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


Partial differential equations, often abbreviated as PDEs, are differential equations that involve several independent variables and their partial derivatives. They are fundamental in the mathematical modeling of a variety of physical systems, including those that describe phenomena such as the behavior of heat, sound, fluid flow, and electromagnetic fields.

Understanding the basic concepts

At the core of partial differential equations is the concept of a "partial derivative." The partial derivative is similar to an ordinary derivative, but it treats all variables as constants except for one. Let's look at a simple function:

f(x, y) = x^2 + y^2

In this function, there are two variables x and y. While calculating the partial derivative with respect to a variable, say x, we consider y as a constant.

∂f/∂x = 2x

Similarly, the partial derivative of the function f with respect to y is:

∂f/∂y = 2y

Classification of PDEs

Partial differential equations can be classified into several types based on various criteria such as linearity, order, and degree. The most common types of PDEs found in undergraduate studies are:

1. Order of the PDE

The order of the PDE is determined by the highest order of the partial derivative present in the equation. For example, in the equation:

∂^2u/∂x^2 + ∂^2u/∂y^2 = 0

The highest order of the derivative is 2. Therefore, this is a second order PDE.

2. Linearity of PDE

A PDE is called linear if all the terms relating the dependent variable and its derivatives are linear. Consider the equation:

a(x, y) ∂u/∂x + b(x, y) ∂u/∂y = c(x, y)

This is a linear PDE because the dependent variable u and its derivatives appear in linear form.

If any of the terms is nonlinear then the PDE is nonlinear. For example:

(∂u/∂x)^2 = u * ∂u/∂y

This is a nonlinear PDE because of the squared derivative and product terms.

Examples of general PDEs

Let us look at some traditional types of partial differential equations that are frequently found in applications.

Heat equation

The heat equation models the distribution of heat (or temperature variation) over time in a given region. The one-dimensional form of the heat equation is:

∂u/∂t = α ∂^2u/∂x^2

Here, u(x, t) is the temperature at position x and time t, and α is a constant known as the thermal diffusivity.

Wave equation

The wave equation is important in fields such as acoustics, electromagnetics, and fluid dynamics. It describes the behavior of waves in space and time. In one dimension, it can be written as:

∂^2u/∂t^2 = c^2 ∂^2u/∂x^2

Here, c is the wave speed, and u(x, t) represents the wave displacement at point x and time t.

Laplace equation

The Laplace equation is fundamental to potential theory and is used to describe steady state heat conduction, electrostatics, and incompressible fluid flow. It is given as:

∂^2u/∂x^2 + ∂^2u/∂y^2 = 0

It presents a scenario in which there is no change over time; thus, it involves only spatial variables.

Solution techniques for P.D.E.

Solving partial differential equations can be quite complicated, especially for nonlinear PDEs. However, many methods have been developed to find solutions. We discuss some important techniques below.

Separation of variables

This is a powerful method used to reduce a PDE into one or more ordinary differential equations (ODEs). The basic idea is to assume that the solution can be written as a product of functions, each of which depends on a single variable. Consider the equation:

∂u/∂t = k ∂^2u/∂x^2

Assume a solution of the form u(x, t) = X(x)T(t) Substituting this into the PDE gives:

X(x) dT/dt = k T(t) d^2X/dx^2

Dividing both sides by X(x)T(t) gives two different ODEs:

dT/dt = -λT, and d^2X/dx^2 = - (λ/k) X

The value of λ is determined by the marginal conditions or constraints.

Characterization method

Some PDEs can be transformed into ordinary differential equations using characteristic curves. Consider a first-order PDE such as:

a(x, y) ∂u/∂x + b(x, y) ∂u/∂y = c(x, y)

By solving the system of ordinary differential equations:

dx/dt = a(x, y), dy/dt = b(x, y), du/dt = c(x, y)

This method converts the PDE into a set of ODEs with characteristic curves in the (x, y) plane.

Visual examples of P.D.E.

Consider the conventional heat equation:

∂u/∂t = α ∂^2u/∂x^2

This equation can be visualized as follows:

heat source

The circle represents an initially hot region whose temperature distribution changes with time.

Applications of PDE

The applications of PDEs are very wide and spread across various domains. Some of these applications are as follows:

1. Physics and engineering

PDEs are very important in modeling the physical world. The wave equation models the propagation of waves such as sound waves in air, water waves in the ocean, and vibrations in structures. The heat equation models the distribution of temperature in a given region over time.

2. Economics

In finance, PDEs are used to model option pricing, such as the famous Black–Scholes equation, which helps to value European options.

3. Biology

PDEs model biological processes such as population dynamics, where they help understand how population density changes over time and space.

Challenges in resolving P.D.E.

Although there are many ways to solve PDEs, these equations can be quite challenging due to their complexity. Here are some of the difficulties:

1. Nonlinearity

Nonlinear PDEs can exhibit complex behavior, including chaotic dynamics, making them difficult to solve analytically.

2. Boundary and initial conditions

Finding solutions in PDEs often requires appropriate boundary and initial conditions, which can sometimes be difficult to determine or satisfy.

3. Numerical stability

When PDEs are solved using numerical methods, ensuring the stability and accuracy of the solution can be a significant challenge.

Conclusion

Partial differential equations are an essential aspect of mathematics that has profound applications in a wide variety of fields. While challenging, understanding their formulation, classification, and solution techniques opens up a way to model and analyze the complexities of the world around us. With advances in computational power, the ability to numerically solve complex PDEs will continue to improve, expanding the horizons for research and practical applications.


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