Abstract

A non-orthogonal predefined exponential basis set is used to handle half-bounded domains in multi domain spectral method (MDSM). This approach works extremely well for real-valued semi-infinite differential problems. It spans simultaneously wide range of exponential decay rates with multi scaling and does not suffer from zero crossing. These two conditions are necessary for many physical problems. For comparison, the method is used to solve different problems and compared with analytical and published results. The comparison exhibits the strengths and accuracy of the presented basis set.

Highlights

  • With the growing complexities of the numerically studied problems in natural and applied sciences, spectral method (SM) starts gaining more attention mainly because of its high level of analyticity

  • Multi domain spectral method (MDSM) is developed where the computational window is divided into homogeneous domains where the discontinuities lie at the boundaries

  • The first one is to approximate a wide range of exponentially decaying functions using the same basis sets, while the second is to obtain the quantized energy states in semiconductor quantum well (QW)

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Summary

Introduction

With the growing complexities of the numerically studied problems in natural and applied sciences, spectral method (SM) starts gaining more attention mainly because of its high level of analyticity. This is resulted from its meshfree nature, which reduces the computational memory and time requirements where a major part of the problem is solved analytically. Spectral methods work very well for homogeneous and smooth computational windows It suffers from the Gibbs phenomenon if the structural function of the studied problem is not analytical. The Gibbs phenomenon is generally a peculiarity of any functional approximation at simple discontinuity To avoid this problem, multi domain spectral method (MDSM) is developed where the computational window is divided into homogeneous domains where the discontinuities lie at the boundaries. These are joined by applying the proper boundary conditions between domains [1,2,3,4]

The Exponential Basis Set
Coordinate Transformation
Comparisons
Approximating Wide Range of Exponentially Decaying Functions
Single Semiconductor QW without Biasing Field
Conclusions

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