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**Unformatted text preview: **Chapter 11. Partial Differential Equations 11.1 Differential Equations Mathematical models of physical phenomena often involve differential equations with one or more in- dependent variable. For instance, a model of heat conduction in a region (e.g. an ocean, the atmo- sphere, or a system of pipes) will normally involve three-space variables for coordinates of points in the region and one time variable, as well as other physical information. 11.1.1 Ordinary differential equation An ordinary differential equation (o.d.e.) is an equation that involves an unknown function y ( x ) of 2 MA1505 Chapter 11. Partial Differential Equations exactly one independent variable x and derivatives of y . We also call y the dependent variable. 11.1.2 Example (i) y- xy = 0 where y = dy dx is the derivative of y w.r.t. x . This is an o.d.e. that involves the function y ( x ) with one independent variable x . (ii) y 00- 3 y + 2 y = 0 where y = dy dx and y 00 = d 2 y dx 2 . Again, in this o.d.e., x is the only independent vari- able of the function y . 2 3 MA1505 Chapter 11. Partial Differential Equations 11.1.3 Partial differential equation A partial differential equation (p.d.e.) is an equation containing an unknown function u ( x,y,... ) of two or more independent variables x,y,... and its partial derivatives with respect to these variables. We also call u the dependent variable. 11.1.4 Example (i) u xy- 2 x + y = 0 This is a p.d.e. that involves the function u ( x,y ) with two independent variables x and y . (ii) w xy + x ( w z ) 2 = yz This is a p.d.e. that involves the function w ( x,y,z ) with three independent variables x , y and z . 3 4 MA1505 Chapter 11. Partial Differential Equations 11.1.5 Solutions of Differential Equations A solution of a differential equation is any function which satisfies the equation identically. There are usually one or more family of solutions for a differential equation. We call such a family of solu- tions a general solution of the differential equation. A specific function from the general solution is called a particular solution of the differential equation. 11.1.6 Example Let us substitute y = e x 2 / 2 in the o.d.e of example 11.1.2 (i): 4 5 MA1505 Chapter 11. Partial Differential Equations By differentiating y = e x 2 / 2 using chain rule, we have y = e x 2 / 2 d dx x 2 2 ¶ = xe x 2 / 2 . On the other hand, xy = xe x 2 / 2 . So y = e x 2 / 2 satisfies the o.d.e. and hence is a par- ticular solution of the o.d.e....

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