{"id":1084,"date":"2025-10-15T14:04:22","date_gmt":"2025-10-15T14:04:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1084"},"modified":"2025-10-27T13:08:36","modified_gmt":"2025-10-27T13:08:36","slug":"introduction-to-partial-derivatives-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-partial-derivatives-learn-it-4\/","title":{"raw":"Introduction to Partial Derivatives: Learn It 4","rendered":"Introduction to Partial Derivatives: Learn It 4"},"content":{"raw":"

Partial Differential Equations<\/h2>\r\n

In earlier coursework, we studied differential equations where the unknown function had one independent variable. A partial differential equation<\/strong> (PDE) is an equation involving an unknown function of more than one independent variable and one or more of its partial derivatives.<\/p>\r\n\r\n

\r\n

Examples of partial differential equations include:<\/p>\r\n\r\n

    \r\n \t
  • [latex]u_t=c^{2}(u_{xx}+u_{yy})[\/latex] (heat equation in two dimensions)<\/li>\r\n \t
  • [latex]u_{tt}=c^{2}(u_{xx}+u_{yy})[\/latex] (wave equation in two dimensions)<\/li>\r\n \t
  • [latex]u_{xx}+u_{yy}=0[\/latex] (Laplace's equation in two dimensions)<\/li>\r\n<\/ul>\r\n<\/section>\r\n

    In the first two equations, the unknown function [latex]u[\/latex] has three independent variables\u2014[latex]t[\/latex], [latex]x[\/latex], and [latex]y[\/latex]\u2014and [latex]c[\/latex] is an arbitrary constant. The independent variables [latex]x[\/latex] and [latex]y[\/latex] represent spatial variables (position in space), and [latex]t[\/latex] represents time. In Laplace's equation, the unknown function [latex]u[\/latex] has two independent variables [latex]x[\/latex] and [latex]y[\/latex].<\/p>\r\n\r\n

    Verify that\r\n

    [latex]\\large{u\\,(x,\\ y,\\ t)=5\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)}}[\/latex]<\/p>\r\nis a solution to the wave equation\r\n

    [latex]\\large{u_{tt}=4\\,(u_{xx}+u_{yy})}.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1577773930324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1577773930324\"]\r\n

    First, we calculate [latex]u_{tt},\\ u_{xx}[\/latex], and [latex]u_{yy}[\/latex]:<\/div>\r\n
    [latex]\\begin{array}{ccc}\\hfill {u_{tt}} & =\\hfill & {\\frac{\\partial}{\\partial t}\\left[\\frac{\\partial u}{\\partial t}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial t}[5\\sin{(3\\pi x)}\\sin{(4\\pi y)}(-10\\pi\\sin{(10\\pi t)})]}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial t}[-50\\pi\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\sin{(10\\pi t)}]}\\hfill \\\\ \\hfill & =\\hfill & {-500\\pi^{2}\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)}}\\hfill \\\\ \\hfill {u_{xx}} & =\\hfill & {\\frac{\\partial}{\\partial x}\\left[\\frac{\\partial u}{\\partial x}\\right]} \\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial x}[15\\pi\\cos{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)}]}\\hfill \\\\ \\hfill & =\\hfill & {-45\\pi^{2}\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)}}\\hfill \\\\ \\hfill {u_{yy}} & =\\hfill & {\\frac{\\partial}{\\partial y}\\left[\\frac{\\partial u}{\\partial y}\\right]}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial y}[5\\sin{(3\\pi x)}(4\\pi\\cos{(4\\pi y)})\\cos{(10\\pi t)}]}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\partial}{\\partial y}[20\\pi\\sin{(3\\pi x)}\\cos{(4\\pi y)}\\cos{(10\\pi t)}]}\\hfill \\\\ \\hfill & =\\hfill & {-80\\pi^{2}\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)}.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n
    Next, we substitute each of these into the right-hand side of our wave equation and simplify:<\/div>\r\n
    \r\n

    [latex] \\begin{align} 4(u_{xx}+u_{yy}) &= 4(-45\\pi^2\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)} + -80\\pi^2\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)}\\\\ &=4(-125\\pi^2\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)} &\\quad\\\\ &=-500\\pi^2\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)} \\\\ &=u_{tt} \\end{align}[\/latex]<\/p>\r\nThis verifies the solution.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

    Visualizing Solutions<\/h3>\r\n

    Since the solution to the two-dimensional heat equation is a function of three variables, creating a visual representation is challenging. We can graph the solution for fixed values of [latex]t[\/latex], which gives us snapshots of the heat distribution at specific times. These snapshots show how heat spreads over a two-dimensional surface as time progresses.<\/p>\r\n

    The graph of the solution at time [latex]t=0[\/latex] appears in Figure 1. As time progresses, the extreme temperatures level out, approaching zero as [latex]t[\/latex] approaches infinity.<\/p>\r\n\r\n\r\n[caption id=\"attachment_985\" align=\"aligncenter\" width=\"456\"]\"A Figure 1.[\/caption]\r\n

    If we consider the heat equation in one dimension, then graphing the solution over time becomes more manageable. The heat equation in one dimension is:<\/p>\r\n

    [latex]u_{t}=c^{2}u_{xx}[\/latex],<\/p>\r\n

    where [latex]c^{2}[\/latex] represents the thermal diffusivity of the material. A solution of this differential equation can be written as:<\/p>\r\n

    [latex]u_{m}(x,t)=e^{-\\pi^{2}m^{2}c^{2}t}\\sin(m\\pi x)[\/latex]<\/p>\r\n

    where [latex]m[\/latex] is any positive integer.<\/p>\r\n

    Figure 2 shows a graph of this solution using [latex]m=1[\/latex], where the initial temperature distribution over a wire of length [latex]1[\/latex] is given by [latex]u(x,0)=\\sin(\\pi x)[\/latex]. Notice that as time progresses, the wire cools off. From left to right, the highest temperature (which occurs in the middle of the wire) decreases and changes color from red to blue.<\/p>\r\n\r\n\r\n[caption id=\"attachment_986\" align=\"aligncenter\" width=\"707\"]\"A Figure 2.\u00a0Graph of a solution of the heat equation in one dimension over time.[\/caption]","rendered":"

    Partial Differential Equations<\/h2>\n

    In earlier coursework, we studied differential equations where the unknown function had one independent variable. A partial differential equation<\/strong> (PDE) is an equation involving an unknown function of more than one independent variable and one or more of its partial derivatives.<\/p>\n

    \n

    Examples of partial differential equations include:<\/p>\n

      \n
    • [latex]u_t=c^{2}(u_{xx}+u_{yy})[\/latex] (heat equation in two dimensions)<\/li>\n
    • [latex]u_{tt}=c^{2}(u_{xx}+u_{yy})[\/latex] (wave equation in two dimensions)<\/li>\n
    • [latex]u_{xx}+u_{yy}=0[\/latex] (Laplace’s equation in two dimensions)<\/li>\n<\/ul>\n<\/section>\n

      In the first two equations, the unknown function [latex]u[\/latex] has three independent variables\u2014[latex]t[\/latex], [latex]x[\/latex], and [latex]y[\/latex]\u2014and [latex]c[\/latex] is an arbitrary constant. The independent variables [latex]x[\/latex] and [latex]y[\/latex] represent spatial variables (position in space), and [latex]t[\/latex] represents time. In Laplace’s equation, the unknown function [latex]u[\/latex] has two independent variables [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n

      Verify that<\/p>\n

      [latex]\\large{u\\,(x,\\ y,\\ t)=5\\sin{(3\\pi x)}\\sin{(4\\pi y)}\\cos{(10\\pi t)}}[\/latex]<\/p>\n

      is a solution to the wave equation<\/p>\n

      [latex]\\large{u_{tt}=4\\,(u_{xx}+u_{yy})}.[\/latex]<\/p>\n