Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Mathematics - iii (Differential Calculus) - Vol 2
Explore and master the fundamentals of Mathematics - iii (Differential Calculus) - Vol 2
You've not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.Chapter 1
Formation of Partial Differential Equations
Partial Differential Equations (PDEs) are essential for modeling phenomena in various fields, involving equations with partial derivatives of functions of multiple variables. The chapter discusses methods for forming PDEs by eliminating arbitrary constants and functions, providing systematic techniques and examples for both methods. Understanding these formation processes is crucial for solving PDEs and applying them in real-world scenarios.
Chapter 2
Classification of PDEs (Elliptic, Parabolic, Hyperbolic)
Partial Differential Equations (PDEs) are crucial in modeling physical phenomena and are categorized into elliptic, parabolic, and hyperbolic types based on their coefficients and discriminant. The classification relies on the discriminant formula Δ = B² - 4AC, leading to different behaviors and solution methods. Understanding PDE types aids in determining appropriate numerical approaches and initial or boundary conditions necessary for solving complex problems.
Chapter 3
Linear and Non-linear PDEs
Partial Differential Equations (PDEs) are pivotal in modeling various physical phenomena involving multiple variables and partial derivatives. The chapter distinguishes between linear and non-linear PDEs, classifies them into parabolic, hyperbolic, and elliptic types, and discusses their characteristics and implications for solving real-world problems. Understanding these classifications is essential for applying appropriate mathematical methods in various scientific fields.
Chapter 4
First-Order PDEs
First-order partial differential equations (PDEs) are essential in the mathematical modeling of various physical phenomena, capturing first derivatives with respect to multiple independent variables. The chapter explores the formation, solutions, and classifications of first-order PDEs, detailing methods such as Lagrange's and Charpit's approaches for linear and non-linear equations. A clear understanding of types of solutions—complete, particular, singular, and general—enables better categorization and application in complex problem-solving scenarios.
Chapter 5
Lagrange’s Linear Equation
Lagrange's Linear Equation is a crucial method for solving first-order partial differential equations, showcasing a structured approach in mathematical modeling. The technique involves transforming complex PDEs into simpler ordinary differential equations through characteristic equations. The chapter illustrates various methods, including the formulation of characteristic equations, integration steps, and providing general solutions. Examples clarify the application of Lagrange’s method in diverse scenarios, highlighting its effectiveness when the coefficients of the equations are known functions.
Chapter 6
Charpit’s Method
Charpit's Method is a systematic approach designed to solve first-order non-linear partial differential equations (PDEs), converting them into a system of ordinary differential equations (ODEs). The method facilitates the finding of complete integrals by utilizing auxiliary equations derived from the original PDEs. It proves especially useful for non-linear equations where traditional methods might not apply effectively.
Chapter 7
Method of Separation of Variables
The Method of Separation of Variables is an essential technique for solving linear partial differential equations (PDEs) by transforming them into simpler ordinary differential equations (ODEs). This method relies on the assumption that solutions can be expressed as a product of functions, each depending on a single variable. It requires appropriate boundary conditions and can effectively address problems such as the heat and wave equations through Fourier series and superposition principles.
Chapter 8
Homogeneous Linear PDEs with Constant Coefficients
Homogeneous Linear PDEs with Constant Coefficients describe equations critical in various scientific fields including engineering. This unit focuses on the definitions, general forms, and solving methods for these equations, particularly using the Operator method to develop solutions through auxiliary equations. The chapter emphasizes the importance of root types in determining the solution forms and stresses the systematic nature of the operator method for solving homogeneous equations.
Chapter 9
Non-Homogeneous Linear PDEs
Non-Homogeneous Linear Partial Differential Equations (PDEs) feature a non-zero function on their right-hand side, essential for modeling physical phenomena under external forces. The general solution combines the complementary function (CF) of the homogeneous equation with a particular integral (PI). Various solving techniques include the operator method, method of undetermined coefficients, and variation of parameters, which are crucial for tackling advanced engineering problems.
Chapter 10
Solution of PDEs by Direct Integration
Partial Differential Equations (PDEs) serve as essential tools in modeling various physical phenomena through the application of direct integration techniques. The method focuses on solving first-order PDEs by integrating partial derivatives step-by-step, enabling a clearer approach to finding solutions for simpler equations. Critical insights on the importance of arbitrary functions during integration highlight the straightforward nature of this technique, setting the foundation for more advanced PDE methods.
Chapter 11
One-Dimensional Wave Equation
The one-dimensional wave equation describes the propagation of wave phenomena in a medium along a single spatial dimension. Deriving from Newton's laws, this second-order linear partial differential equation showcases critical aspects including boundary and initial conditions necessary for unique solutions. D'Alembert's formula provides a general solution, while the method of separation of variables aids in solving complex problems involving fixed and free boundary conditions.
Chapter 12
One-Dimensional Heat Equation
The One-Dimensional Heat Equation is a critical model for understanding heat diffusion in materials. It highlights the importance of boundary and initial conditions in deriving solutions through methods such as separation of variables. The equation also finds applications in various fields, from engineering to financial mathematics, underscoring its broad relevance.
Chapter 13
Two-Dimensional Laplace Equation
The two-dimensional Laplace equation is a critical second-order partial differential equation reflecting steady-state phenomena across various fields such as physics and engineering. Central to the equation are its properties, boundary value problems, and methods for solution, particularly through the separation of variables. Analytical techniques prevail, while numerical methods provide alternatives for complex geometries and scenarios where analytical solutions are intractable.
Chapter 14
D’Alembert’s Solution of Wave Equation
D’Alembert’s solution is an analytical method for solving the one-dimensional wave equation, providing insight into wave propagation in various physical systems. The method involves the formulation of the wave equation, the derivation of its solution, and an application of initial conditions to derive a complete solution. Key aspects such as linear superposition and non-dispersive wave properties are highlighted.
Chapter 15
Fourier Series Solutions to PDEs
The Fourier Series technique is essential in solving Partial Differential Equations (PDEs), particularly for scenarios involving heat flow, vibrations, and potential theory. This method transforms PDEs into solvable ordinary differential equations (ODEs) using an infinite series of sine and cosine functions. By leveraging orthogonality and convergence under Dirichlet conditions, the Fourier Series allows for practical applications in various engineering and physics problems.
Chapter 16
Boundary and Initial Conditions
Boundary and initial conditions play a crucial role in defining unique and stable solutions of Partial Differential Equations (PDEs). Different types of PDEs—elliptic, parabolic, and hyperbolic—require specific conditions based on the physical context. Furthermore, understanding how to classify and apply these conditions is vital for solving real-world problems efficiently using PDEs.
Chapter 17
Applications in Engineering Problems
Partial Differential Equations (PDEs) are essential in modeling and analyzing various engineering phenomena where changes occur over time and space. The chapter highlights the different types of PDEs, including elliptic, parabolic, and hyperbolic forms, and their significant applications in fields such as heat conduction, fluid dynamics, wave propagation, and electromagnetic field analysis. Various solution methods, both analytical and numerical, are also discussed to equip engineers with tools for addressing complex engineering problems.
Chapter 18
Eigenfunction Expansion Method
The Eigenfunction Expansion Method provides a systematic approach for solving linear partial differential equations (PDEs) by utilizing the properties of eigenfunctions from Sturm–Liouville problems. It allows the representation of solutions as infinite series, connecting concepts from linear algebra, differential equations, and Fourier analysis. This method is particularly effective for boundary value problems, enabling efficient solution derivation under various conditions.
Chapter 19
Use of Laplace Transforms in Solving PDEs
Laplace Transforms provide a powerful method for solving linear partial differential equations (PDEs), particularly in scenarios involving time-dependent processes by transforming them into ordinary differential equations (ODEs). The method simplifies the resolution of complex PDEs, allowing for efficient retrieval of solutions through inverse transforms. This technique is instrumental across various applications in physics and engineering, including heat conduction, wave propagation, and fluid dynamics.
Chapter 20
Numerical Methods for PDEs (basic overview)
Numerical methods are crucial for approximating solutions to Partial Differential Equations (PDEs) that model various phenomena in engineering and science. Key numerical methods include Finite Difference Method (FDM), Finite Element Method (FEM), and Finite Volume Method (FVM), each offering unique advantages based on problem characteristics. The choice of method depends on factors such as the type of PDE, geometrical complexity, and conservation requirements, guiding effective simulation in real-world applications.