Analysis And Computation Of Caputo-Hadamard Fractional Differential Equations

Fractional calculus(including fractional differentiation and fractional integration)can be considered a branch of mathematical analysis,which is primarily used to characterize the history dependance and nonlocal effect in space.Although it has almost the same history as that of integer-order calculus,fractional calculus was studied mainly by mathematicians as an abstract area containing only pure mathematical manipulations of little or no use,except for some sporadic applications in rheology.In recent decades,fractional calculus gradually attracts applied scientists and engineer's attention and interest mostly due to its potential applications in modeling history dependance,nonlocal effects in space,Levy flights,heredity,anomalous convection,anomalous diffusion,etc.This greatly advances the mathematicians to recognize and study fractional calculus.In recent several years,it has been found that Caputo-Hadamard fractional derivative can well characterize fatigue and fracture of materials,and Lomnitz logarithmic creep law.The corresponding mathematical models are Caputo-Hadamard fractional differential equa-tions(FDEs).From the references available,pure theoretical studies for Caputo-Hadamard fractional ordinary/partial differential equations(FODEs/FPDEs)are rather limited.And numerical methods for Caputo-Hadamard fractional ordinary/partial differential equations have not been appeared.Such a topic has been studied in this dissertation.The innova-tive research results in this thesis have three aspects:(?)The continuation theorem of the solution to the Caputo-Hadamard FODE is proved.The corresponding technique can be applied other types of FODEs.(?)By using the "first order" polynomial interpolation in the sense of logarithmic function,the effective numerical schemes for the Caputo-Hadamard FODE are established.As far as we know,such an idea has never been seen which can be extended other types of FODEs.(?)We firstly derive the L1 scheme for the Caputo-Hadamard fractional derivative,where the local truncated error is investigated.Such an L1 scheme,together with the central difference scheme,is successfully applied to numerical solving Caputo-Hadamard fractional partial differential equation.The main contents are introduced in details in the following three chapters(Chapters 2-4).In Chapter 2,we study the initial value problem of Caputo-Hadamard FODE.We first change the Caputo-Hadamard FODE into an equivalent Volterra integral equation with a logarithmic kernel.By using the fixed point theorem,the existence theorem of the solution to the equivalent integral equation is given,then uniqueness theorem and continuation the-orem of solution are completed under the Lipschitz condition.All these theorems are the theoretic bases for the following numerical methods.In Chapter 3,we construct the finite difference algorithms for the Caputo-Hadamard FDEs,which is the core content of this thesis.In details,there are totally three parts in this chapter.The first two parts for Caputo-Hadamard FODE.And the third part is for Caputo-Hadamard fractional partial differential equation(FPDE).(1)We first transform the Caputo-Hadamard FODE into an equivalent Volterra integral equation with a logarithmic kernel.By choosing the different approximations for the inte-grand in every subinterval,the fractional rectangular,trapezoidal,and predictor-corrector methods are constructed.Using the derived fractional Gronwall inequality,numerical sta-bility,convergence,and error estimates of these numerical methods are studied.(2)For the Volterra integral equation in(1),by taking the place of the integrand by the"first order" polynomial interpolation in the sense of logarithmic function in every subin-terval,an implicit scheme with a higher accuracy is obtained.Such an implicit scheme has good stability but with large amount of calculation.Applying the technique of predictor-corrector to the derived implicit scheme,an explicit scheme based on the "first order"polynomial interpolation is established.Such a scheme has somewhat a bit lower accuracy but greatly reduce the amount of calculation.Numerical stability,convergence,and error estimates for these two methods are carefully investigated.(3)Next,we numerically study the initial and boundary value problem of Caputo-Hadamard FPDE.We first establish the L1 scheme for numerical approaching Caputo-Hadamard derivative where the local truncated error is derived.Using this L1 scheme,the semi-discrete scheme is proposed.Numerical stability and convergence for the semi-discrete scheme are studied.Furthermore,we utilize the central difference scheme to ap-proximate the spatial derivative,then we obtain a fully discrete scheme for the Caputo-Hadamard FPDE.And numerical stability,convergence,and error estimate are presented.Numerical examples are also displayed which are in line with the theoretical results.In Chapter 4,we consider the optimal homotopy asymptotic method(OHAM)for the Riccati differential equation with Caputo derivative,where the approximate analytical solution is obtained.Numerical experiments are displayed to show the effectiveness of this method.The last chapter concludes the main results of this dissertation.