Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 1 / 19
Introduction The derivation of the heat equation is based on a more general principle called the conservation law. It is also based on several other experimental laws of physics. We will derive the equation which corresponds to the conservation law. Then, we will state and explain the various relevant experimental laws of physics. Finally, we will derive the one dimensional heat equation. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 2 / 19
Conservation Law Many PDE models involve the study of how a certain quantity changes with time and space. This change follows a basic law called the conservation law which states that the rate at which a quantity changes in a given domain must equal the rate at which the quantity flows across the boundary of that domain plus the rate at which the quantity is created or destroyed, inside the domain. One can see how this would apply to the study of a certain population in a given area. Similar statements can be made about many other quantities such as heat energy, the mass of a chemical, the number of automobiles on a freeway,... Our study focuses on a quantity which only changes in one direction, say the x-direction. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 3 / 19
Conservation Law Figure: Tube with cross-sectional area A Consider this to be our domain. It has a constant cross-sectional area we call A. In it, we study how a certain quantity (mass, energy, species,...) changes. We let u (x, t) denote the density of the quantity. Recall that density is measured in amount of quantity per unit volume or per unit length. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 4 / 19
Conservation Law Let us make some remarks and introduce further notation: 1 The amount of the quantity at time t in a small section of width dx will be u (x, t) Adx for each x. The amount of the quantity in an arbitrary section a x b will be b a u (x, t) Adx. 2 φ = φ (x, t) denotes the flux of the quantity. It measures the amount of the quantity crossing the section at x, at time t. Its units are amount of quantity per unit area, per unit time. The amount of the quantity crossing the section at x, at time t is Aφ (x, t). By convention, flux is positive if the flow is to the right. 3 f (x, t) denotes the given rate at which the quantity is created (source) or destroyed (sink) per unit volume within the section at x, at time t. It is measured in amount of quantity per unit volume, per unit time. The amount of the quantity being created in a small section of width dx for each x is f (x, t) Adx per unit time and the amount of the quantity being created in an arbitrary section a x b will be b a f (x, t) Adx. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 5 / 19
Conservation Law Write the corresponding equation for the conservation law in an arbitrary section a x b of our domain. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 6 / 19
Conservation Law Write the corresponding equation for the conservation law in an arbitrary section a x b of our domain. We should have obtained d b dt a u (x, t) dx = φ (a, t) φ (b, t) + b a f (x, t) dx Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 6 / 19
Conservation Law In order to go further, we need some results from mathematics. Theorem (Leibniz Rule) If a (t), b (t), and F (x, t) are continuously differentiable then d dt b(t) a(t) F (y, t) dy = b(t) a(t) F t (y, t) dy+f (b (t), t) b (t) F (a (t), t) a (t) This is often known as "taking the derivative inside the integral". Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 7 / 19
Conservation Law Using Leibniz rule, explain how we can obtain that for every a and b, b a What can be concluded? [u t (x, t) + φ x (x, t) f (x, t)] dx = 0 Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 8 / 19
Conservation Law: Summary We study how a certain quantity changes with time in a given region. We make the following assumptions: 1 u (x, t) denotes the density of the quantity being studied. u is assumed to be continuously differentiable. 2 φ (x, t) is the flux of the quantity at time t at x. It measures the amount of the quantity crossing a cross section of our region at x. φ is assumed to be continuously differentiable. 3 f (x, t) is the rate at which the quantity is created or destroyed within our region. f is assumed to be continuous. 4 We assume the quantity being studied only varies in the x direction. 5 Then, the equation describing how our quantity changes with time in the given region (fundamental conservation law) is u t (x, t) + φ x (x, t) = f (x, t) Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 9 / 19
Conservation Law: Remarks Remark Let us make a few remarks before looking a specific examples. 1 The above equation is often written as u t + φ x = f for simplicity. 2 The functions φ and f are functions of x and t. That dependence may be through the function u as in f = f (u) or φ = φ (u) which would lead to a nonlinear model. We will see one in the examples. 3 The above equation involves two unknown functions: u and φ, usually the source f is assumed to be given. This means that another equation relating u and φ is needed. We will see various examples. 4 The above equation is in its most general form. As we look at specific models, it will take on different forms. In the next sections we will consider various possibilities including advection and diffusion. Also, the source term can take on different forms. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 10 / 19
Advection Equation Advection refers to transport of a certain substance in a fluid (water, any liquid, air,...) such as transport of a pollutant in a river. Definition A model where the flux is proportional to the density is called an advection model. It is easy to understand why. Thinking of the example of the river carrying a pollutant, the amount of pollutant which crosses the boundary of a given region in the river clearly depends on the density of the pollutant. In this case, we have for some constant c: φ = cu The constant c is the speed of the fluid. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 11 / 19
Advection Equation Rewrite the fundamental conservation law in the case of an advection model with no source. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 12 / 19
Advection Equation Rewrite the fundamental conservation law in the case of an advection model with no source. We should have obtained u t + cu x = 0 Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 12 / 19
Advection Equation Rewrite the fundamental conservation law in the case of an advection model with no source. We should have obtained u t + cu x = 0 Rewrite the fundamental conservation law in the case of an advection model with source f (x, t). Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 12 / 19
Advection Equation Rewrite the fundamental conservation law in the case of an advection model with no source. We should have obtained u t + cu x = 0 Rewrite the fundamental conservation law in the case of an advection model with source f (x, t). We should have obtained u t + cu x = f (x, t) Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 12 / 19
Diffusion Diffusion is the transport by molecular motion. From observations and experiments, we know that: Flow is always from more agitated molecules (higher kinetic energy) to less agitated molecules. The steeper the density gradient of the substance being studied, the greater the flow. Hence, with diffusion we have: φ (x, t) = Du x where D > 0 is called the diffusion constant, u is the density of the quantity being studied. What can we say about φ in the case the density u increases from left to right? Does it agree with what we said above? Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 13 / 19
Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Rewrite the conservation law when there is diffusion and advection with no source. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Rewrite the conservation law when there is diffusion and advection with no source. We should have obtained u t + cu x Du xx = 0 Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Rewrite the conservation law when there is diffusion and advection with no source. We should have obtained u t + cu x Du xx = 0 Rewrite the conservation law when there is diffusion and advection with a source f (x, t). Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Rewrite the conservation law when there is diffusion and advection with no source. We should have obtained u t + cu x Du xx = 0 Rewrite the conservation law when there is diffusion and advection with a source f (x, t). We should have obtained u t + cu x Du xx = f (x, t) Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
Heat Equation Consider the heat equation with only diffusion present and no source. Consider a thin rod having a constant density ρ and specific heat C. The specific heat of a substance is the amount of energy needed to raise the temperature of a unit mass of the substance by one degree. Both ρ and C are known for known substances, they can be found in engineering and physics handbooks. If u (x, t) is the energy density and θ the temperature, then u (x, t) = ρcθ (x, t) Also, heat flow follows a diffusion model, hence φ = Kθ x. Combine these two equation with the conservation law to derive the heat equation. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 15 / 19
Heat Equation Consider the heat equation with only diffusion present and no source. Consider a thin rod having a constant density ρ and specific heat C. The specific heat of a substance is the amount of energy needed to raise the temperature of a unit mass of the substance by one degree. Both ρ and C are known for known substances, they can be found in engineering and physics handbooks. If u (x, t) is the energy density and θ the temperature, then u (x, t) = ρcθ (x, t) Also, heat flow follows a diffusion model, hence φ = Kθ x. Combine these two equation with the conservation law to derive the heat equation. We should have obtained θ t kθ xx = 0 where k = K, it is called ρc the diffusivity or thermal diffusivity. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 15 / 19
Heat Equation Remark In deriving this equation, we made some assumptions about the rod. These assumptions lead us to assume that ρ and C and K were constants. If the rod is not homogeneous, K will also depend on x. In addition, if we consider wide ranges of temperatures, K may also depend on θ. So, the heat equation would become This is a non-linear model. ρcθ t (K (x, θ) θ x ) x = 0 Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 16 / 19
Steady State Solution Many PDE models, in particular diffusion problems, have the property that after a long time, they approach a steady state, that is a solution which is no longer evolving with time. In other words, for large t, u (x, t) becomes a function of x only and u t = 0. Even if we do not yet know how to solve PDEs, we can, in most cases, find the steady state solution. To do so, we set u t = 0 in the diffusion equation. The resulting equation is a second-order linear ODE, which we know how to solve. Remember that the solution should be a function of x only. The examples below illustrate how. Find the steady state solution for the problem PDE u t = cu xx 0 < x < L 0 < t < BC1 u (0, t) = T 1 0 < t < BC2 u (L, t) = T 2 0 < t < IC u (x, 0) = f (x) 0 x L Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 17 / 19
Steady State Solution Many PDE models, in particular diffusion problems, have the property that after a long time, they approach a steady state, that is a solution which is no longer evolving with time. In other words, for large t, u (x, t) becomes a function of x only and u t = 0. Even if we do not yet know how to solve PDEs, we can, in most cases, find the steady state solution. To do so, we set u t = 0 in the diffusion equation. The resulting equation is a second-order linear ODE, which we know how to solve. Remember that the solution should be a function of x only. The examples below illustrate how. Find the steady state solution for the problem PDE u t = cu xx 0 < x < L 0 < t < BC1 u (0, t) = T 1 0 < t < BC2 u (L, t) = T 2 0 < t < IC u (x, 0) = f (x) 0 x L We should have found u (x) = T 2 T 1 x + T 1. L Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 17 / 19
Heat Equation:Summary 1 We derived the conservation law formula u t + φ x = f (x, y) 2 The flux, φ, takes on different forms depending on what kind of model corresponds to the situation being studied. 1 Advection model: φ = cu hence the conservation law becomes: u t + cu x = f (x, y) 2 Diffusion model: φ = Du x hence the conservation law becomes: u t Du xx = f (x, y) 3 Advection plus diffusion model: φ = cu Du x hence the conservation law becomes: u t + cu x Du xx = f (x, y) 3 For each equation above, f (x, t) is the source. When there is no source, f (x, t) = 0. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 18 / 19
Exercises See the problems at the end of my notes on derivation of the one-dimensional heat equation. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 19 / 19