scholarly journals p-Trigonometric andp-Hyperbolic Functions in Complex Domain

2016 ◽  
Vol 2016 ◽  
pp. 1-18
Author(s):  
Petr Girg ◽  
Lukáš Kotrla
Keyword(s):  
Differential Equations ◽  
Complex Domain ◽  
Trigonometric Functions ◽  
Maclaurin Series ◽  
Hyperbolic Functions

We study extension ofp-trigonometric functionssinpandcospand ofp-hyperbolic functionssinhpandcoshpto complex domain. Our aim is to answer the question under what conditions onpthese functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example,sin(z)=-i·sinh⁡i·z. In particular, we prove in the paper that forp=6,10,14,…thep-trigonometric andp-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series forp-trigonometric andp-hyperbolic functions.

scholarly journals Exponentials of general multivector in 3D Clifford algebras

2022 ◽  
Vol 27 (1) ◽  
pp. 179-197
Author(s):  
Adolfas Dargys ◽  
Artūras Acus
Keyword(s):  
Image Processing ◽  
Differential Equations ◽  
Automatic Control ◽  
Closed Form ◽  
Series Expansion ◽  
Clifford Algebras ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Geometric Algebras ◽  
And Robotics

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

scholarly journals (Kink; Kink; Kink; Kink) and (Pulse; Pulse; Pulse; Pulse) Solutions of a Set of Four Equations Modeled in a Nonlinear Hybrid Electrical Line with Crosslink Capacitor

2019 ◽  
pp. 1-14
Author(s):  
Tiague Takongmo Guy ◽  
Jean Roger Bogning
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Solitary Waves ◽  
Nonlinear Partial Differential Equations ◽  
Mathematical Methods ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
Hybrid Parts ◽  
Nonlinear Hybrid

The physics system that helps us in the study of this paper is a nonlinear hybrid electrical line with crosslink capacitor. Meaning it is composed of two different nonlinear hybrid parts Linked by capacitors with identical constant capacitance. We apply Kirchhoff laws to the circuit of the line to obtain new set of four nonlinear partial differential equations which describe the simultaneous dynamics of four solitary waves. Furthermore, we apply efficient mathematical methods based on the identification of coefficients of basic hyperbolic functions to construct exact solutions of those set of four nonlinear partial differential equations. The obtained results have enabled us to discover that, one of the two nonlinear hybrid electrical line with crosslink capacitor that we have modeled accepts the simultaneous propagation of a set of four solitary waves of type (Pulse; Pulse; Pulse; Pulse), while the other accepts the simultaneous propagation of a set of four solitary waves of type (Kink; Kink; Kink; Kink) when certain conditions we have established are respected. We ameliorate the quality of the signals by changing the sinusoidal waves that are supposed to propagate in the hybrid electrical lines with crosslink capacitor to solitary waves which are propagating in the new nonlinear hybrid electrical lines; we therefore, facilitate the choice of the type of line relative to the type of signal that we want to transmit.

scholarly journals Generalized trigonometric functions in complex domain

2015 ◽  
Vol 140 (2) ◽  
pp. 223-239
Author(s):  
Petr Girg ◽  
Lukáš Kotrla
Keyword(s):  
Complex Domain ◽  
Trigonometric Functions

Complex Wave Solutions to Mathematical Biology Models I: Newell–Whitehead–Segel and Zeldovich Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
Alper Korkmaz
Keyword(s):  
Gordon Equation ◽  
Reaction Diffusion ◽  
Expansion Method ◽  
Reaction Diffusion Equations ◽  
Trigonometric Functions ◽  
Sine Gordon Equation ◽  
Hyperbolic Functions ◽  
Complex Wave ◽  
Complex Valued ◽  
Homogeneous Balance

Complex and real valued exact solutions to some reaction-diffusion equations are suggested by using homogeneous balance and Sine-Gordon equation expansion method. The predicted solution of finite series of some hyperbolic functions is determined by using some relations between the hyperbolic functions and the trigonometric functions based on Sine-Gordon equation and traveling wave transform. The Newel–Whitehead–Segel (NWSE) and Zeldovich equations (ZE) are solved explicitly. Some complex valued solutions are depicted in real and imaginary components for some particular choice of parameters.

scholarly journals Ordinary and partial difference equations

1986 ◽  
Vol 27 (4) ◽  
pp. 488-501 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Equations ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
Partial Difference Equations ◽  
Partial Difference

AbstractOrdinary difference equations (OΔE's), mostly of order two and three, are derived for the trigonometric, Jacobian elliptic, and hyperbolic functions. The results are used to derive partial difference equations (PΔE's) for simple solutions of the wave equation and three nonlinear evolutionary partial differential equations.

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