Effective: 2018-01-01

Course Description

The course outline below was developed as part of a statewide standardization process.

General Course Purpose

The purpose of the course is to provide for a smooth transition of STEM students to 4-year colleges and introduce them to advanced topics of mathematics, physics and engineering: numerical methods for solving differential equations, classical partial differential equations, methods for solving PDEs and boundary-value problems (BVPs).

Course Prerequisites/Corequisites

Prerequisite: Completion of MTH 267 with a grade of C or better or equivalent.

Course Objectives

• System of Linear First Order Differential Equations
• Define system of linear first-order differential equations, Initial value problem (IVP) and its solution vector, linear dependence/independence, fundamental set of solutions
• Check that a vector of functions is a solution of a system or an initial value problem (IVP)
• Apply criterion for linearly independent solutions and find general solution for homogeneous and nonhomogeneous systems (for the 3 types of eigenvalues: distinct real, complex, repeated)
• Solve nonhomogeneous linear systems by the methods of undetermined coefficients and variation of parameters
• Numerical solutions of Ordinary Differential Equations
• Understand the concept of local and global truncation errors, stability of numerical method
• Use single-step and multistep methods( Euler's Method, Improved Euler's Method, 1st, 2nd and 4th-order Runge-Kutta Method, Adams-Bashforth-Moulton Method) and finite difference method to approximate ivp and bvp solutions
• Plane Autonomous Systems
• Explain the concept of autonomous systems of first-order des (linear and nonlinear)
• Find critical points and classify critical points of linear and nonlinear systems (stable/unstable nodes, saddle point, degenerate unstable node, center, stable/unstable spiral points), identify equilibrium solution and periodic solution
• Use stability criterion for plane autonomous systems
• Apply the concept of linearization of differentiable function and classify stable and unstable critical points
• Perform stability analysis for linear/nonlinear systems for various applications
• Orthogonal Functions
• Define orthogonal functions and sets of orthogonal functions
• Write the definition of the Fourier Series and expansion of functions in a Fourier Series
• Define Sturm-Liouville problem and solve it
• Write the definitions and expand the function in Fourier-Bessel Series
• Boundary-Value Problems in Rectangular Coordinates
• Define linear/nonlinear, homogeneous/nonhomogeneous partial differential equations
• Classify the linear second-order pdes as hyperbolic, parabolic or elliptic
• Use the method of separation of variables to find particular solution of pdes
• Identify classical and modified pdes and bvps (1d heat equation, 1d wave equation and 2d form of Laplace?s Equation) and solve them
• Use the concept of orthogonal series expansions or generalized Fourier Series and solve bvps by using orthogonal series expansions
• Integral Transforms
• Find the Laplace transform of partial derivatives of functions of two variables, use Laplace transform to solve bvps
• Define a Fourier integral of function and conditions for convergence, the Fourier integral of even/odd functions
• Use the definitions of three Fourier transform pairs (direct and inverse integral transforms)
• Solve bvps using the Fourier transform

Major Topics to be Included

• System of Linear First Order Differential Equations
• Numerical solutions of Ordinary Differential Equations
• Plane Autonomous Systems
• Orthogonal Functions
• Boundary-Value Problems in Rectangular Coordinates
• Integral Transforms