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Related Concept Videos

Singularity Functions for Bending Moment01:18

Singularity Functions for Bending Moment

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Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented...
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Magnetostatic Boundary Conditions01:28

Magnetostatic Boundary Conditions

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An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...
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Consider an external electric field propagating through a homogeneous medium. When the electric field crosses the surface boundary of the medium, it undergoes a discontinuity. The electric field can be resolved into normal and tangential components. The amount by which the field changes at any boundary is given by the difference between the field components above and below the surface boundary.
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Accurately determining beam deflection and slope under various loading conditions in structural engineering is crucial for ensuring safety and structural integrity. Singularity functions offer a streamlined approach to analyzing beams, especially when multiple loading functions complicate the bending moment equation.
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A Coupled Experiment-finite Element Modeling Methodology for Assessing High Strain Rate Mechanical Response of Soft Biomaterials
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Graded mesh B-spline collocation method for two parameters singularly perturbed boundary value problems.

Fellek Sabir Andisso1, Gemechis File Duressa2

  • 1Department of Mathematics, Arba Minch University, Arba Minc, 21, Ethiopia.

Methodsx
|September 11, 2023
PubMed
Summary
This summary is machine-generated.

This study presents a stable and accurate numerical method using cubic B-splines on a graded mesh for singularly perturbed boundary value problems. The method effectively resolves boundary layers, offering superior efficiency compared to existing techniques.

Keywords:
B-Spline collocationBoundary layersCubic B-spline collocation methodGraded meshLayer adaptedSingularly perturbedparameters uniform convergent

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Differential Equations

Background:

  • Singularly perturbed boundary value problems often feature boundary layers, which pose significant challenges for standard numerical approximations.
  • Existing numerical methods struggle to accurately capture the behavior of solutions in the presence of these boundary layers.

Purpose of the Study:

  • To introduce and analyze a novel numerical treatment for a class of two-parameter singularly perturbed boundary value problems.
  • To address the limitations of conventional methods in accurately approximating solutions with boundary layer phenomena.

Main Methods:

  • A collocation method utilizing cubic B-splines is employed on a graded mesh specifically designed to resolve boundary layers.
  • The proposed approach simplifies the resulting system of equations to a tri-diagonal linear system.
  • Stability and parameter-uniform convergence of the method are rigorously examined.

Main Results:

  • The method demonstrates stability and parameter-uniform convergence, with numerical findings corroborating theoretical predictions.
  • Experimental results indicate a high degree of accuracy in approximating the solution, achieving a convergence rate of order two in the maximum norm.
  • The cubic B-spline collocation method on a graded mesh outperforms other existing methods, including those on Shishkin meshes.

Conclusions:

  • The proposed numerical method provides a stable and efficient solution for two-parameter singularly perturbed boundary value problems.
  • The use of cubic B-splines on a graded mesh is highly effective in capturing boundary layer behavior and achieving accurate approximations.
  • This method represents a significant improvement over existing techniques for this class of problems.