scholarly journals Singular Perturbation of Nonlinear Systems with Regular Singularity

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Domingos H. U. Marchetti ◽  
William R. P. Conti
Keyword(s):  
Nonlinear Systems ◽  
Power Series ◽  
Bessel Functions ◽  
Formal Solution ◽  
Nonlinear Partial Differential Equations ◽  
Power Series Expansion ◽  
Singularly Perturbed ◽  
Type Condition ◽  
Regular Singularity ◽  
Evolution Type

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.

Semi-analytic treatment of mixed hyperbolic–elliptic Cauchy problem modeling three-phase flow in porous media

2021 ◽  
Author(s):  
Emad Az-Zo’bi ◽  
Ahmet Yildirim ◽  
Lanre Akinyemi
Keyword(s):  
Power Series ◽  
Applied Mathematics ◽  
Nonlinear Partial Differential Equations ◽  
Power Series Expansion ◽  
Elliptic Systems ◽  
Approximate Solutions ◽  
Flow In Porous Media ◽  
Cauchy Problems ◽  
Algebraic Equations ◽  
Power Series Method

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

scholarly journals Näherungslösungen für die Form kernmagnetischer Signale bei Impulsexperimenten in Festkörpern

1968 ◽  
Vol 23 (8) ◽  
pp. 1194-1201
Author(s):  
G. Siegle
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Phase Difference ◽  
Pulse Sequence ◽  
Formal Solution ◽  
Power Series Expansion ◽  
Maximum Amplitude ◽  
Experimental Results ◽  
Longitudinal Relaxation ◽  
Pulse Distance

NMR transients in solids have been calculated using the method of LOWE and NORBERG 1 and its extension by POWLES and STRANGE 2. Some remarks are given concerning the evaluation of the formal solution for the signal after several pulses.So far the influence of the pulse durations and of the longitudinal relaxation is neglected usually, higher terms of the power series expansion of the signals were discussed only with respect to corrections of their amplitudes. Some remarkable differences between these calculations and new experimental results to be reported here are explained — at least in principle — avoiding these limitations:The maximum amplitude of the echo after two 90°-pulses (phase difference of their modulations Δφ=90°) occurs not exactly at a time twice the pulse distance, the deviation is explained considering the duration of the pulses and the decay of the echo envelope.After two pulses with Δφ=0° an echo was found which should disappear only in an ideal twospin system. For a three pulse sequence the influence of the longitudinal relaxation is described; and in a multi-pulse sequence the calculated modulation of the amplitude of successive echoes was checked by experiments.

The solution of an infinite set of differential-difference equations occurring in polymerization and queueing problems

1963 ◽  
Vol 59 (1) ◽  
pp. 117-124 ◽  
Author(s):  
A. Wragg
Keyword(s):  
Difference Equations ◽  
Queueing Theory ◽  
Bessel Functions ◽  
Formal Solution ◽  
Monomer Concentration ◽  
Arrival Rate ◽  
Time Dependent ◽  
Type Integral ◽  
Dependent Parameter ◽  
Infinite Set

AbstractThe time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

Power Spectral Density of Nonlinear System Response: The Recursion Method

1993 ◽  
Vol 60 (2) ◽  
pp. 358-365 ◽  
Author(s):  
R. Vale´ry Roy ◽  
P. D. Spanos
Keyword(s):  
Broad Class ◽  
Formal Solution ◽  
Power Series Expansion ◽  
Kolmogorov Equation ◽  
System Response ◽  
Lanczos Algorithm ◽  
Spectral Densities ◽  
Recursion Method ◽  
Approximate Form ◽  
Single Well

Spectral densities of the response of nonlinear systems to white noise excitation are considered. By using a formal solution of the associated Fokker-Planck-Kolmogorov equation, response spectral densities are represented by formal power series expansion for large frequencies. The coefficients of the series, known as the spectral moments, are determined in terms of first-order response statistics. Alternatively, a J-fraction representation of spectral densities can be achieved by using a generalization of the Lanczos algorithm for matrix tridiagonalization, known as the “recursion method.” Sequences of rational approximations of increasing order are obtained. They are used for numerical calculations regarding the single-well and double-well Duffing oscillators, and Van der Pol type oscillators. Digital simulations demonstrate that the proposed approach can be quite reliable over large variations of the system parameters. Further, it is quite versatile as it can be used for the determination of the spectrum of the response of a broad class of randomly excited nonlinear oscillators, with the sole prerequisite being the availability, in exact or approximate form, of the stationary probability density of the response.

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