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Lecture# Ordinary Differential Equations: Error Analysis

Description

This lecture covers error analysis in ordinary differential equations, focusing on the convergence of numerical methods for approximating the Cauchy problem. It discusses truncation errors, local truncation errors, and proof for the forward Euler method. The lecture also explores the Lipschitz continuity and convergence criteria for numerical methods.

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In course

MATH-251(a): Numerical analysis

This course presents numerical methods for the solution of mathematical problems such as systems of linear and non-linear equations, functions approximation, integration and differentiation, and diffe

Instructors (2)

Related concepts (49)

Related lectures (37)

Euler method

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .

Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE).

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Numerical Methods: Euler and Crank-Nicolson

Covers Euler and Crank-Nicolson methods for solving differential equations.

Runge-Kutta Methods: Approximating Differential EquationsMATH-351: Advanced numerical analysis

Covers the stages of the explicit Runge-Kutta method for approximating y(t) with detailed explanations.

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Explores numerical methods for solving ODE systems, stability regions, and absolute stability importance.

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Covers advanced topics in numerical analysis, focusing on techniques for solving complex mathematical problems.

High Order Methods: Space DiscretisationMATH-351: Advanced numerical analysis

Covers high order methods for space discretisation in linear differential systems.