MAT 5673 - Revision history

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==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Order of Differential Equations]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Explain the basic notion of the order of a differential equation.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Solutions of Differential Equations]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Explain the basic notion of solutions of differential equations.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Initial Value Problem|Initial Value Problem (IVP)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Explain the basic notion of the initial values problem.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Cauchy Problem]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Explain the Cauchy Problem
* Explain the basic notion of existence and uniqueness of a solution to the Cauchy Problem.
|-
|Week I
||
* Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3
||
* [[Separation of Variables (1st Order)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Determine separable differential equations of the first order.
* Apply direct methods to evaluate exact solutions of separable differential equations of the first order.
|-
|Week II
||
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
||
* [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Determine homogeneous differential equations of the first order.
* Apply direct methods to evaluate exact solutions of homogeneous differential equations of the first order (substitutions).
* Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
|-
|Week II
||
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
||
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Determine linear differential equations of the first order.
* Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
|-
|Week II
||
* Ahmad and Ambrosetti 2014, Chaps. 1 and 3
||
* [[Integrating Factor]]
||
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
||
* Apply integrating factor to solve linear differential equations of the first order.
* Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
|-
|Week III
||
* Ahmad and Ambrosetti 2014, Ch. 3
||
* [[Bernoulli Equations (1st Order)]]
||
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
||
* Determine Bernoulli of the first order.
* Apply direct methods to evaluate exact solutions of Bernoulli of the first order.
|-
|Week III
||
* Ahmad and Ambrosetti 2014, Ch. 3
||
* [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
||
* [[Integrating Factor]] for exact equations.
* [[Partial Derivatives]]
||
* Determine Exact Differential Equations of the first order.
* Apply direct methods to evaluate exact solutions of Exact Differential Equations of the first order.
* Use the integrating factor technique for exact equations.
|-
|Week IV
||
* Ahmad and Ambrosetti 2014, Chaps. 1-3
||
* Overview of the solutions methods discussed so far (Chapters 1-3).
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
* [[Partial Derivatives]]
* First-order differential equations:
:- [[Separation of Variables (1st Order)]]
:- [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
:- [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
:- [[Bernoulli Equations (1st Order)]]
:- [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
||
* Determine the type of different classes of differential equations of the first order: separable, linear, homogeneous, Bernoulli, exact.
* Use direct methods to solve first order differential equations solved and not solved for the first derivative.
|-
|Week V
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Linear Independence of Functions]].
||
* [[Linear Independence of Vectors]].
||
* Understanding of Linear Independence of Functions.
|-
|Week V
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Linear Independence of Functions|Linear Dependence of Functions]].
||
* [[Linear Dependence of Vectors]].
||
* Understanding of Linear Dependence of Functions.
|-
|Week V
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Wronskian]]
||
* [[Linear Independence of Functions]].
* [[Linear Independence of Functions|Linear Dependence of Functions]].
* [[Determinant]].
||
* Showing linear independence of two functions using the Wronskian.
* Showing linear independence of two solutions of Linear Second-Order ODEs using the Wronskian.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Reduction of the Order]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Apply of the reduction of the order technique for second-order ODEs with a given solution.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Homogeneous Differential Equations|Linear Homogeneous Equations]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Determine homogeneous classes of differential equations of the second and higher order.
* Determine linear and non-linear classes of differential equations of the second and higher order.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Abel’s Theorem]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Determine Wronskian for a second-order ODE with 2 given solutions.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Fundamental Solutions]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Determine fundamental solutions.
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Linear Differential Equations|Linear Non-homogeneous Equations]]
||
* [[Wronskian]].
* [[Quadratic Equations]].
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]].
* [[Solutions of Linear Systems]].
||
* Determine non-homogeneous classes of differential equations of the second and higher order.
* Determine linear and non-linear classes of differential equations of the second and higher order
|-
|Week VI
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Variation Of Parameters (2nd Order)|Variation of Parameters (2nd Order)]]
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
* [[Quadratic Equations]].
* [[Solutions of Linear Systems]].
||
* Apply of the variation of parameters technique for second-order ODEs.
|-
|Week VII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Variation Of Parameters (2nd Order)|Variation of Parameters (2nd Order)]] (continued)
||
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
* [[Quadratic Equations]].
* [[Solutions of Linear Systems]].
||
* Apply variation of parameters technique for second-order ODEs.
|-
|Week VII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Method of Undetermined Coefficients (2nd Order)]]
||
* [[Quadratic Equations]].
* [[Systems of Linear Equations]].
||
* Apply method of undetermined coefficients technique for second-order ODEs.
|-
|Week VIII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Non-linear 2nd Order ODEs]]
||
* [[Algebraic Equations]]
* [[Reduction of the Order]]
* Integration techniques
:- [[Direct Integration]]
:- [[Integration by Substitution]]
:- [[Integration by Parts]]
:- [[Partial Fractions]]
||
* Methods for nonlinear second-order ODEs.
* Apply reduction of the order method to some nonlinear second-order ODEs.
|-
|Week VIII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
||
* [[Variation Of Parameters|Variation of Parameters (2nd Order)]]
||
* Apply variation of parameters technique for higher-order ODEs
|-
|Week VIII
||
* Ahmad and Ambrosetti 2014, Ch. 5
||
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
||
* [[Method of Undetermined Coefficients (2nd Order)]]
||
* Apply method of undetermined coefficients technique for higher-order ODEs
|-
|Week IX
||
* Ahmad and Ambrosetti 2014, Ch. 6
||
* [[Linear Differential Equations|Linear Differential Equations (Higher Order)]]
||
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
* [[Variation Of Parameters|Variation of Parameters (Higher Order)]].
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]].
||
* Methods for linear higher-order ODEs
|-
|Week X
||
* Ahmad and Ambrosetti 2014, Chaps. 5, 6
||
* Overview of the solutions methods for second and higher order differential equations.
||
* [[Algebraic Equations]]
* Direct methods for second and higher-order ODEs:
:- [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
:- [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
||
* Evaluate the exact solutions of important classes of differential equations such as second order differential equations as well as some higher order differential equations.
|-
|Week X
||
* Ahmad and Ambrosetti 2014, Chaps. 10
||
* [[Power Series Solutions]]
||
* [[Power Series Induction]]
||
Apply power series method to evaluate solutions of first-order and second-order ODEs.
|-
|Week XI
||
* Ahmad and Ambrosetti 2014, Chaps. 10
||
* [[Power Series Solutions]] (continued)
||
* [[Power Series Induction]]
||
Apply power series method to evaluate solutions of first-order and second-order ODEs.
|-
|Week XII
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[Laplace Transform]]
||
* [[Functions]] of Single Variable.
* [[Continuity]] of functions of single variables.
* [[Derivatives]] of functions of single variables.
* [[Improper Integrals]] of functions of single variables with infinite limits.
||
* Definition and main properties of the L-transform.
|-
|Week XIII
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[Inverse Laplace Transform]]
||
* [[Laplace Transform]]
* [[Complex Derivatives]]
||
* Apply the theorem(s) for inverse L-transform.
|-
|Week XIV
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[Laplace Transform to ODEs]]
||
* [[Linear Differential Equations|Linear Equations]]
* [[Laplace Transform]]
* [[Inverse Laplace Transform]]
||
* Apply the Laplace transform as solution technique.
|-
|Week XIV
||
* Ahmad and Ambrosetti 2014, Ch. 11
||
* [[Laplace Transform to ODEs|Laplace Transform to Systems of ODEs]]
||
* [[Solutions of Linear Systems]].
* [[Laplace Transform]].
* [[Inverse Laplace Transform]].
||
* Apply the Laplace transform as solution technique.
|-
|Week XV
||
* Ahmad and Ambrosetti 2014
||
* Overview of the solutions methods discussed.
||
* [[Separation of Variables (1st Order)]]
* [[Homogeneous Differential Equations|Homogeneous Differential Equations (1st Order)]]
* [[Linear Differential Equations|Linear Differential Equations (1st Order)]]
* [[Integrating Factor]]
* [[Bernoulli Equations (1st Order)]]
* [[Exact Differential Equations|Exact Differential Equations (1st Order)]]
* [[Reduction of the Order]]
* [[Method of Undetermined Coefficients (2nd Order)]]
* [[Non-linear 2nd Order ODEs]]
* [[Variation Of Parameters|Variation of Parameters (Higher Order)]]
* [[Method of Undetermined Coefficients|Method of Undetermined Coefficients (Higher Order)]]
* [[Linear Differential Equations|Linear Differential Equations (Higher Order)]]
* [[Power Series Solutions]]
* [[Laplace Transform to ODEs]]
* [[Laplace Transform to ODEs|Laplace Transform to Systems of ODEs]]
||
* Apply all solutions methods discussed.
|}

@@ Line 8: / Line 8: @@
 Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are
 an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.
 ==Topics List==
 {| class="wikitable sortable"

@@ Line 104: / Line 104: @@
 |Week 10
 ||
-Complex numbers
 ||
 Qualitative properties of PDE's
@@ Line 118: / Line 118: @@
 Introduction to numerical methods for PDE (optional)
 ||
-Derivatives, Calculus
+Derivatives, Calculus, Matrices, Linear Algebra
 ||
 * Basic finite difference schemes for first-order quasilinear equations, CFL condition
@@ Line 124: / Line 124: @@
 |Week 12
 ||
-Matrices, Linear Algebra
 ||
 Introduction to the Laplace and Poisson equation

@@ Line 88: / Line 88: @@
 Introduction to Fourier series
 ||
+Infinite series
 ||
 * Orthonormal systems of functions, spectral method for the wave and heat equation
@@ Line 98: / Line 98: @@
 Schroedinger equation
 ||
+Complex numbers
 ||
 * Basic properties of Schroedinger equation, particle in a potential well
@@ Line 104: / Line 104: @@
 |Week 10
 ||
+Complex numbers
 ||
 Qualitative properties of PDE's
@@ Line 118: / Line 118: @@
 Introduction to numerical methods for PDE (optional)
 ||
+Derivatives, Calculus
 ||
 * Basic finite difference schemes for first-order quasilinear equations, CFL condition
@@ Line 124: / Line 124: @@
 |Week 12
 ||
+Matrices, Linear Algebra
 ||
 Introduction to the Laplace and Poisson equation
@@ Line 138: / Line 138: @@
 Introduction to the Calculus of Variations
 ||
-Differentiation of an integral with respect to a parameter
+Differentiation of an integral with respect to a parameter, parametric surfaces
 ||
 * Compute the variational derivative of a functional

@@ Line 80: / Line 80: @@
 Partial derivatives, chain rule
 ||
-* Infinite superposition of basic solutions found by separation to form more general solutions
+* Forming more general solutions out of infinite superposition of basic solutions
 |}

@@ Line 1: / Line 1: @@
 ==Topics List==
 {| class="wikitable sortable"
@@ Line 6: / Line 15: @@
 ||
+*
 ||
+*
 ||
+*
 ||
-* Learning outcome
+Initial-boundary value problem for heat and wave equation I
 |}