Nonlinear Ordinary Differential Equations

Hermann, Martin; Saravi, Masoud

doi:10.1007/978-81-322-2812-7

Cited by 18 publications

(10 citation statements)

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“…It is assumed that Peano's existence theorem and the uniqueness theorem of Picard and Lindelöf are fulfilled, and the Picard iteration converges; see (Hermann and Saravi, 2016). Above all, this means that left and right parts of a differential equation are continuous and bounded for all points of the argument.…”

Section: Mol and Non-linearitymentioning

confidence: 99%

“…However, here's to do with non-linear processes: The result of the calculation -e.g., a spatial electric field distribution for each FD step -depends in principle on "itself". According to (Hermann and Saravi, 2016), it is hypothesized that the equations have one and nonsingular solutions. This is finally confirmed by means of an iterative algorithm with a self-consistent, convergent solution.…”

Section: Introductionmentioning

confidence: 99%

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Optimized Method of Lines for non-linear waveguide problems

Spiller

2022

Preprint

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The Method of Lines is a semi-analytical versatile tool for the solution of partial differential equations. For the analysis of spatial complex linear waveguide structures, this method is combined with impedance/admittance and field transformation, as well as with finite differences. This paper extends this approach to the treatment of structures with non-linear dielectric materials. The non-linear generalized transmission line equations are derived. An iterative algorithm based on the impedance/admittance transformation with the field transformation obtains efficient and self-consistent solutions. The specific limiting factors for the algorithm and how to overcome them were investigated. A bidirectional, spatially, and temporally periodic energy exchange between the harmonics were found. For demonstration, a stripe waveguide with the non-linear core is considered. The Kerr non-linearity was investigated, though the general case is treatable. The approach can be used for any spatial structure, including, for instance, photonic crystal waveguides and metamaterials.

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Section: Mol and Non-linearitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimized Method of Lines for non-linear waveguide problems

Spiller

2022

Preprint

View full text Add to dashboard Cite

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“…However, it should be ensured that the NGTL -as a non-linear differential equation -has an existing and non-singular solution to the initial value problem. According to (Hermann and Saravi, 2016), the following hypothesis can be considered.…”

Section: Mol and Non-linearitymentioning

confidence: 99%

“…However, here's to do with non-linear processes: The result of the calculation -e.g., a spatial electric field distribution for each FD step -depends in principle on "itself." According to (Hermann and Saravi, 2016), it is hypothesized that the equations have one and non-singular solutions. This is finally confirmed by means of an iterative algorithm with a self-consistent, convergent solution.…”

Section: Introductionmentioning

confidence: 99%

Optimized method of lines for non-linear waveguide problems

Spiller

2022

Opt Quant Electron

View full text Add to dashboard Cite

The Method of Lines is a semi-analytical versatile tool for the solution of partial differential equations. For the analysis of spatial complex linear waveguide structures, this method is combined with impedance/admittance and field transformation, as well as with finite differences. This paper extends this approach to the treatment of structures with non-linear dielectric materials. The non-linear generalized transmission line equations are derived. An iterative algorithm based on the impedance/admittance transformation with the field transformation obtains efficient and self-consistent solutions. For demonstration, a non-linear stripe waveguide is considered. The Kerr non-linearity was investigated, though the general case is treatable. As a criterion for the correctness of the algorithm, a second harmonic generation as well as a bidirectional, spatially, and temporally energy exchange between the harmonics was examined. The specific limiting factors for the algorithm were explored. The approach can be used for any spatial structure, including, for instance, photonic crystal waveguides and metamaterials.

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“…Nonlinear ordinary differential equations play an essential role in scientific, technical, economic, and mathematical modeling. Applications of nonlinear ordinary differential equations include the calculation of trajectories of space gliders and airplanes, forecast calculations, spread of AIDS, and automated parking maneuvering based on mechanical systems [1] . Boundary layer equations are an important class of nonlinear ordinary differential equations that have many applications in physics and fluid mechanics [2][3] .…”

Section: Introductionmentioning

confidence: 99%

Numerical solution to the Falkner-Skan equation: a novel numerical approach through the new rational a-polynomials

Abbasbandy

Hajishafieiha

2021

Appl. Math. Mech.-Engl. Ed.

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The new rational a-polynomials are used to solve the Falkner-Skan equation. These polynomials are equipped with an auxiliary parameter. The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients. To find the unknown coefficients and the auxiliary parameter contained in the polynomials, the collocation method with Chebyshev-Gauss points is used. The numerical examples show the efficiency of this method.

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Nonlinear Ordinary Differential Equations

Cited by 18 publications

References 0 publications

Optimized Method of Lines for non-linear waveguide problems

Optimized Method of Lines for non-linear waveguide problems

Optimized method of lines for non-linear waveguide problems

Numerical solution to the Falkner-Skan equation: a novel numerical approach through the new rational a-polynomials

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