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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
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Smart speakers process voice commands by modeling audio inputs as piecewise functions and analyzing them through integration against trigonometric functions, such as cosine. This mathematical approach is fundamental in signal processing, where complex sound waves are decomposed into simpler frequency components.Consider a definite integral involving a piecewise function multiplied by a cosine function. Because the function is defined differently over separate intervals, the integral is split...
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Differential Equations: Problem Solving01:21

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When analyzing the motion of falling objects, it is essential to consider not only the force of gravity but also the opposing force of air resistance. A practical example involves releasing a heavy test weight during a safety check on a ship. As the weight falls from rest, gravity accelerates it downward while air resistance exerts an upward force that increases with velocity. This dynamic interplay of forces is well described by differential equations, which provide a mathematical framework...
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The shape of a suspension bridge cable hanging under its own weight is described by a catenary curve, which is modeled using the hyperbolic cosine function. This mathematical model accurately captures the balance between gravity and tension acting along the cable. When a particular vertical position on the cable is known, the corresponding horizontal position can be determined using the inverse hyperbolic cosine function, allowing for a detailed analysis of the cable's geometry.Inverse...
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Trigonometric Equations01:30

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Trigonometric equations involve one or more trigonometric functions and arise frequently in mathematical modeling. These equations may be either identities, which are valid for all values of the variable, or conditional equations, which hold true only for specific values. The process of solving trigonometric equations typically involves both algebraic techniques and the use of fundamental properties of trigonometric functions.Some trigonometric equations resemble standard algebraic forms and...
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Separable Differential Equations01:20

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A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...
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Related Experiment Video

Updated: Apr 19, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Numerical algorithm based on Haar-Sinc collocation method for solving the hyperbolic PDEs.

A Pirkhedri1, H H S Javadi2, H R Navidi2

  • 1Department of Computer Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran.

Thescientificworldjournal
|December 9, 2014
PubMed
Summary
This summary is machine-generated.

This study introduces the Haar-Sinc collocation method for solving hyperbolic partial telegraph equations. The method offers exponential convergence and high computational speed, confirmed by four examples.

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Area of Science:

  • Numerical analysis
  • Computational mathematics

Background:

  • Hyperbolic partial telegraph equations present significant challenges in numerical solutions.
  • Existing methods may lack efficiency or high convergence rates.

Purpose of the Study:

  • To introduce and evaluate the Haar-Sinc collocation method for solving hyperbolic partial telegraph equations.
  • To demonstrate the method's efficiency, precision, and performance.

Main Methods:

  • The Haar-Sinc collocation method combines Sinc functions in space and Haar functions in time.
  • The method transforms the partial differential equation into a system of linear algebraic equations.
  • Unknown coefficients are determined by expanding approximations using Sinc and Haar functions.

Main Results:

  • The Haar-Sinc method achieves exponential convergence rates.
  • The use of Haar operational matrices ensures high computational speed.
  • The method's effectiveness was validated through four illustrative examples.

Conclusions:

  • The Haar-Sinc collocation method is an efficient and precise technique for solving hyperbolic partial telegraph equations.
  • The proposed method offers a favorable balance of accuracy and computational performance.