Chapter 1 Partial Differential Equations UniMAP Chapter 1 Partial Differential Equations Introduction Solution of PDE Applied Partial Differential Equations -Heat & Wave Equations EUT 203

Introduction Consider the following function f (x1, x2,…, xn) UniMAP Consider the following function f (x1, x2,…, xn) where x1, x2,…, xn are independent variables. If we differentiate f with respect to variable xi , then we assume a) xi as a single variable, b) as constants. EUT 203

Example 1.1 UniMAP Write down all partial derivatives of the following functions: EUT 203

1.6 Partial Differential Equations What is a PDE? Given a function u = u(x1,x2,…,xn), a PDE in u is an equation which relates any of the partial derivatives of u to each other and/or to any of the variables x1,x2,…,xn and u. Notation EUT 203

Example of PDE EUT 203

Focus  first order with two variables PDEs We can already solve  By integration Example Solution EUT 203

(b). Solve PDE in (a) with initial condition u(0,y) = y Solution EUT 203

Separation of Variables Given a PDE in u = u (x,t). We say that u is a product solution if for functions X and T. How does the method work? Let’s look at the following example. EUT 203

KUKUM Example 1.10 EUT 203

Solution KUKUM EUT 203

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KUKUM EUT 203

KUKUM EUT 203

KUKUM EUT 203

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KUKUM Exercise 1.7 Answer EUT 203

Partial Differential Equations for Heat Equation (one dimensional heat equation) Example : Find the solution to the one dimensional heat equation EUT 203

Solution : when t = 0, when n = 5, when n = 2, EUT 203

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Partial Differential Equations for Wave Equation (one dimensional wave equation) Example : Find the solution to the one dimensional wave equation EUT 203

Solution when t = 0, when n = 3, when n =10, EUT 203

when t = 0, when n = 4, when n = 6, EUT 203

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