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Math433课程简介
In this course, we further explore the world of differential equations. Mainly, we cover topics in partial differential equations. Partial Differential Equations (PDEs) are fundamental to the modeling of many natural phenomena, arising in many fields, including fluid mechanics, heat and mass transfer, electromagnetic theory, finance, elasticity, and more. The goals of this course are to discuss the following topics: classification of PDEs in terms of order, linearity and homogeneity; physical interpretation of canonical PDEs; solution techniques, including separation of variables, series solutions, integral transforms, and the method of characteristics.
Prerequisites
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This is an advanced mathematics class dedicated to the study of partial differential equations (PDEs). These equations are the most important mathematical tools for the study of complex physical phenomena such as waves and fluids (including both air and water), heat transfer, electromagnetism, and finance.
Math433 Partial Differential Equations HELP（EXAM HELP， ONLINE TUTOR）
Inviscid Burgers’ equation: $u(x, t)$ solves $u_t+u u_x=0$
i. This is a first-order, quasilinear, homogeneous equation.
ii. The Inviscid Burgers’ equation describes the conservation of energy in a system $u$ defined over the spatial variable $x$ and temporal variable $t$.
iii. The solutions of the partial differential equation are straight lines where the intersection represent the location of shockwaves in the system.
The Inviscid Burgers’ equation is a first-order, quasilinear, homogeneous partial differential equation.
ii. This is not entirely accurate. The Inviscid Burgers’ equation is a simplified model for fluid flow, and it describes the propagation of shock waves in a one-dimensional fluid flow. It is not a conservation law in the strict sense of energy conservation, although it does conserve mass.
iii. This is not correct. The solutions of the Inviscid Burgers’ equation can exhibit shock waves, which are not straight lines but rather curves that connect different regions of the solution. The equation is known to have a characteristic form, and its solutions can develop discontinuities, such as shock waves, which are represented by the intersection of characteristic curves.
Viscous Burgers’ equation: $u(x, t)$ solves $u_t+u u_x=\mu u_{x x}$
i. This is a second-order, linear, nonhomogeneous equation.
ii. The Viscous Burgers’ equation describes the energy present in a viscous system $u$ defined over the spatial variable $x$ and the temporal variable $t$.
iii. The partial differential equation captures shock waves of energy present in viscous systems.
The Viscous Burgers’ equation is a second-order, nonlinear, and nonhomogeneous partial differential equation, which includes a diffusion term that describes the dissipation of energy due to viscosity.
ii. This is not entirely accurate either. The Viscous Burgers’ equation is a model for fluid flow that takes into account the effects of viscosity on the flow, and it describes the evolution of a one-dimensional fluid flow. The equation does not describe the total energy present in the system, but rather the flow of energy due to the fluid motion.
iii. This is partially correct. The Viscous Burgers’ equation can exhibit shock waves, which are a result of the nonlinear advection term. However, the presence of the viscosity term tends to smooth out the shock waves, resulting in a gradual transition from one region of the solution to another, rather than a sharp jump as in the inviscid case. The equation captures both the propagation and dissipation of energy due to the fluid motion.
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