本文選題：Volterra積分-微分方程 + 積分變換�。� 參考：《西安理工大學(xué)》2017年碩士論文

【摘要】：Volterra積分-微分方程頻繁出現(xiàn)在生物學(xué)、物理學(xué)、工程等實(shí)際問(wèn)題的數(shù)學(xué)建模中。由于該類數(shù)學(xué)模型帶有未知核函數(shù)的積分項(xiàng),可更好的反映系統(tǒng)的非局部及記憶反饋性質(zhì),相比傳統(tǒng)的偏微分方程似乎更接近模擬實(shí)際問(wèn)題。因此對(duì)Volterra型積分-微分方程的理論與解法研究為當(dāng)今的一個(gè)熱點(diǎn)課題。本文對(duì)一類帶有廣義Mittag-Leffler函數(shù)型、冪律函數(shù)型及指數(shù)因子型記憶核的Volterra型積分-微分方程的解析解展開研究:(1)在無(wú)界區(qū)域上分別討論了帶有三種記憶核的高維非齊次拋物型Volterra積分-微分(Parabolic Volterra Integro-Differential,PVI-D)方程的解析解。基于積分變換及特殊函數(shù)得到了包含廣義Mittag-Leffler函數(shù)、Fox-H函數(shù)、積分算子以及積分形式的無(wú)窮級(jí)數(shù)解表達(dá)式。其次得到了帶有冪律型記憶核的一維齊次PVI-D方程在初值為狄克拉-?函數(shù)下的解析解。最后對(duì)帶有冪律型記憶核的齊次PVI-D方程的解析解進(jìn)行數(shù)值模擬,模擬結(jié)果表明解析解在x(28)0處達(dá)到峰值,其圖像呈Gaussian對(duì)稱形態(tài)且具有Gaussian緩慢衰減分布特征。(2)在半無(wú)界區(qū)域上考慮了帶有三種記憶核的一維非齊次PVI-D方程的解析解�；贔ourier-Sine變換、Laplace變換、Fourier-Cosine變換及Mittag-Leffler函數(shù)和Fox-H函數(shù)的性質(zhì)得到了由廣義Mittag-Leffler函數(shù)、Fox-H函數(shù)組成的無(wú)窮級(jí)數(shù)解與無(wú)限域邊界條件下的解析形式相似。(3)研究了帶有三種記憶核的一維、二維、三維非齊次PVI-D方程分別在有界限區(qū)間、圓域、球域上的解析解�；诜蛛x變量、積分變換及特殊函數(shù)得到了由帶有三角函數(shù)、積分算子、廣義Mittag-Leffler函數(shù)、Bessel函數(shù)及Legendre函數(shù)的多重?zé)o窮級(jí)數(shù)解析表達(dá)式。最后對(duì)帶有冪律型記憶核的二維齊次PVI-D方程在圓域上的解析解進(jìn)行數(shù)值模擬,模擬結(jié)果表明解析圖像呈帽狀且具有緩慢耗散的特征,同時(shí)給出曲面等高線變化圖可清楚看到能量耗散過(guò)程,等高線濃密與稀疏的分布決定能量耗散程度。(4)在無(wú)界、有界區(qū)域上分別考慮了一類帶有三種時(shí)間記憶核的非齊次Fokker-Planck方程的解析解,基于分離變量法和積分變換得到了相應(yīng)的解析表達(dá)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.
【學(xué)位授予單位】：西安理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175.6

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本文編號(hào)：1950350