一类Volterra积分—微分方程的解析解研究

发布时间：2018-05-29 09:22

本文选题：Volterra积分-微分方程 + 积分变换　；参考：《西安理工大学》2017年硕士论文

【摘要】：Volterra积分-微分方程频繁出现在生物学、物理学、工程等实际问题的数学建模中。由于该类数学模型带有未知核函数的积分项,可更好的反映系统的非局部及记忆反馈性质,相比传统的偏微分方程似乎更接近模拟实际问题。因此对Volterra型积分-微分方程的理论与解法研究为当今的一个热点课题。本文对一类带有广义Mittag-Leffler函数型、幂律函数型及指数因子型记忆核的Volterra型积分-微分方程的解析解展开研究:(1)在无界区域上分别讨论了带有三种记忆核的高维非齐次抛物型Volterra积分-微分(Parabolic Volterra Integro-Differential,PVI-D)方程的解析解。基于积分变换及特殊函数得到了包含广义Mittag-Leffler函数、Fox-H函数、积分算子以及积分形式的无穷级数解表达式。其次得到了带有幂律型记忆核的一维齐次PVI-D方程在初值为狄克拉-?函数下的解析解。最后对带有幂律型记忆核的齐次PVI-D方程的解析解进行数值模拟,模拟结果表明解析解在x(28)0处达到峰值,其图像呈Gaussian对称形态且具有Gaussian缓慢衰减分布特征。(2)在半无界区域上考虑了带有三种记忆核的一维非齐次PVI-D方程的解析解。基于Fourier-Sine变换、Laplace变换、Fourier-Cosine变换及Mittag-Leffler函数和Fox-H函数的性质得到了由广义Mittag-Leffler函数、Fox-H函数组成的无穷级数解与无限域边界条件下的解析形式相似。(3)研究了带有三种记忆核的一维、二维、三维非齐次PVI-D方程分别在有界限区间、圆域、球域上的解析解。基于分离变量、积分变换及特殊函数得到了由带有三角函数、积分算子、广义Mittag-Leffler函数、Bessel函数及Legendre函数的多重无穷级数解析表达式。最后对带有幂律型记忆核的二维齐次PVI-D方程在圆域上的解析解进行数值模拟,模拟结果表明解析图像呈帽状且具有缓慢耗散的特征,同时给出曲面等高线变化图可清楚看到能量耗散过程,等高线浓密与稀疏的分布决定能量耗散程度。(4)在无界、有界区域上分别考虑了一类带有三种时间记忆核的非齐次Fokker-Planck方程的解析解,基于分离变量法和积分变换得到了相应的解析表达式。
[Abstract]:Volterra integro-differential equations frequently appear in mathematical modeling of practical problems such as biology, physics, engineering and so on. Because this kind of mathematical model has the integral term of unknown kernel function, it can better reflect the nonlocal and memory feedback property of the system. Compared with the traditional partial differential equation, it seems to be closer to the practical problem of simulation. Therefore, the research on the theory and solution of Volterra type integral-differential equation is a hot topic. In this paper, we consider a class of functions with generalized Mittag-Leffler functions. Study on Analytical Solutions of Volterra Type Integro-differential equations with Power Law function and exponential Factor Type memory kernels; (1) in unbounded domain, we discuss the analytic solutions of Volterra integro-differential parabolic Volterra Integro-Differential Volterra PVI-D equations with three kinds of memory kernels, respectively. Based on the integral transformation and special functions, the expressions of infinite series solutions including generalized Mittag-Leffler functions, integral operators and integral forms are obtained. Secondly, the one-dimensional homogeneous PVI-D equation with power-law memory kernel is obtained. The analytic solution under the function. Finally, the analytical solutions of homogeneous PVI-D equation with power-law memory kernel are numerically simulated. The simulation results show that the analytical solution reaches its peak value at x ~ (28) ~ 0. The image is Gaussian symmetric and has the characteristic of Gaussian slow attenuation distribution. In the semi-unbounded region, the analytical solution of one-dimensional nonhomogeneous PVI-D equation with three kinds of memory kernels is considered. Based on the properties of Fourier-Sine transform, Fourier-Cosine transform and Mittag-Leffler function and Fox-H function, the infinite series solution composed of generalized Mittag-Leffler function and Fox-H function is obtained. The analytical form is similar to that in infinite domain boundary condition. The one-dimensional and two-dimensional memory kernels with three kinds of memory kernels are studied. The analytic solutions of three dimensional nonhomogeneous PVI-D equations in bounded interval, circle domain and sphere domain, respectively. Based on the separation of variables, integral transformations and special functions, the analytic expressions of multiple infinite series with trigonometric functions, integral operators, generalized Mittag-Leffler functions and Legendre functions are obtained. Finally, the analytical solutions of two-dimensional homogeneous PVI-D equation with power-law memory kernel are numerically simulated in a circular domain. The simulation results show that the analytical images are cap shaped and slow dissipative. At the same time, the energy dissipation process can be clearly seen in the curve contour diagram. The energy dissipation degree is determined by the density and sparsity of the contour line, and the energy dissipation degree is determined by the density and sparsity of the contour line. The analytic solutions of a class of nonhomogeneous Fokker-Planck equations with three kinds of time memory kernels are considered in the bounded domain, and the corresponding analytical expressions are obtained based on the method of separating variables and the integral transformation.
【学位授予单位】：西安理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.6

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