--- title: Partial Differential Equations description: Partial differential equations involve functions of multiple variables and their partial derivatives. Learners will study classical equations like the heat, wave, and Laplace equations, using methods such as separation of variables to solve them. category: mathematics subcategory: differential-equations difficulty: beginner, intermediate, advanced url: /subject/partial-differential-equations --- # Partial Differential Equations Partial differential equations involve functions of multiple variables and their partial derivatives. Learners will study classical equations like the heat, wave, and Laplace equations, using methods such as separation of variables to solve them. ## Available Resources 6 Books • 5 Courses • 2 Websites ## Websites ### 1. nLab - Partial Differential Equation Explore partial differential equations with nLab! Dive into theory & applications on this comprehensive resource. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/partial+differential+equation **Tags:** partial-differential-equations, mathematics ### 2. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Principles of Applied Mathematics (MIT 18.311) **Author:** Rodolfo Rosales Continuum applied mathematics through examples such as traffic flow, fluids and granular flows: conservation laws, kinematic waves, characteristics and shocks, diffusion, finite differences and stability, and Fourier and spectral methods. Provides 41 lecture note files and problem sets for modeling wave and diffusion problems. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/ **Tags:** partial-differential-equations, conservation-laws, shock-waves, diffusion, numerical-methods ### 2. Differential Analysis (MIT 18.155) **Author:** Richard Melrose First semester of graduate differential analysis: fundamental solutions for elliptic, hyperbolic and parabolic operators, the method of characteristics, Lebesgue integration, distributions, homogeneous distributions, the Fourier transform and asymptotic methods. Provides 18 lecture note files and problem sets with solutions, preparing readers for modern PDE theory. **Difficulty:** Advanced | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/ **Tags:** distribution-theory, partial-differential-equations, fourier-transform, lebesgue-integration, fundamental-solutions ### 3. Mathematical Methods for Engineers II (MIT 18.086) **Author:** Gilbert Strang Continuation of 18.085 covering numerical methods for initial-value problems and partial differential equations, finite differences, network flows and optimization. Includes 29 lecture videos, problem sets with solutions and example projects, for learners ready to build and analyze numerical solvers. **Difficulty:** Advanced | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-086-mathematical-methods-for-engineers-ii-spring-2006/ **Tags:** finite-difference-methods, initial-value-problems, partial-differential-equations, numerical-linear-algebra, optimization ### 4. Computational Science and Engineering I (MIT 18.085) **Author:** Gilbert Strang Applied linear algebra for networks, structures and estimation, followed by equilibrium differential equations, Laplace's equation, boundary-value problems, calculus of variations, Fourier series and the discrete Fourier transform. Includes 50 lecture videos, problem sets and exams with solutions, and programming assignments. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-fall-2008/ **Tags:** applied-linear-algebra, boundary-value-problems, calculus-of-variations, fourier-analysis, differential-equations ### 5. Differential Equations for Engineers This course is all about differential equations and covers both theory and applications. In the first five weeks, students will learn about ordinary differential equations, while the sixth week is an introduction to partial differential equations. The course includes 56 concise lecture videos, with a few problems to solve after each lecture. After each major topic, there is a short practice quiz. At the end of each week, there is an assessed quiz. Solutions to the problems and practice quizzes can be found in the instructor-provided lecture notes. Download the lecture notes from the link https://www.math.hkust.edu.hk/~machas/differential-equations-for-engineers.pdf Watch the promotional video from the link https://youtu.be/eSty7oo09ZI **Difficulty:** Beginner | **Language:** English | **Duration:** 6 weeks of study, 4 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/differential-equations-engineers **Tags:** courses, mathematics-statistics, calculus ## Books ### 1. Green's Functions and Boundary Value Problems **Author:** Ivar Stakgold, Michael J. Holst The graduate reference on the subject. The canonical treatment of the Green's-function half of this topic, at the level where BVPs become operator theory. Stakgold and Holst develop distributions, one-dimensional boundary value problems, Hilbert space and operator theory, spectral theory of differential operators, nonlinear problems, and a long chapter on finite-element approximation and iterative solvers. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0470609702?tag=edmonddante07-20 **Tags:** greens-functions, boundary-value-problems, operator-theory, spectral-theory, distributions ### 2. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems **Author:** Richard Haberman The standard course text for the engineering BVP sequence. Haberman derives the heat, wave and Laplace equations physically, then works separation of variables, Sturm-Liouville theory, eigenfunction expansions, Green's functions and finite differences with unusually patient exposition. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0321797051?tag=edmonddante07-20 **Tags:** partial-differential-equations, fourier-series, separation-of-variables, sturm-liouville, boundary-value-problems ### 3. Finite Difference Methods for Ordinary and Partial Differential Equations **Author:** Randall J. LeVeque Unifies stability theory for ordinary and partial differential equations. Part one treats two-point boundary value problems, elliptic equations and iterative solvers for sparse systems; part two covers initial value problems and parabolic and hyperbolic PDEs, proving consistency, stability and convergence. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0898716292?tag=edmonddante07-20 **Tags:** finite-difference-methods, partial-differential-equations, stability-analysis, ordinary-differential-equations, sparse-linear-systems ### 4. Partial Differential Equations **Author:** Lawrence C. Evans A standard graduate textbook covering existence, uniqueness, and regularity theory for linear and nonlinear PDEs, including Sobolev spaces, weak solutions, and techniques for elliptic, parabolic, and hyperbolic equations. Prepares readers for research-level PDE analysis. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0821807242?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, differential-equations ### 5. Introduction to Partial Differential Equations **Author:** Peter J. Olver An undergraduate-level introduction covering the heat, wave, and Laplace equations through separation of variables, Fourier series, and Green's functions, building toward first- and second-order linear and nonlinear PDE theory with worked examples. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/3319347446?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, differential-equations ### 6. Partial Differential Equations for Scientists and Engineers **Author:** Stanley J. Farlow A problem-driven introduction showing how partial differential equations arise from physical models in diffusion, wave, and elliptic problems, then how to solve them with separation of variables, Fourier methods, and numerical and approximate techniques, aimed at applied practitioners. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/048667620X?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, differential-equations --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/partial-differential-equations