Lecture

Solving the Quintic: Dominant Balance Analysis

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In course

ME-201: Continuum mechanics

Continuum conservation laws (e.g. mass, momentum and energy) will be introduced. Mathematical tools, including basic algebra and calculus of vectors and Cartesian tensors will be taught. Stress and deformation tensors will be applied to examples drawn from ...

Description

This lecture delves into the dominant balance analysis of solving the quintic polynomial, exploring the non-dimensionalization process and the identification of distinct limits. By using rigorous expansion methods and series expansions, the coefficients in the expansion are calculated order-by-order. The lecture demonstrates the process of balancing terms and evaluating the accuracy of approximate solutions. The analysis reveals the behavior of roots as the parameter varies, highlighting the importance of symbolic expressions in understanding the solution's behavior. The lecture concludes with insights into the significance of dominant balance analysis in obtaining a deeper understanding of complex problems.

Instructor

John Martin Kolinski

Dr. Kolinski studied Applied Mathematics (Sc.M.) and Applied Physics (Ph.D.) at Harvard University, completing a PhD under the supervision of L. Mahadevan and Shmuel Rubinstein on the role of air in droplet impact. John did his post-doc at the Hebrew University of Jerusalem in Israel supported by the Fulbright post-doctoral fellowship. At HUJI, he worked on interfacial instabilities in soft matter in the labs of Eran Sharon and Jay Fineberg. John continues his research into interfacial mechanics at EPFL in EMSI, his newly founded laboratory for the study of Engineering Mechanics of Soft Interfaces.

Official source

Ontological neighbourhood

Mathematics

Analysis: Complex analysis, Numerical analysis

Algebra: Equations

Related concepts (32)

Series expansion

In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division). The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.

Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

Quintic function

In mathematics, a quintic function is a function of the form where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function.

Degree of a polynomial

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see Order of a polynomial (disambiguation)).

Polynomial

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.

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