Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Differential Equations → Partial Differential Equations ↓


Heat Equation


The heat equation is one of the most important partial differential equations (PDEs) in mathematical physics. It describes how the distribution of heat (or change in temperature) in a given region changes over time. The heat equation is important to understand because it models situations where heat transfer is involved, from simple cases like heating a metal rod to complex systems like climate models.

The general form of the heat equation is as follows:

∂u/∂t = α ∇²u

Where:

  • u(x, t) is the temperature at position x and time t.
  • α is the thermal diffusivity of the material, a constant that describes how quickly heat spreads through the material.
  • ∇² is the Laplace operator, which describes how temperature changes in space.

Understanding equation components

The left side of the equation, ∂u/∂t, represents the rate of change of temperature relative to time. It tells us how quickly the temperature is changing at a specific point.

The right side of the equation, α ∇²u, represents how heat spreads through space. The Laplace operator, ∇², is important because it deals with how the temperature at a point is affected by the temperature at neighboring points.

Physical meaning of the heat equation

Consider a simple metal rod with one end hot. Over time, heat will tend to spread through the rod. The heat equation helps determine the temperature at different points of the rod at any given time.

Imagine if we apply different initial heats to two rods of the same material, but one rod is thin and the other is thick. The heat will spread faster in the thin rod than in the thick rod. This shows why thermal expansion (α) is important in the heat equation.

A simple example: Solving the heat equation

Let's consider a simple one-dimensional heat equation problem. Suppose you have a rod with ends at x = 0 and x = L. You want to determine the temperature distribution along this rod over time.

We assume the following initial and marginal conditions:

  • Initial condition: The initial temperature distribution is given by u(x, 0) = f(x).
  • Boundary conditions: The ends of the rod are placed at zero degrees, which means u(0, t) = 0 and u(L, t) = 0.

Solving with separation of variables

One way to solve this PDE is to use separation of variables. This technique consists in assuming that the solution u(x, t) can be written as the product of two functions, one of which depends only on x and the other only on t:

u(x, t) = X(x)T(t)

Substitute this product into the heat equation:

X(x) ∂T/∂t = α T(t) ∂²X/∂x²

After dividing both sides by α X(x) T(t), separating the variables gives you:

(1/T) ∂T/∂t = α (1/X) ∂²X/∂x² = -λ

Here, λ is a dissociation constant. This leads to two ordinary differential equations:

∂T/∂t = -λ T(t) ∂²X/∂x² = (-λ/α) X(x)

The solutions to these ODEs can be found using standard methods, resulting in:

T(t) = e^(-λt) X(x) = A sin(√(λ/α) x) + B cos(√(λ/α) x)

Apply the marginal conditions on X(x): X(0) = 0 and X(L) = 0. The marginal condition X(0) = 0 implies that the condition B = 0 gives X(L) = 0 sin(√(λ/α)L) = 0, which means that:

√(λ/α) L = nπ, n = 1, 2, 3, ...

This results in discrete values for λ:

λ_n = (n²π²α) / L², n = 1, 2, 3, ...

The general solution is the sum of these solutions:

u(x, t) = Σ C_n e^(-λ_nt) sin(nπx/L)

where C_n are constants determined by the initial condition. Each term in this series represents a harmonic mode, and their corresponding coefficients are found through Fourier series analysis of the initial condition f(x).

Visual representation of harmonic modes

It is useful to look at the various harmonic modes. Below are the first few sinusoidal modes of a rod of length L:

Each curve with a different color represents a different harmonic mode. The red curve is the first mode, the green is the second mode, and the blue is the third mode.

Practical applications of the heat equation

The heat equation applies to many areas beyond simple heat flow. For example:

  • In engineering, it is used to model heat conduction in various materials and environments.
  • In financial mathematics, some form of the heat equation appears in option pricing models.
  • In biology, the heat equation helps understand processes such as nutrient distribution in tissues.

Expansion of the heat equation

The simple heat equation can also be adapted to more complex systems. For example:

  • In higher dimensions, such as 2D or 3D spaces, the heat equation involves additional spatial variables.
  • Non-linear variants consider scenarios where thermal properties change depending on temperature.
  • The introduction of source terms allows scenarios of internal heat generation to be modelled.

Closing thoughts

The heat equation is an important concept in mathematics and physics because it is widely applicable to a variety of fields. Through understanding its structure and solutions, one can gain insights about how heat and similar dispersive processes behave across time and space. These insights are not only applicable in academic settings, but also provide tools for solving real-world problems involving heat transfer and related phenomena.

Further study often involves exploring numerical methods for solving the heat equation, given that analytical solutions may be challenging or impossible for complex boundary conditions and geometries.

Ultimately, mastery of the heat equation establishes the basis for further exploration in the mathematical modeling of physical systems.


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Heat Equation

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