The asymptotic expansion of solutions of the differential equation u iv +λ 2 [(z 2 + c ) u" + azu '+ bu ]=0 for large IzI

1980 ◽  
Vol 293 (1404) ◽  
pp. 511-533 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Order Differential Equation ◽  
Fundamental System ◽  
Whittaker Function ◽  
Fundamental Solutions ◽  
Gauss Hypergeometric Function ◽  
Linear Combinations ◽  
Asymptotic Expansion Of Solutions ◽  
Asymptotic Character

The asymptotic expansion of solutions of the fourth order differential equation u iv + λ 2 [( z 2 + c ) u " + azu ' + bu ] = 0 are investigated for | z | -> oo where the parameters a, b, c and λ are supposed to be arbitrary complex constants with λ, b ^ 0. Exact solutions in the form of Laplace and Mellin-Barnes integrals, involving a Whittaker function and a Gauss hypergeometric function respectively, are used to define a fundamental system of solutions. The asymptotic expansion of these solutions is obtained in a full neighbourhood of the point at infinity and their asymptotic character is found to be either exponentially large or algebraic in certain sectors of the z -plane. The expansions corresponding to certain special values of the parameters a and b which yield logarithmic expansions are also treated. Linear combinations of these fundamental solutions which possess an exponentially small expansion for | z |->oo in a certain sector are discussed.

The asymptotic form of the Titchmarsh–Weyl m-function associated with a second order differential equation with locally integrable coefficient

1986 ◽  
Vol 102 (3-4) ◽  
pp. 243-251 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential ◽  
Image Position

SynopsisWe derive an asymptotic expansion for the Titchmarsh–Weyl m-function associated with the second order linear differential equationin the case where the only restriction on the real-valued function q is

scholarly journals The Asymptotic Expansion Method via Symbolic Computation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Juan F. Navarro
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Perturbation Method ◽  
Symbolic Computation ◽  
Order Differential Equation ◽  
Expansion Method ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Asymptotic Expansion Method

This paper describes an algorithm for implementing a perturbation method based on an asymptotic expansion of the solution to a second-order differential equation. We also introduce a new symbolic computation system which works with the so-called modified quasipolynomials, as well as an implementation of the algorithm on it.

scholarly journals On Some Properties of Solutions of Polyharmonic Equation in Polyhedral Angles

2007 ◽  
Vol 14 (3) ◽  
pp. 565-580
Author(s):  
Ilia Tavkhelidze
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Fundamental Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Uniqueness Theorems ◽  
Energy Integral ◽  
Polyharmonic Operator ◽  
Polyharmonic Equation

Abstract For a higher order differential equation with the polyharmonic operator, the Dirichlet and Riquier boundary value problems are studied in some polyhedral angles. Uniqueness theorems for solutions with a bounded “energy integral” of the corresponding BVPs are proved. Recurrent formulas are constructed for representation of fundamental solutions and Green's functions. The asymptotic behavior of solutions at infinity is studied.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

scholarly journals Even-order differential equation with continuous delay: nonexistence criteria of Kneser solutions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ali Muhib ◽  
M. Motawi Khashan ◽  
Osama Moaaz
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Delay Argument ◽  
Even Order ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this paper, we study even-order DEs where we deduce new conditions for nonexistence Kneser solutions for this type of DEs. Based on the nonexistence criteria of Kneser solutions, we establish the criteria for oscillation that take into account the effect of the delay argument, where to our knowledge all the previous results neglected the effect of the delay argument, so our results improve the previous results. The effectiveness of our new criteria is illustrated by examples.

scholarly journals A New Approach for Approximate Solution of ADE: Physical-Based Modeling of Carriers in Doping Region

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 458
Author(s):  
Leobardo Hernandez-Gonzalez ◽  
Jazmin Ramirez-Hernandez ◽  
Oswaldo Ulises Juarez-Sandoval ◽  
Miguel Angel Olivares-Robles ◽  
Ramon Blanco Sanchez ◽  
...  
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytic Solution ◽  
Order Differential Equation ◽  
Electric Circuit ◽  
New Approach ◽  
In Doping ◽  
Electric Behavior ◽  
Spatio Temporal ◽  
Physical Phenomena

The electric behavior in semiconductor devices is the result of the electric carriers’ injection and evacuation in the low doping region, N-. The carrier’s dynamic is determined by the ambipolar diffusion equation (ADE), which involves the main physical phenomena in the low doping region. The ADE does not have a direct analytic solution since it is a spatio-temporal second-order differential equation. The numerical solution is the most used, but is inadequate to be integrated into commercial electric circuit simulators. In this paper, an empiric approximation is proposed as the solution of the ADE. The proposed solution was validated using the final equations that were implemented in a simulator; the results were compared with the experimental results in each phase, obtaining a similarity in the current waveforms. Finally, an advantage of the proposed methodology is that the final expressions obtained can be easily implemented in commercial simulators.

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