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Related Concept Videos

Partial Fractions01:28

Partial Fractions

389
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...
389
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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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...
268
Integration of Rational Functions Using Partial Fractions01:29

Integration of Rational Functions Using Partial Fractions

362
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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Fundamental Theorem of Calculus I: Problem Solving01:22

Fundamental Theorem of Calculus I: Problem Solving

233
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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Inverse z-Transform by Partial Fraction Expansion01:20

Inverse z-Transform by Partial Fraction Expansion

821
The inverse z-transform is a crucial technique for converting a function from its z-domain representation back to the time domain. One effective method for finding the inverse z-transform is the Partial Fraction Method, which involves decomposing a function into simpler fractions with distinct coefficients. These fractions correspond to known z-transform pairs, facilitating the inverse transformation process.
To begin the process, the poles of the function are identified and the function is...
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FRACTIONAL INTEGRATION TOOLBOX.

Toma M Marinov1, Nelson Ramirez2, Fidel Santamaria1

  • 1Department of Biology University of Texas at San Antonio San Antonio, 78249 TX, USA.

Fractional Calculus & Applied Analysis
|May 10, 2014
PubMed
Summary
This summary is machine-generated.

This study introduces a new Fractional Integration Toolbox (FIT) for efficient numerical fractional calculus on large datasets. FIT enables parallel processing on CPUs and GPUs, overcoming limitations of existing tools for complex fractional calculus problems.

Keywords:
Riemann-Liouville fractional derivativeRiemann-Liouville fractional integralfractional calculusfractional reaction-diffusion equationnumerical quadrature

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Area of Science:

  • Applied Mathematics
  • Computational Science
  • Control Engineering

Background:

  • Fractional calculus problems frequently necessitate numerical fractional integration and differentiation of extensive datasets.
  • Existing fractional control toolboxes lack efficiency in handling numerical integration for multiple large data sequences.

Purpose of the Study:

  • To develop a specialized toolbox for efficient numerical fractional integration and differentiation.
  • To address the computational bottleneck in processing large datasets for fractional calculus applications.

Main Methods:

  • Development of a Fractional Integration Toolbox (FIT).
  • Implementation of Riemann-Liouville type fractional numerical integration/differentiation.
  • Design for parallelization on both CPU and GPU platforms.

Main Results:

  • FIT demonstrates efficient performance on large data sequences.
  • The toolbox successfully performs fractional numerical integration and differentiation.
  • Parallelization capabilities enhance computational speed and scalability.

Conclusions:

  • The Fractional Integration Toolbox (FIT) provides an efficient solution for numerical fractional calculus on large datasets.
  • FIT's parallel processing on CPU/GPU platforms overcomes limitations of previous tools.
  • This toolbox is valuable for researchers and engineers working with fractional calculus models.