The Positive Definite Solutions Of Matrix Equations And Spectral Problems Of Fractional Differential Equations

The research on differential equations has been developed along with the appear-ance of Calculus with a history of more than300years. There are plentiful research fields about differential equations which are considered one of the most powerful math-ematic tools in studying natural phenomena. The research in this field is closely related with other subjects and widely applied in the fields of natural science, ecological en-vironment, engineering technology and social economy. This paper mainly focuses on two aspects closely related to the theory and application of differential equations, which refers to positive definite solutions and perturbation analysis of matrix equations as well as the spectral problems of fractional differential equations. The motive and content of the research are detailed as follows.On the one hand, as to many mathematical physics problems originated from prac-tical issues, which are depicted by ordinary differential equations, integral equations and integral differential equations, the usually applied method is to discretize them through the difference method and therefore the original problem can be transformed into a certain kind of matrix equations to be researched. Matrix equation is one of the important substance of the matrix theory and numerical algebra. In recent years there has been a constantly increasing interest in developing the theory and numerical approaches of the positive definite solutions for the nonlinear matrix equation of the form X-∑i=1m Ai*X-pi Ai=Q. This kind of matrix equations arises in solving a large-scale system of linear equations in many physical calculations, extremal interpolation problem, differential equations, control theory, ladder network, dynamic programming, stochastic filtering and so on. From the application point of view, the positive definite solutions are more important. In the sequel, a solution always means a positive definite one. The research results of this kind of matrix equations mainly concentrate on three basic problems:(i) Necessary and sufficient conditions for the existence of positive definite solutions; (ⅱ) Effective numerical ways for obtaining the positive definite solutions;(ⅲ) Perturbation analysis of the positive definite solutions, including perturbation bound, condition number, backward error, residual bound,where the solvability is the theoretical basis of the numerical solution for the nonlinear matrix equation, and numerical solution methods provide a feasible calculation process. Since nonlinear matrix equations stem from calculation problems with mass data in engineering and physics, there usually exist two kinds of errors during solution proce-dure influencing the accuracy of the results which refer to truncation errors caused by the numerical calculation method and rounding errors caused by calculating environ-ment. In order to analyze the impact of errors on the solutions of original problems, one needs to study the impact on the solution from the turbulence of original data, therefore the perturbation analysis of positive definite solutions for matrix equations is required. The perturbation bound and condition number will be applied to illustrate the stability of matrix equations, while the backward error and residual bound will be applied to check the numeral stability of the algorithm and the accuracy of the ap-proximately estimated solutions. Motivated by the work and applications of this kind of matrix equations, we study the following nonlinear matrix equations.1. For the nonlinear matrix equation X-∑Ai*X-pi Ai=Q(pi>0), we consider two cases:the case m=1and the case m>1.(i) When m=1, the above equation can be reduced to X-A*X-p A=Q(p>0). Two cases are considered here:the case p≥1and the case0<p<1. In the case p≥1, a new sufficient condition for the existence of a unique positive definite solution of the matrix equation is obtained. Based on the integral representation of matrix function X-p (p>0), we derive a perturbation es-timate for the positive definite solution. Moreover, explicit expressions of the condition number defined by Rice for the positive definite solution are giv-en. In the case0<p<1, applying the operator theory, we obtain a new and improved perturbation bound for the unique positive definite solution. A new sharper residual bound of an approximate solution is obtained using the Schauder’s fixed point theorem. The existing results are generalized and im-proved. These results are compared numerically against other known results from the literature. (ii) When m>1, some necessary and sufficient conditions for the existence of pos-itive definite solutions of the matrix equation X-∑Ai*X-pi Ai=Q(pi>0) are obtained. Simultaneously, the condition that the equation has a unique positive definite solution is also given. An effective iterative method to obtain the unique solution is established. By using Brouwer’s fixed point theorem, the properties of the Kronecker product and matrix norm, we evaluate a perturba-tion bound and a backward error of an approximate solution to this equation. Moreover, based on the integral representation of matrix function X-p (p>0) and operator theory, we obtain the explicit expressions of the condition number for the positive definite solution. When0<pi<1, Q=I, using the principle for the monotonic and bounded sequence in the Banach space, we prove the existence of a unique positive definite solution for this equation. Applying the Schauder’s fixed point theorem and the operator theory, we evaluate two per-turbation bounds of the unique positive definite solution to the equation. One perturbation bound does not use any knowledge of the actual solution, another perturbation bound is more sharper. A residual bound of an approximate so-lution to the equation is evaluated by means of fixed point theorem and norm inequalities. Moreover, using the integral representation of matrix function X-p (0<p<1) and operator theory, we obtain the expression of condition number defined by Rice.The correctness and effectiveness are illustrated by numerical examples.2. For the nonlinear matrix equation X-∑Ai*X-pi Ai=Q(pi>0), we obtain necessary and sufficient conditions for the existence of positive definite solutions of this matrix equation. we consider two cases:the case0<pi<1and the case pi>0.(i) In the case0<pi<1, necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are obtained. Moreover, using the principle for the monotonic and bounded sequence in the Banach space, we prove the existence of a unique positive definite solution for this equation. Based on the fixed point theorem and operator theory, we derive two perturbation bounds and a residual bound of an approximate solution to the equation. Applying the integral representation of matrix function Xp (0<p<1) and operator theory, we obtain the explicit expressions of the condition number for the equation. The theoretical results are illustrated by numerical examples.(ii) In the case pi>1, sufficient conditions for the existence of a unique positive definite solutions for the matrix equation are obtained. Some properties of positive definite solutions are derived. By using the elegant properties of the Frobenius norm, we derive a perturbation bound for this matrix equation.On the other hand, one important method to deal with differential equations is to study their solutions through the research on the nature of their spectrums, i.e. the spectral theory of differential equations. The Sturm-Liouville problem is the basic problem in the spectral theory of differential equations and its related theories were raised170years ago, and since then its related theories have played important roles in scientific, engineering and mathematical fields.Sturm-Liouville problem is originated from the boundary value problem of ordi-nary differential equations which are partly from practical issues directly and partly from the problems of partial differential equations, such as the heat conduction (or dif-fusion) problem, the string (film) vibration problem and the Maxwell equation problem in electromagnetics. At the beginning of the19th century, Fourier systematically put forward the method of separation of variables and applied this method to problems of partial differential equations equations with boundary value conditions caused by more complicated physical phenomena. From operator point of view, Sturm-Liouville operator is an extremely important kind which has significant application background in both classical differential operators and modern quantum physics. Besides, since the end of the20th century, the rapid development and widespread application of the fractional calculus theory have promoted the appearance and growth of fractional dif-ferential equations. It has been discovered that to introduce the concept of fractional calculus can describe the change rule and essential attribute of objects, which brought to the wide application of fractional differential equations in reality, such as in the fields of fractal dynamics, continuum mechanics, auto control, hydrodynamics, biomechanics, viscoelastic mechanics, quantum mechanics, statistics, engineering science, Brownian motion, earthquake analysis, fractional models of nerve and mathematical models de-scribing the population reproduction. Therefore, fractional differential equations have more and more drawn the interest of mathematicians. The research on the theory of fractional differential equations can both enrich existing mathematical theories and provide better mathematical models for studies in physical, biological and economical process and phenomena. In many cases, the consideration in the spectrum problem of fractional differential equations is needed to deal with practical issues. The study on spectrum problem of fractional equations can satisfy practical requirement as well as enrich and perfect related theories of fractional differential equations. Since the spec-trum problem of fractional differential equations was raised in early research papers, it has never been researched intensively. Recently, many scholars have applied the method of numerical calculation to study this problem, whereas even the natures of eigenvalue and eigenfunction have not been theoretically illustrated. Until now, there have been very few articles theoretically referring to the natures of eigenvalue and eigenfunction in fractional differential equations. Based on these findings, this paper will mainly focus on following spectrum problems of fractional differential equations:1. The spectral problem is considered, where q∈L2(0,1) is a real-valued function, D0+α and D1-α are the left and right Riemann-Liouville fractional derivatives of order α, respectively,1<α<3/2, μ is a real constant and λ is the spectral parameter. This equa-tion aries from the non-local continuum mechanics. It is a governing equilibrium equation of an elastic bar of finite length L with long-range interactions among non-adjacent particles. In this study, by using the spectral theory of self-adjoint compact operators in Hilbert spaces, we prove that the spectrum of the spectral problem associated to this equation with1<α<3/2consists of only countable and real eigenvalues with finite multiplicity and the orthogonal completeness of the corresponding eigenfunction system in the Hilbert spaces. Furthermore, we obtain a lower bound of the eigenvalues.2. The spectral problem is considered, where q∈L2(0,1) is a real-valued function, D0+α, D1-α are respectively the left and right Riemann-Liouville fractional derivatives of order α,0<α<1/2, μ is a real constant and A is the spectral parameter. We prove that the spectrum of this problem consists of only real eigenvalues and its eigenfunctions corresponding to different eigenvalues are orthogonal by the spectral theory of self-adjoint compact operators in Hilbert spaces. Furthermore, we demonstrate that the real eigenvalues of this equation are countable and each eigenvalue has finite multiplicity. We also prove that the set of all corresponding orthogonal eigenfunctions forms a complete system. A lower bound of the eigenvalues is obtained.The theory about initial value problems of differential equations involving only left or right fractional derivatives has been considerably perfect. The existence of solutions, the consistent dependence on parameters of solutions, the differentiability of solutions and the continuation theorem have been established. However, the initial value prob-lems of differential equations with both left and right fractional derivatives have not been put forward clearly enough, and seldom mentioned in the existing publications and papers on fractional differential equations. Nevertheless, on the research of bound-ary problems of differential equations with integer orders, the corresponding theory of initial value problems proves to be a very effective method and tool, such as the Prufer transformation and the differentiability of solutions on parameters. Therefore, in this paper, we first put forward the initial value problems of differential equations with both left and right fractional derivatives. Under proper conditions, we can prove the existence and uniqueness of solutions for this kind of initial value problems. Then, ap-plying results above, we specifically research the geometric multiplicity of eigenvalues for eigenvalue problems and form a series of problems with single eigenvalue.1. First, the initial value problem is considered, where q∈L(0,1) is a real-valued function, D0+α and D1-α are the left and right Riemann-Liouville fractional derivatives of order α, respectively,0<α <1, μ and λ are real constants. The uniqueness of the solution for this kind of initial value problem is proved. And then, by means of the above results, we prove the eigenvalue of eigenvalue problem is simple. 2. The initial value problem is considered, where q∈L(0,1) is a real-valued function, D0+α is the left Riemann-Liouville fractional derivatives of order a and D1-α is the right Caputo fractional derivatives of order a, respectively,0<α<1/2,μ and λ are real constants. The uniqueness of the solution for this kind of initial value problem is proved. And then, by means of the above results, we prove the eigenvalue of eigenvalue problem is simple.