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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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Types of Functions III01:28

Types of Functions III

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Logarithmic and piecewise functions play central roles in mathematical modeling, particularly when capturing nonlinear or segmented behaviors in real-world phenomena. Although these functions differ fundamentally in structure and application, both serve to represent complex relationships in simplified mathematical terms.A logarithmic function is defined as the inverse of an exponential function, expressed as These functions grow quickly for small values of x but slow down as x increases,...
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Piecewise-Defined Functions01:28

Piecewise-Defined Functions

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Piecewise defined functions are mathematical models where different expressions define a function over distinct intervals of the domain. These functions are useful for representing systems with varying behaviors depending on input values.For example, the function:  uses a linear rule for inputs less than or equal to –1 and a quadratic rule for values greater than –1. Although it has two formulas, it still defines a single function.Another common type is the absolute value...
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Difference Equation Solution using z-Transform01:24

Difference Equation Solution using z-Transform

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The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
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Inverse z-Transform by Partial Fraction Expansion01:20

Inverse z-Transform by Partial Fraction Expansion

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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.
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Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

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The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
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Related Experiment Videos

Exp-function method for solving fractional partial differential equations.

Bin Zheng1

  • 1School of Science, Shandong University of Technology, Zibo, Shandong 255049, China. zhengbin2601@126.com

Thescientificworldjournal
|July 3, 2013
PubMed
Summary
This summary is machine-generated.

Researchers developed a new Exp-function method for fractional differential equations using a modified Riemann-Liouville derivative. This approach successfully found novel exact solutions for the space-time fractional Fokas and Sharma-Tasso-Olver equations.

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Fractional Calculus

Background:

  • Fractional partial differential equations (FPDEs) model complex phenomena.
  • Existing analytical methods face challenges with FPDEs.

Purpose of the Study:

  • To extend the Exp-function method for solving FPDEs.
  • To introduce a nonlinear fractional complex transformation.

Main Methods:

  • Modified Riemann-Liouville derivative for fractional calculus.
  • Exp-function method adapted for FPDEs.
  • Application to space-time fractional Fokas and Sharma-Tasso-Olver equations.

Main Results:

  • Successfully established new exact solutions for the studied FPDEs.
  • Demonstrated the validity and effectiveness of the extended Exp-function method.

Conclusions:

  • The extended Exp-function method is a powerful tool for solving FPDEs.
  • The method provides novel analytical solutions for nonlinear fractional models.