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Related Concept Videos

Partial Fractions01:28

Partial Fractions

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A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
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Fundamental Theorem of Calculus II01:29

Fundamental Theorem of Calculus II

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In calculus, the computation of the area under a continuous curve has been fundamentally simplified by applying the Fundamental Theorem of Calculus, Part 2. Rather than relying on the limiting process of summing infinitely many infinitesimal rectangles, this theorem permits direct evaluation using antiderivatives, thereby streamlining the process of definite integration.The Fundamental Theorem of Calculus, Part 2, states that if a function f(x) is continuous on a closed interval [a, b], then...
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Integration of Rational Functions Using Partial Fractions01:29

Integration of Rational Functions Using Partial Fractions

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Rational functions are expressions written as the ratio of two polynomials, and their integrals are evaluated by simplifying the integrand into manageable parts. These functions are classified as proper or improper based on the degrees of the numerator and denominator.A rational function is proper when the degree of the numerator is less than the degree of the denominator. In this case, partial fraction decomposition is used to rewrite the function as a sum of simpler rational terms. The...
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The Quotient Rule01:30

The Quotient Rule

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The quotient rule is a fundamental differentiation technique in calculus used to differentiate functions expressed as a ratio of two differentiable functions. Given a function of the form:Where g(x) and h(x) are both differentiable and h(x) ≠ 0, the derivative of f(x) is given by:Example:The quotient rule is beneficial when differentiating rational functions, trigonometric ratios, and exponential functions. For example, given:applying the quotient rule,This rule is essential in solving...
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Fundamental Theorem of Calculus I: Problem Solving01:22

Fundamental Theorem of Calculus I: Problem Solving

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In many engineering and environmental applications, accumulated quantities are determined from rates that vary over time. A common example arises in water management, where a supply system pumps water into a storage tank at a rate that changes with time. Accurately determining how much water has entered the tank over a given period is essential for maintaining proper pressure, scheduling operations, and ensuring system safety.The flow rate of water into the tank is described by a time-dependent...
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Integrals of Powers of Secant and Tangent01:18

Integrals of Powers of Secant and Tangent

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Integrals involving powers of tangent and secant are commonly evaluated using substitution, with the strategy determined by the parity of the exponents. The method relies on pairing part of the integrand with the derivative of a suitable trigonometric function and rewriting the remaining factors using trigonometric identities.When the power of secant is even, tangent is chosen as the substitution variable. Since the derivative of tangent is secant squared, a factor of sec⁡2x can be...
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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180&#176; Curved Artery Test Section
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TEMPERED FRACTIONAL CALCULUS.

Mark M Meerschaert1, Farzad Sabzikar2, Jinghua Chen3

  • 1Department of Statistics and Probability, Michigan State University, East Lansing MI 48823 mcubed@stt.msu.edu URL: http://www.stt.msu.edu/users/mcubed/

Journal of Computational Physics
|June 19, 2015
PubMed
Summary
This summary is machine-generated.

Tempered fractional calculus extends fractional calculus with exponential tempering, modeling phenomena like financial data and wind speed. This approach offers new tools for diffusion equations and time series analysis.

Keywords:
Fractional calculusanomalous diffusionrandom walk

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Area of Science:

  • Mathematics
  • Physics
  • Finance
  • Geophysics
  • Time Series Analysis

Background:

  • Fractional derivatives and integrals are defined by power-law convolutions.
  • Tempered fractional calculus introduces exponential factors, modifying these power laws.
  • This modification is crucial for modeling real-world phenomena with tempered distributions.

Purpose of the Study:

  • To explore the mathematical framework of tempered fractional calculus.
  • To demonstrate its applications in diffusion equations, finance, geophysics, and time series analysis.
  • To introduce tempered fractional Brownian motion and tempered fractional Gaussian noise as novel stochastic models.

Main Methods:

  • Utilizing the concept of tempered fractional derivatives and integrals.
  • Analyzing random walk models with exponentially tempered power law jump distributions.
  • Developing tempered difference methods for solving tempered fractional diffusion equations.
  • Investigating the properties of tempered fractional Brownian motion and its increments.

Main Results:

  • Tempered fractional diffusion equations arise as limits of tempered random walks.
  • Limiting probability densities exhibit semi-heavy tails, relevant to financial modeling.
  • Tempered fractional time derivatives are useful in geophysics.
  • Tempered fractional Brownian motion can display semi-long range dependence.
  • Tempered fractional Gaussian noise provides a new stochastic model for wind speed.
  • Tempered differences offer a basis for numerical solutions and time series correlation models.

Conclusions:

  • Tempered fractional calculus provides a powerful generalization with broad applicability.
  • It offers new mathematical models for complex systems in science and finance.
  • The study highlights the utility of tempered fractional calculus in diverse fields, from geophysics to time series analysis.