biharmonic function (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun

Phonetic Transcription

/bɪˈhɑːrmənɪk ˈfʌŋkʃən/

Meaning and Usage

A biharmonic function is a type of function in mathematics, specifically in the field of partial differential equations and mathematical physics. It is a function that satisfies the biharmonic equation, which is the Laplace equation applied twice. In simpler terms, a function ( u ) is called biharmonic if it is twice continuously differentiable and satisfies the equation ( \Delta^2 u = 0 ), where ( \Delta ) is the Laplace operator.

Frequency of Use: The term is primarily used in advanced mathematics, physics, and engineering, making it more prevalent in written academic contexts rather than in everyday spoken language.

Example Sentences

1. The mathematician demonstrated that the biharmonic function could be used to model the deflection of a thin plate under pressure.

2. Математик продемонстрировал, что бигамоническая функция может быть использована для моделирования прогиба тонкой пластины под давлением.

3. In fluid dynamics, biharmonic functions play a crucial role in describing potential flows.

4. В гидродинамике бига́рмонические функции играют ключевую роль в описании потенциальных потоков.

5. The solutions to the biharmonic equation are often related to various physical phenomena, such as heat conduction.

6. Решения бига́рмонического уравнения часто связаны с различными физическими явлениями, такими как теплопроводность.

Idiomatic Expressions

The term "biharmonic function" is not commonly found in idiomatic expressions, given its specialized nature in mathematics. However, here are few instances of how it may appear in contextual phrases relating to mathematics and functions:

1. When using the biharmonic function, one must be careful about boundary conditions.

2. При использовании бига́рмонической функции необходимо проявлять осторожность относительно граничных условий.

3. Many physical systems can be represented by biharmonic functions, showcasing their versatility.

4. Многие физические системы могут быть представлены бига́рмоническими функциями, демонстрируя их универсальность.

5. In theoretical physics, studying biharmonic functions can lead to deeper insights into fluid flows.

6. В теоретической физике изучение бига́рмонических функций может привести к более глубоким пониманиям потоков жидкости.

Etymology

The term "biharmonic" is derived from the prefix "bi-" meaning two and "harmonic," which is derived from the Greek word "harmonia," meaning harmony or fitting together. The "harmonic" reference indicates a relationship to harmonics or the second order of the Laplace equation.

Synonyms and Antonyms

Synonyms: - Twice harmonic function - Biharmonic equation solutions

Antonyms: - Non-harmonic function - Singular function

This detailed breakdown provides a comprehensive overview of the term "biharmonic function" including its application and relevance in mathematical discourse.