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Related Concept Videos

Difference Equation Solution using z-Transform01:24

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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.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
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In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
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A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a prescribed path. If the prescribed path is closed, the integrals reduce to a closed-line integral. The closed-contour integral of the vector field is referred to in terms of the circulation of the vector field around the closed path. A vector with zero circulation around every closed path is called a conservative field, while one with non-zero...
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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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The important convolution properties include width, area, differentiation, and integration properties.
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The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
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A new general integral transform for solving integral equations.

Hossein Jafari1,2,3,4

  • 1Department of Mathematics, University of Mazandaran, Babolsar, Iran.

Journal of Advanced Research
|September 6, 2021
PubMed
Summary
This summary is machine-generated.

A new general integral transform unifies existing Laplace-type transforms, simplifying differential and integral equations. This unified transform offers no inherent advantage over specialized ones but broadens their applicability.

Keywords:
26A3331B1044A10Fractional order integral equationsIntegral equationLaplace transformOrdinary differential equations

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Area of Science:

  • Mathematics
  • Applied Mathematics
  • Integral Transforms

Background:

  • Integral transforms are crucial for solving differential and integral equations by converting them into algebraic forms.
  • Numerous Laplace-transform-class integral transforms have been developed in recent decades, including Sumudu, Elzaki, and G-transforms.

Purpose of the Study:

  • To introduce a novel general integral transform within the Laplace transform family.
  • To investigate the properties of this new transform and compare its efficacy against existing transforms.
  • To demonstrate the transform's utility in solving various mathematical problems.

Main Methods:

  • Introduction of a new general integral transform.
  • Analysis of the transform's fundamental properties.
  • Comparative study with established integral transforms like Laplace, Sumudu, and Elzaki transforms.
  • Application of the new transform to solve higher-order initial value problems, integral equations, and fractional integral equations.

Main Results:

  • The newly introduced general integral transform encompasses several previously developed transforms as special cases.
  • It was demonstrated that the general transform covers Laplace, Elzaki, and Sumudu transforms for specific parameter choices.
  • The transform proved effective in solving ordinary differential equations, integral equations, and fractional integral equations.

Conclusions:

  • The general integral transform provides a unifying framework for various Laplace-type transforms.
  • While not offering a universal advantage, it simplifies the introduction of new transforms and broadens the scope of existing ones.
  • Its application in solving diverse mathematical problems, including fractional calculus, highlights its significance.