**Workspace**: [[MA265 Methods of Modelling 3 Workspace.canvas|MA265 Methods of Modelling 3 Workspace]]. **Summary**: - We use the **method of characteristics** to solve **transport equation** and find the general solution to **wave equation**. - We derive **d'Alembert's formula** for initial value problems for wave equation. Use energy methods to prove uniqueness of solutions. Use method of separation of variables to find PDEs solutions satisfying homogenous boundary conditions and know sufficient conditions for existence of solution to satisfying initial conditions. --- # 2. Introduction to Partial Differential Equations We define [[Partial differential equations|partial differential equations]] and classify them by their *order*, the *linearity* of their *associated differential operator*; and *homogeneity*. We [[Classification of linear second order PDEs|classify linear second order PDEs]] as *hyperbolic*, *elliptic* and *parabolic*. Boundary conditions: Cauchy, Dirichlet, Neumann, Mixed (Robin) & periodic. We discuss [[Well-Posedness|well-posedness]] in the context of boundary-value problems for PDEs. # 3. The Transport Equation & The Method of Characteristics 3.1 [[Linear Advection Equation]]. 3.2 [[Method of Characteristics for Transport Equation]]. # 4. The Wave Equation 4.1.1 [[Derivation of 1D Wave Equation From Acoustics]] \[Non-examinable\]. 4.1.2 [[Derivation of 1D Wave Equation From String Vibration]] \[Non-examinable\]. 4.1.3 [[1D Wave Operator is a Composite of Two Opposite Direction Linear Advection Operators]]. 4.2 [[General Solution to 1D Wave Equation]]/Coordinate Method. 4.2.1 Initial Value Problem for 1D Wave Equation. 4.2.2 [[Solution to Cauchy Problem for 1D Wave Equation (d'Alembert's Formula)]] \[Uniqueness of solutions is non-examinable\]. **Principle of causality**: be able to define the domains of dependence and influence for a point $(x,t)$ 4.3.1 [[Uniqueness of Solution to Initial Boundary Value Problem for 1D Wave Equation with Homogenous Dirichlet Boundary Condition]]. 4.3.2 [[Solution to 1D Wave Equation with Homogenous Dirichlet Boundary Condition by Separation of Variables]]. 4.8 [[Solution to 1D Wave Equation with Neumann Boundary Condition by Separation of Variables]]. # 5. Fourier Series 5.1 [[Equivalence of Real and Complex Trigonometric Polynomials]]. 5.2 [[Complex exponentials form orthonormal basis of square-integrable functions on (-pi, pi)]]. ### Approximation of Functions 5.3 [[Optimality of Fourier Coefficients]]. 5.4 [[Riemann-Lebesgue Lemma]]. 5.5 [[Fourier Transform of Even and Odd 2Pi-Periodic Functions]]. ### Convergence of Fourier Series 5.7 [[Fourier Series in Terms of Dirichlet Kernel]]. 5.8 [[Fourier Series of 2Pi-Periodic Continuously Differentiable Function is Pointwise Convergent]]. 5.9 [[Decay of Fourier coefficients]]. 5.10 [[Uniform Convergence of Fourier Series]]. 5.11 [[Regularity of Fourier Coefficients]]. 5.12 [[Sufficient Condition for Existence of Separated Solution to Initial Boundary Value Problem for 1D Wave Equation with Homogenous Dirichlet Boundary Condition]]. # 6. The Heat Equation 6.1 [[Derivation of 1D Heat Equation From Heat Diffusion in Uniform Rod]] [[Solution to 1D Heat Equation with Homogenous Dirichlet Boundary Conditions by Separation of Variables]] 6.2 [[Duhamel's Principle for 1D Heat Equation]] 6.3 [[Maximum principle for heat equation]]. # 7. Laplace Equation # 8. The Finite Difference Method