic solutions, one classbing regular, including , another class beingsingular, .In Chapter 3, we try to extend the analytical method in studying the stability ofperiodic solutions of Lagrange equations to the nonlinear planar Hamiltonian systems.First, we establish two important facts on linear planar Hamiltonian systems. Oneis the reduction from ellipticity to R-ellipticity. Another is the relation between thestability of linear systems and the existence of periodic solutions of the generalizedErmakov-Pinney equations. Based on these two basic facts and the Birkhoff normalforms of area-preserving mappings, we compute the twist coefficients of planar non-linear Hamiltonian systems. Such twist coefficients play an important role in studyingthe Lyapunov stability of periodic solutions. For some special nonlinear systems, westate and prove the stability results. As an example, the stability of the equilibrium ofthe one-dimensionalΦ-Laplacian is given.As can be seen in Chapter 2 and 3, the existence and the estimates of periodic so- lutions of singular equations play important roles in the stability theory. Therefore, wedevelop some existence results for second order non-autonomous dynamical systems inChapter 4. The first one is based on a nonlinear alternative principle of Leray–Schauderand the result is applicable to the case of a strong singularity as well as the case of aweak singularity. The second one is based on Schauder’s fixed point theorem and theresult sheds some new light on problems with weak singularities and proves that insome situations weak singularities may help create periodic solutions. The third exis-tence result is concerned with the nontrivial periodic solutions and the proof is basedon a well-known fixed point theorem in cones.
Jul 17
adminPh.D. Thesis Caputo derivative, diffusion equation, finite difference methods, Fractional calculus, integral transform, Riemann-Liouville derivative, Spectral methods
Fractional calculus is a branch of studying the property of any order integral or derivative.Fractional order differential equation is the equation containing the noninteger order derivative,raising from the standard differential equations by replacing the integer-order derivatives with fractional-order derivatives.Its application is very broad, many researchers find that the fractional differential equations more precisely describe the property of some materials with memory and heredity.Fractional order differential equations are playing an increasingly important role in engineering,physics and other fields,such as the fractal theory and the diffusion in porous media,fractional capacitance theory,electrolysis chemical,fractional biological neurons,condensate physics,vibration control of viscoelastic system,statistical mechanics and so on.In this paper,we mainly consider the time-fractional anomalous diffusion equation, discuss its analytic solution,numerical solution and its application.In Chapter 1,the developmental history of fractional calculus and the existing work about fractional calculus are reviewed.We also recall some definitions and properties of the fractional derivatives used in this paper.In Chapter 2,two time-fractional anomalous diffusion equations are deduced from the random walk and a stable law.These two equations will be investigated numerically in the next two chapters.In Chapter 3,the solution of time fractional anomalous diffusion equation is discussed. Using separation of variable methods and Laplace transform,the analytical solutions of a non-homogeneous anomalous sub-diffusion equation with Dirichlet,Neumann and Robin boundary conditions are derived respectively.The solution is expressed in terms of the Mittag-Leffler function.These techniques can be applied to solve other kinds of anomalous diffusion problems.In Chapter 4,we consider a time fractional anomalous diffusion equation on a finite domain.We propose an efficient finite difference/spectral method to solve the time fractional diffusion equation.Stability and convergence of the method are rigourously established.We prove that the full discretization is unconditionally stable,and the numerical solution converges to the exact one with order O(△t~(2-α)+N~(-m)),where△t,N and m are the time step size,polynomial degree,and regularity of the exact solution respectively.Numerical experiments are carried out to support the theoretical claims.In Chapter 5,we generalize the method that we have proposed in the Chapter 4 to the time fractional Cable equation for modeling neuronal dynamics.Numerical results are presented to show the applicability of the method.In Chapter 6,we discuss one class of nonlinear time fractional Fokker-Planck equation with initial-boundary value on a finite domain.The stability and convergence of a finite difference method are discussed by energy methods.A numerical example is presented to compare with the exact analytical solution.
Jul 17
adminPh.D. Thesis anomalous subdiffusion equation, Fractional ordinary differential equation, Implicit numerical method, Predictor-Corrector method, Stability and convergence
Fractional calculus has a long history, the application is very comprehensive, including the memory of many kinds of materials, anomalous diffusion,signal pro-cessing.control theory,vibration control of viscoelastic system and pliable structure ob-jects.fractional biological neurons, advection-diffusion in porous or fractured medium, chaotic, etc. Comparing with the classical inter-order differential equation, the new fractional order differential equation which is containing the non-integer order derivative, is more adequate to simulate practical problems. It can effectively describe the memory and transmissibility of many kinds of materials, and play an increasingly important role in engineering, physics, finance, hydrology and other fields. For inter-order differential equations, correlative numerical arithmetics are mature relatively, but for fractional differential equations in the fractional models, the investigation of numerical methods is underway, theoretical analysis is limited especially.Fractional ordinary differential equations can describe many physical phenomena, investigated widely. For example, dynamical controlled systems, chaotic model, fractional PI~λD~μcontroller simulation investigation. However they are limited in some applied fields. In recent years, researchers have proposed some numerical methods, but there are some difficulties in the error analysis. At present, the numerical methods, theoretical analysis and applications are just under exploration. Developing computationally efficient solution method and theoretical analysis of fractional ordinary differential, exploring further application of fractional ordinary differential will be very significative, which engineers are interested in.Fractional dynamic equations are very useful in describing power transmission phenomenon of complex systems, for example, a modified anomalous subdiffusion equation, etc. But it is difficult to solve such problems. For anomalous diffusion model with different situations, many investors proposed different numerical methods, and improved the error analysis of theoretical study continually. This thesis focuses on two kinds of problems: numerical methods and application of fractional ordinary equations, modified anomalous subdiffusion problem.Introduction gives some concerning fractional calculus to prepare the knowledge and present basic definitions and properties of fractional calculus. It describes numerical methods of fractional ordinary equations and anomalous subdiffusion problem comprehensively.The first kind of problems, we consider numerical methods and application of fractional ordinary equations, which are consisted of Chapters 2 to 4.In Chapter 2, we discuss the fractional Relaxation-Oscillation equation (FROE). The existence and uniqueness of solution for FROE is proven, and its analytic solution is given. A computationally effective fractional Predictor-Corrector method is proposed, and a detailed error analysis is derived. Finally, we give some numerical examples, and show the characteristic phenomena of fractional Relaxation-Oscillation equation\’s solution.In Chapter 3, we consider the fractional-order dynamical controlled systems. The multi-order fractional differential equation is transferred into a system of fractional-order differential equations. A new computationally effective fractional Predictor-Corrector method is proposed for simulating the fractional order systems and controllers. A detailed error analysis is derived. Finally, we give some numerical examples.In Chapter 4, we consider the application in practical physical models. We consider four chaotic models: fractional chaotic oscillator model, Chaotic “jerk” model, Chen system, Chaotic systems using state feedback controller. A computationally effective fractional Predictor-Corrector method is proposed for simulating the fractional order Chaotic systems. Finally, we give some numerical examples. The numerical results are in agreement with chaotic physical phenomena.The second kind of problems, we consider the fractional anomalous modified anomalous subdiffusion problem, which are consisted of Chapters 5 and 6.In Chapter 5, we consider a modified anomalous subdiffusion equation with nonlinear source terms for describing processes that become less anomalous as time pro- gresses by the inclusion of the second fractional time derivative acting on the diffusion term. It is an open problem. An implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, we give some numerical examples. The numerical results demonstrate the effectiveness of theoretical analysis.In Chapter 6, we propose a new implicit fractional Predictor-Corrector Trapezoidal method for the modified anomalous subdiffusion equation. Firstly, we give out the numerical approximation of time fractional Riemann-Liouville derivative. Using numerical techniques, the modified anomalous subdiffusion equation is transformed into a system of ordinary differential equations (ODE). The implicit fractional Predictor-Corrector Trapezoidal method for the ODE is proposed. There are some advantages: no need for iterative, high-precision, having same coefficient matrix in predictor and corrector methods. Finally, numerical results are given to demonstrate the effectiveness of this method. This technique can also be applied to solve other types of fractional partial differential equations.
Jul 17
adminPh.D. Thesis anomalous subdiffusion equation, convergency, finite difference method, finite element method, fractional kinetic equation, stability, the energy method
Fractional kinetic equations have been of great interest recently.It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics,chemistry,biology, environmental sciences,engineering and finance.Fractional kinetic equations provide a powerful instrument for the description of memory and hereditary properties of different substances.However,many analytical solutions for the fractional kinetic equations are complicated, which include the complicated series or especial function.Moreover,analytic solutions of most fractional kinetic equations cannot be obtained explicitly.At present numerical methods and analysis of stability and convergence for fractional partial differential equations are quite limited and difficult to derive.This motivates us to develop effective numerical methods for the fractional differential equations.In this thesis,we consider two kind of fractional kinetic equations.The first kind of the fractional kinetic equations is the fractional kinetic equations of the diffusion,diffusion-advection, and Fokker-Planck type.Numerical methods and theoretical analysis for the fractional kinetic equations are discussed in Chapters 2,3 and 4,respectively.The second kind of the fractional kinetic equations is the fractional kinetic equations of anomalous subdiffusion type,such as the anomalous subdiffusion equation,a nonlinear fractional reaction-subdiffusion process and the fractional cable equation.Numerical methods and theoretical analysis for the fractional kinetic equations are discussed in Chapters 5,6 and 7,respectively.These fractional kinetic equations above-mentioned have been presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.These fractional equations can be derived asymptotically from basic random walk models,and from a generalised master equation.In the first chapter,we summarize the history of the theory of fractional calculus, the background and significance of this dissertation,and the previous works about the fractional kinetic equations.Our research group and the framework of this thesis are given.In Chapter 2,we consider a space-time fractional diffusion equation on a finite domain.The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann-Liouville fractional derivative of orderβ∈(1,2],and the first-order time derivative by a Caputo fractional derivative of orderα∈(0,1].An implicit and an explicit difference approximations for the spacetime fractional diffusion equation with initial and boundary values are investigated. Stability and convergency results for the methods are discussed.Using mathematical induction,we prove that the implicit difference method is unconditionally stable and convergent,but the explicit difference method is conditionally stable and convergent. Some numerical results show the system exhibits anomalous diffusive behaviour.In this chapter,we also consider a two-dimensional fractional diffusion equation on a finite domain.We examine an implicit difference approximation to solve the space-time fractional diffusion equation.Stability and convergency of the method are discussed. Some numerical examples are presented to show the application of the present technique.In Chapter 3,we consider a space-time fractional advection dispersion equation on a finite domain.This equation is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of orderα∈(0,1],and the first-order and second-order space derivatives by the Riemman-Liouville fractional derivatives of orderβ∈(0,1]and of orderγ∈(1,2], respectively,n implicit and an explicit difference approximations is proposed.Using mathematical induction,we prove that the implicit difference method is unconditionally stable and convergent,but the explicit difference method is conditionally stable and convergent.Numerical results are in good agreement with theoretical analysis.In Chapter 4,we consider a space-time fractional Fokker-Planck equation on a finite domain.This equation is obtained from the standard Fokker-Planck equation by replacing the first-order time derivative by the Caputo fractional derivative,the second- order space derivative by the left and right Riemann-Liouville fractional derivatives. We propose a computationaUy effective implicit numerical method to solve this equation. Stability and convergence of the methods are discussed.Numerical example is given,which is in good agreement with the exact solution.In Chapter 5,we consider anomalous subdiffusion equation.A new implicit numerical method and two solution techniques for improving the order of convergence of the implicit numerical method for solving the anomalous subdiffusion equation are proposed.The stability and convergence of the new implicit numerical method are investigated by the energy method.Some numerical examples are given.The numerical results demonstrate the effectiveness of theoretical analysis.These methods and supporting theoretical results can also be applied to other fractional integro-differential equations and higher-dimensional problems.In Chapter 6,a nonlinear fractional reaction-subdiffusion process is considered. We propose a new computationally efficient numerical method to simulate the process. Firstly,the nonlinear fractional reaction-subdiffusion equation is decoupled,which is equivalent to solving a nonlinear fractional reaction-subdiffusion equation.Secondly, we propose an implicit numerical method to approximate this equation.Thirdly,the stability and convergence of the method are discussed using a new energy method.Finally, some numerical examples are presented to show the application of the present technique.This method and supporting theoretical results can also be applied to fractional integro-differential equations.In Chapter 7,a fractional cable equation is discussed.An implicit difference method is proposed.The stability and convergence of the method are discussed using an energy method.Moreover,we also propose the finite element approximation of the fractional cable equation.The stability and error estimates are established.We derive the convergent order of the method.Numerical examples are presented which demonstrate the effectiveness of the methods and confirm the theoretical analysis.
Jul 17
adminPh.D. Thesis Convergence, fractional advection-dispersion equation, fundamental solution, numerical solution, random walk model, stability
The characteristic of fractional order differential equation is containing the noninteger order derivative.It can effectively describe the memory and transmissibility of many kinds of material,and plays an increasingly important role in physics,mathematics, mechanical engineering,biology,electrical engineering,control theory,finance and other fields.All kinds of fractional models have close relation with chaotic dynamics. Anomalous diffusion in physics were originally developed from stochastic random walk models.Fractional advection-dispersion equations is powerful tool to simulate all kinds of anomalous diffusion phenomena.They are a subset of fractional kinetic equations that allow fractional derivatives in both the space and time operators.We discuss the time,space,space-time Fractional advection-dispersion equations respectively in this paper.The spatial derivatives discussed in the paper are all Riesz space fractional derivative,which include the left and right Riemann-Liouville fractional derivatives. The notable merit of Riesz space fractional derivative lies in its applicability to higher dimensional space.This thesis consists of the four chapters.Introduction presents the developmental history of fractional calculus and some important previous works at first.Then,gives some concerning fractional calculus to prepare the knowledge and present basic definitions and properties of fractional calculus.In Chapter 2,starting from the time fractional diffusion equation,we present an explicit conservative difference approximation,and give the stability and convergency analysis.Then,we extend the obtained results to the time fractional advectiondispersion equation.For the explicit conservative difference approximation of the time fractional advection-dispersion equation,we analyge the stability and convergency by using mathematical induction,and interpret it as a particle random walk.Random walks have proven to be a useful model in understanding processes across a wide spec- trum of scientific disciplines.In Chapter 3,we consider the Riesz space fractional advection-dispersion equation. It has three components.At first,we consider the case of initial value problem. Using the method of the Laplace and Fourier transforms,we obtain the fundamental solution of the equation with initial condition.The fundamental solution is represented by Green function,and can be intergreted the probability interpretation.We construct an explicit finite difference approximation for the equation by using the equivalence relation between Riemann-Liouville fractional derivative and Grünwald-Letnikovmake fractional derivative.The discrete scheme can be interpreted as a discrete random walk model,and the random walk model converges to a stable probability distribution.Secondly, we consider the case of initial-boundary problem.For the Riesz space fractional derivative can be expressed by a fractional power of the Laplacian operator,the numerical solution of our equation can be obtained by recur to matrix transfer technique and fractional method of lines.We also derive the new analytic solution by utilizing the property of eigenfunction and Laplace transform.Furthermore we compare the analytic solution and the numerical solution.Finally,we discuss the finite difference approximations in the case of initial-boundary problem.The explicit and implicit difference approximations are presented and the error analysis is also given.In Chapter 4,we consider the Riesz space-time fractional advection-dispersion equation.At first,we consider the case of initial value problem.We obtain the fundamental solution by using the method of the Laplace and Fourier transforms.The fundamental solution also be represented by Green function,and also can be proposed the probability interpretation.Using the equivalence relation between Riemann-Liouville fractional derivative and Grünwald-Letnikovmake fractional derivative,an explicit finite difference approximation for the equation is presented.The discrete scheme can be interpreted as a discrete random walk model.Then,the case of initial-boundary problem are discussed.The explicit and implicit finite difference approximations are proposed and the error analysis are also given.The non-local structure of fractional derivatives is one reason,why numerical methods for fractional differential equations are much more costly in computational time and storage requirements that their in- teger order counterparts.Thus,we propose the Richardson extrapolation which can promote the accuracy and “short-memory” principle which reduce the computational cost finally,these two methods are used to improve our numerical methods.Some numerical examples are presented in each chapter,which show the efficiency of our numerical methods.
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