Analysis And Computation For Caputo-Hadamard Type Fractional Evolution Equations

In recent decades,fractional calculus has been widely used in various fields of natural science and engineering technology such as physics,chemistry,materials,dynamic system,quantum mechanics,and so on.Fractional calculus operators,which have nonlocality and singularity,can describe some complex physical processes with historical dependence and spatial global correlation more accurately.Therefore,they are favored by many researchers,and have obtained fruitful results.Compared with Riemann-Liouville fractional operator,it has been found that CaputoHadamard fractional operator can better describe the complex processes such as Lomnitz logarithmic creep law and ultra slow dynamics.Its corresponding models are CaputoHadamard fractional differential equations.The theoretical analysis and numerical method for this kind of fractional ordinary/partial differential equations are very limited,and this thesis is devoted to a comprehensive study of this topic.The innovative results of this thesis contain the following three aspects:(1)The logarithmic asymptotics of solutions to the initial value problem for Caputo-Hadamard fractional evolution equations(FEEs)are proved,which differ from the algebraic asymptotics of solutions to the initial value problem for the integer order or Caputo type FEEs;(2)The existence,uniqueness and regularity of solutions for the initial boundary value problem of time fractional evolution equation with Caputo-Hadamard derivative are studied,which provides theoretical basis for solving this kind of equation numerically.(3)The L1 method is constructed to discretize the CaputoHadamard fractional derivative on non-uniform meshes in the logarithmic sense,and it is successfully applied to solving the initial boundary value problem of time FEEs with Caputo-Hadamard derivative.The main contents are introduced in the following chapters in more detail.Chapter 2 is devoted to studying the asymptotics of solutions to the initial value problem for time-space fractional evolution equations with Caputo-Hadamard derivative and fractional Laplace operator.First of all,we consider a parabolic equation with CaputoHadamard derivative of order ? ?(0,1).By means of integral transforms,we obtain the fundamental and analytical solutions to the equation,where the special functions such as Fox H-function are applied.Then using the estimation of the fundamental solution and the Young inequality for convolution,we derive the asymptotic behavior of the solution in the sense of Lp(Rd)and Lp,?(Rd)norm.Furthermore,the gradient estimation and large time behavior of the solution are displayed.Following the same idea as the parabolic equation,we can obtain similar results for the corresponding hyperbolic equation(i.e.? ?(1,2)).Chapter 3 is devoted to the LDG method of the initial boundary value problem for the Caputo-Hadamard type time fractional parabolic equation.By using integral transforms,we obtain the series solution to this kind of linear equation.We also prove that it is actually the unique classical solution of the equation,and the regularity of solution is derived at the same time.From regularity analysis,we observe that the solution to the equation usually has weak regularity with respect to time variable at the initial time.Because of this property,we use the L1 method on non-uniform meshes in the logarithmic sense to approximate the Caputo-Hadamard fractional derivative,and apply the LDG method to discretize the second spatial derivative.Further,numerical stability and error estimation are proved in the sense of L2 norm in detail.Finally,numerical examples are provided to verify the theoretical analysis and the effectiveness of the scheme.Chapter 4 studies the finite difference/DDG method of the initial boundary value problem for the Caputo-Hadamard type time fractional hyperbolic equation.Similar to the idea in Chapter 3,we get the the series solution of the equation by integral transforms.We also prove that it is unique classical solution to the equation.We further investigate the regularity of solution of the considered equation and find that the solution also has weak regularity with respect to time variable at the initial time.Therefore,the Caputo-Hadamard fractional derivative is approximated by finite difference method on non-uniform meshes to derive a time semi-discrete scheme,where numerical stability and convergence are proved on uniform meshes.Then the spatial derivative is approximated by using the DDG method.Therefore a fully discrete scheme is proposed by selecting the appropriate numerical fluxes.Finally,we test the stability and accuracy of this scheme by numerical experiments.The last chapter briefly summarizes the research contents of this dissertation,and puts forward some problems that can be further considered.