When Nonlinear Differential Equations are Equivalent to Linear Differential Equations

AbstractUtilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| > |α2| > |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

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Efficient quantum algorithm for dissipative nonlinear differential equations

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.2026805118 ◽

2021 ◽

Vol 118 (35) ◽

pp. e2026805118

Author(s):

Jin-Peng Liu ◽

Herman Øie Kolden ◽

Hari K. Krovi ◽

Nuno F. Loureiro ◽

Konstantina Trivisa ◽

...

Keyword(s):

Differential Equations ◽

Quantum Algorithms ◽

Nonlinear Differential Equations ◽

Quantum Algorithm ◽

Euler Method ◽

Linear Differential Equations ◽

Dimensional System ◽

Worst Case ◽

Potential Applications ◽

Linear Differential

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T2q poly(log⁡T,log⁡n,log⁡1/ϵ)/ϵ, where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R≥2. Finally, we discuss potential applications, showing that the R<1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.

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Sturmian comparison theory for half-linear and nonlinear differential equations via Picone identity

Filomat ◽

10.2298/fil1705185o ◽

2017 ◽

Vol 31 (5) ◽

pp. 1185-1194

Author(s):

Abdullah Özbekler

Keyword(s):

Differential Equations ◽

Nonlinear Differential Equations ◽

Linear Differential Equations ◽

Comparison Theorems ◽

Comparison Theory ◽

Second Order Differential Equations ◽

Nonlinear Version ◽

Linear And Nonlinear ◽

Linear Differential ◽

Picone Identity

In this paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations of the form (m(t)??(y'))' + Xn,i=1 qi(t)??i(y)=0(1) and the second is the half-linear differential equations (k(t)??(x'))'+ p(t)??(x) = 0 (2) where ?*(s) = |s|*-1s and ?1 >...> ?m > ? > ?m+1 > ... > ?n > 0. Under the assumption that the solution of Eq. (2) has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq. (1) by employing the new nonlinear version of Picone?s formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq. (1). Examples are given to illustrate the relevance of the results.

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On the Exponents of Exponential Dichotomies

Mathematics ◽

10.3390/math8040651 ◽

2020 ◽

Vol 8 (4) ◽

pp. 651

Author(s):

Flaviano Battelli ◽

Michal Fečkan

Keyword(s):

Differential Equations ◽

Exponential Dichotomy ◽

Nonlinear Differential Equations ◽

Linear Differential Equations ◽

Constructive Method ◽

Bifurcation Method ◽

Exponential Dichotomies ◽

The Difference ◽

Linear Differential

An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. This roughness result is crucial in developing a Melnikov bifurcation method for either discontinuous or implicit perturbed nonlinear differential equations.

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