1. Chapter 1 Partial Differential Equations
UniMAP
Chapter 1 Partial Differential Equations
Introduction
Solution of PDE
Applied Partial Differential Equations
-Heat & Wave Equations
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2. 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.
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3. Example 1.1
UniMAP
Write down all partial derivatives of the following functions:
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4. 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
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5. Example of PDE
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6. Focus  first order with two variables
PDEs We can already solve
 By integration
Example
Solution
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7. (b). Solve PDE in (a) with initial condition u(0,y) = y
Solution
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8. 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.
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9. KUKUM
Example 1.10
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10. Solution
KUKUM
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11. KUKUM
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12. KUKUM
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13. KUKUM
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14. KUKUM
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15. KUKUM
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16. KUKUM
Exercise 1.7
Answer
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17. Partial Differential Equations for Heat Equation
(one dimensional heat equation)
Example :
Find the solution to the one dimensional heat equation
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18. Solution :
when t = 0,
when n = 5,
when n = 2,
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20. Partial Differential Equations for Wave Equation
(one dimensional wave equation)
Example :
Find the solution to the one dimensional wave equation
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21. Solution
when t = 0,
when n = 3,
when n =10,
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22. when t = 0,
when n = 4,
when n = 6,
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