The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

Chen, P.Y.P.

doi:10.4236/am.2016.79083

Cited by 8 publications

(9 citation statements)

References 25 publications

(38 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The numerical approach is based on the reduction of Eq. ( 1) into a set of simultaneous first-order ordinary differential equations (ODEs) by the Lanczos-Chebyshev pseudospectral (LCP) method [7,8], and the set of simultaneous ordinary differential equations (ODEs) is solved by a stable forward marching procedure. The temporal local time, t, is mapped into a numerical window [À1, 1].…”

Section: Numerical Solution Methodsmentioning

confidence: 99%

“…The temporal local time, t, is mapped into a numerical window [À1, 1]. The solution is written as an economized power series [7,8]:…”

Section: Numerical Solution Methodsmentioning

confidence: 99%

“…Chose N -1 collocation points, x i , within the interval [À1,1], that are the roots of the Chebyshev polynomial T N-2 (x) [7,8]:…”

Section: Numerical Solution Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Wave Propagation Theory Denies the Big Bang

Chen¹

2022

Astronomy and Planetary Science - From Cryovolcanism to Black Holes and Galactic Evolution

View full text Add to dashboard Cite

Problems related to Big Bang because of the Doppler interpretation of cosmological redshift have not been resolved up to recent years. The “tired light” theory proposes an energy loss model that has its own limitations. Chen in 2020 and 2021 proposed to treat light propagation through the space just as a field problem involving electromagnetic waves and governed by the well-known nonlinear Schrödinger (NLS) equation. The space is not a vacuum and is sparsely populated with matters. Electromagnetic waves traveling through the space will undergo changes as predicted by the NLS equation involving a linear dispersion and a nonlinear self-phase focusing terms. Using the cosmological principle, the coefficients associated with these terms could be constants but extremely small in value. Special numerical methods have been developed and could be used to find both bright and dark soliton-like solutions for the NLS equation that are stable and could travel through the extremely long distance involved. These solutions clearly show the redshift is linearly proportional to distance traveled for both bright and dark solitons. The conclusion is that redshift (and blue shift) is an innate nature of light traveling through the space.

show abstract

Section: Numerical Solution Methodsmentioning

confidence: 99%

“…The temporal local time, t, is mapped into a numerical window [À1, 1]. The solution is written as an economized power series [7,8]:…”

Section: Numerical Solution Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Wave Propagation Theory Denies the Big Bang

Chen¹

2022

Astronomy and Planetary Science - From Cryovolcanism to Black Holes and Galactic Evolution

View full text Add to dashboard Cite

show abstract

“…We have used the Lanczos-Chebyshev pseudospectral reduction method [3] [4] to convert Equation (3) into a set of ordinary differential equations (ODE). Because the emission is a soliton pulse, we need to subdivide the computational t-domain into N divisions.…”

Section: The Numerical Solution Methodsmentioning

confidence: 99%

A Mathematical Model for Redshift

Chen¹

2020

Self Cite

View full text Add to dashboard Cite

We have used model scaling so that the propagation of light through space could be studied using the well-known nonlinear Schrödinger equation. We have developed a set of numerical procedures to obtain a stable propagating wave so that it could be used to find out how wavelength could increase with distance travelled. We have found that broadening of wavelength, expressed as redshift, is proportional to distance, a fact that is in agreement with many physical observations by astronomers. There are other reasons for redshifts that could be additional to the transmission redshift, resulting in the deviation from the linear relationship as often observed. Our model shows that redshift needs not be the result of an expanding space that is a long standing view held by many astrophysicists. Any theory about the universe, if bases on an expanding space as physical fact, is open to question.

show abstract

“…The above equation could be solved by the Lanczos-Chebyshev pseudospectral method [8], seeking solutions in the form:…”

Section: Electromagnetic Waves Transmissionmentioning

confidence: 99%

An Energy Balance Simulation of the Universe

Chen¹

2019

View full text Add to dashboard Cite

We have developed an energy balance equation for the universe. The two system parameters involved in the equation could be "fine-tuned" so that the predicted temperature histories all lead to what is observed in the present cosmic microwave background. We have shown that various combinations of these two parameters are possible; in particular, the present background temperature needs not be the remnant of a very hot temperature in the far distance past. We also solved for the propagation of vortex solitons in optical fibres as contrasting examples to show how electromagnetic wave could be transmitted in a particular waveform under strictly controlled conditions. To avoid singularity, all vortexes have a black centre. We conclude that while numerical techniques can be used to account for an infinite quantity, it is unlikely that such a quantity could exist in reality.

show abstract

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

Cited by 8 publications

References 25 publications

Wave Propagation Theory Denies the Big Bang

Wave Propagation Theory Denies the Big Bang

A Mathematical Model for Redshift

An Energy Balance Simulation of the Universe

Contact Info

Product

Resources

About