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    Review of Partial Differential Equations

    Uploaded by

    Patrick Joseph Robles
    This document provides an introduction to partial differential equations. It defines partial derivatives as derivatives of functions with multiple independent variables, where the derivative is taken with respect to one variable while holding the others constant. It presents examples of common partial differential equations like the one-dimensional heat equation and two-dimensional Laplace equation. It also introduces operators like del (∇) that are used to concisely write long partial differential equations.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Review of Partial Differential Equations

    Uploaded by

    Patrick Joseph Robles
    32 views3 pages
    This document provides an introduction to partial differential equations. It defines partial derivatives as derivatives of functions with multiple independent variables, where the derivative is taken with respect to one variable while holding the others constant. It presents examples of common partial differential equations like the one-dimensional heat equation and two-dimensional Laplace equation. It also introduces operators like del (∇) that are used to concisely write long partial differential equations.

    Original Description:

    Wat Res Engg

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    This document provides an introduction to partial differential equations. It defines partial derivatives as derivatives of functions with multiple independent variables, where the derivative is taken with respect to one variable while holding the others constant. It presents examples of common partial differential equations like the one-dimensional heat equation and two-dimensional Laplace equation. It also introduces operators like del (∇) that are used to concisely write long partial differential equations.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    32 views3 pages

    Review of Partial Differential Equations

    Uploaded by

    Patrick Joseph Robles
    This document provides an introduction to partial differential equations. It defines partial derivatives as derivatives of functions with multiple independent variables, where the derivative is taken with respect to one variable while holding the others constant. It presents examples of common partial differential equations like the one-dimensional heat equation and two-dimensional Laplace equation. It also introduces operators like del (∇) that are used to concisely write long partial differential equations.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    Review of Partial Differential Equations

    We are used to the function of a single variable, say


    f = f(x), “function” meaning that for any given
    value of x, there exists one and only one value of f.
    And we know that

    df f  x Δx − f  x 
    = f '  x =lim ………… (1)
    dx Δx 0 Δx

    which is the first derivative of the function. We


    d2 f
    denote the second derivative as 2
    . There are, of
    dx
    course, higher derivatives.

    We also know about the chain rule. If say we have


    g(f) where f = f(x), then

    dg dg df
    g '  x = = ⋅ ……………………….. (2)
    dx df dx

    The chain can of course be longer than that above.

    Whenever we have a function of two independent


    variables, say f = f(x,y), “independent” meaning x is
    not a function of y nor vice versa, then
    ∂f ∂f
    df = dx dy ………………………… (3)
    ∂x ∂y

    ∂f f  x Δx , y − f  x , y 
    where =lim ……… (4)
    ∂ x Δx  0 Δx
    is the “partial derivative of f with respect to x”, that
    is, the derivative of f while keeping y at some
    ∂f
    constant value; similarly, is the partial
    ∂y
    derivative of f with respect to y. We can of course
    extend this to functions of more than two
    independent variables.
    We can manipulate Eq. (3) in the following manner:
    df ∂ f ∂ f dy df ∂ f dx ∂ f
    =  or =  …… (5)
    dx ∂ x ∂ y dx dy ∂ x dy ∂ y
    We have a chain rule for partial derivatives. If
    f = f(u,v) and u = u(x,y) and v = v(x,y), then:
    ∂ f ∂ f ∂u ∂ f ∂v ∂ f ∂ f ∂u ∂ f ∂ v
    = ⋅  and =  ⋅ … (6)
    ∂ x ∂u ∂ x ∂v ∂ x ∂ y ∂u ∂ y ∂v ∂ y
    which chain is again extendable to three or more,
    and the functions can be of three or more variables.
    If we have the special case, f = f(x,y) where x = x(t) and
    y = y(t), then
    df ∂ f dx ∂ f dy
    =  …………………………… (7)
    dt ∂ x dt ∂ y dt
    This form is called the parametric representation of
    the function f, where t is the “parameter” and x and
    y are the “coordinates”.
    Unlike total differentials, partial differentials, e.g.
    ∂ f or ∂ x , have no meaning by themselves, nor are
    they operable like fractions.
    The higher order partial derivatives may be
    described in the following manner:

    2
    ∂ f
    =
    ∂  
    ∂f
    ∂x ; 2
    ∂ f
    =
    ∂f
    ∂
    ∂y  
    ∂x
    2
    ∂x ∂x∂ y ∂x
    Examples of partial differential equations:
    2
    ∂u ∂u
    =c 2⋅ 2 (One-dimensional heat equation)
    ∂t ∂x
    2 2
    ∂u ∂u
    2
     2 (Two-dimensional Laplace equation)
    ∂x ∂ y
    For subsequent applications, we can now consider the
    ∂ ∂ ∂
    operator ∇ =   , shortened verbally
    ∂x ∂ y ∂z
    as “del”, which has no meaning by itself but is used
    merely to shorten writing of long equations.
    Thus, if we take the Laplacian (del-squared) of u in
    2 2 2
    ∂u ∂u ∂u
    three dimensions, then ∇ u= 2  2  2 is
    2

    ∂x ∂ y ∂z
    the Three-dimensional Laplace equation.

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