Solutions Of Linear Differential Equations Research Articles

Remark on the theory of Sergeev frequencies of zeros, signs, and roots for solutions of linear differential equations: I

The theorem that claims that the spectra (ranges) of upper and lower Sergeev frequencies of zeros, signs, and roots of a linear differential equation of order > 2 with continuous coefficients belong to the class of Suslin sets on the nonnegative half-line of the extended numerical line is inverted for the spectra of upper frequencies of third-order equations under the assumption that the spectra contain zero. In addition, we construct examples of third-order equations with continuous coefficients whose Lebesgue sets of the upper Sergeev frequency of signs belong to the exact first Borel class, and the Lebesgue sets of upper Sergeev frequencies of zeros and roots belong to the exact second Borel class.

Lebesgue measure of escaping sets of entire functions of completely regular growth

We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated points and that most zeros are located in neighborhoods of finitely many rays. We apply the result to solutions of linear differential equations.

On non-normal solutions of linear differential equations

Normality arguments are applied to study the oscillation of solutions of f + A f = 0 f+Af=0 , where the coefficient A A is analytic in the unit disc D \mathbb {D} and sup z ∈ D ( 1 − | z | 2 ) 2 | A ( z ) | > ∞ \sup _{z\in \mathbb {D}} (1-|z|^2)^2|A(z)| > \infty . It is shown that such a differential equation may admit a non-normal solution having prescribed uniformly separated zeros.

Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

Nonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc

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Entire and meromorphic solutions of linear difference equations

Abstract In this paper, we shall investigate the existence of finite order entire and meromorphic solutions of linear difference equation of the form $$f^n (z) + p(z)f^{n - 2} (z) + L(z,f) = h(z)$$ where L(z, f) is linear difference polynomial in f(z), p(z) is non-zero polynomial and h(z) is a meromorphic function of finite order. We also consider finite order entire solution of linear difference equation of the form $$f^n (z) + p(z)L(z,f) = r(z)e^{q(z)}$$ where r(z) and q(z) are polynomials.

Presentation of special series with computed recurrently coefficients of solutions of nonlinear evolution equations

One of the analytical methods of presenting solutions of nonlinear partial differential equations is the method of special series in powers of specially selected functions called basis functions. The coefficients of such series are found successively as solutions of linear differential equations. To find recurrence, the coefficient is achieved by the choice of basis functions, which may also contain arbitrary functions. By using such functional arbitrariness, it allows in some cases to prove the global convergence of the corresponding constructed series, as well as the solvability of the boundary value problem.

On the baire classification of Sergeev frequencies of zeros and roots of solutions of linear differential equations

We show that the upper and lower characteristic frequencies of zeros and the upper frequency of roots of a solution of a linear differential equation treated as functions on the direct product of the space of equations with the compact-open topology by the space of initial vectors of solutions belong to the third Baire class and that the lower characteristic frequency of roots belongs to the second Baire class. As a corollary, we show that the ranges of the considered frequencies on the solutions of a given equation are Suslin (analytic) sets. In addition, we prove the Lebesgue measurability and the Baire property of the extreme characteristic frequencies of zeros and roots of an equation treated as functions of a real parameter on which the coefficients of the equation depend continuously.

On Oscillation of Solutions of Linear Differential Equations

An interrelationship is found between the accumulation points of zeros of non-trivial solutions of $$f''+Af=0$$ and the boundary behavior of the analytic coefficient A in the unit disc $$\mathbb {D}$$ of the complex plane $$\mathbb {C}$$ . It is also shown that the geometric distribution of zeros of any non-trivial solution of $$f''+Af=0$$ is severely restricted if * $$\begin{aligned} |A(z)| \left( 1-|z|^2\right) ^2 \le 1 + C \left( 1-|z|\right) , \quad z\in \mathbb {D}, \end{aligned}$$ for any constant $$0<C<\infty $$ . These considerations are related to the open problem of whether (*) implies finite oscillation for all non-trivial solutions.

On compact representations for the solutions of linear difference equations with variable coefficients

A comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients, of both nth and unbounded order, is presented. The equivalence between their celebrated combinatorial and determinantal representations is considered. A corresponding representation is proposed using determined nested sums of their variable coefficients. It makes explicit all the sum of products involved in the previous representations of such solutions. Some basic applications are also illustrated.

Operator Method for Construction of Solutions of Linear Fractional Differential Equations with Constant Coefficients

Abstract One of the effective methods to find explicit solutions of differential equations is the method based on the operator representation of solutions. The essence of this method is to construct a series, whose members are the relevant iteration operators acting to some classes of sufficiently smooth functions. This method is widely used in the works of B. Bondarenko for construction of solutions of differential equations of integer order. In this paper, the operator method is applied to construct solutions of linear differential equations with constant coefficients and with Caputo fractional derivatives. Then the fundamental solutions are used to obtain the unique solution of the Cauchy problem, where the initial conditions are given in terms of the unknown function and its derivatives of integer order. Comparison is made with the use of Mikusinski operational calculus for solving similar problems.

Evaluation of integrated solution numerical method of mathematical model of mechanical oscillatory system dynamics

A comparative analysis of the accuracy solutions of linear differential equations in the time domain to the developed integrated and a number of famous classical numerical methods is done. Test model examples show the high efficiency of the proposed numerical method regarding problems of analysis of dynamic processes of mechanical oscillatory systems

On solutions of linear differential equations of arbitrary fast growth in the unit disc

On solutions of linear differential equations of arbitrary fast growth in the unit disc

Q-Deformation of meromorphic solutions of linear differential equations

In this paper, we consider the behavior, when q goes to 1, of the set of a convenient basis of meromorphic solutions of a family of linear q-difference equations. In particular, we show that, under convenient assumptions, such basis of meromorphic solutions converges, when q goes to 1, to a basis of meromorphic solutions of a linear differential equation. We also explain that given a linear differential equation of order at least two, which has a Newton polygon that has only slopes of multiplicities one, and a basis of meromorphic solutions, we may build a family of linear q-difference equations that discretizes the linear differential equation, such that a convenient family of basis of meromorphic solutions is a q-deformation of the given basis of meromorphic solutions of the linear differential equation.

Effects of Temperature-Dependent Material Properties on Temperature Variation in a Thermoelement

The dependency of temperature variation in a thermoelement on the temperature-dependent material properties was analyzed under the prescribed temperatures at its two ends. Based on the homotopy perturbation method, an analytical solution of the nonlinear governing equation for thermoelectric heat transport was obtained and then used for the analysis. Four different thermoelectric materials were used in the current work to study temperature variation along the thermoelement. The results calculated by the analytical solution of the nonlinear differential equation showed significant differences in temperature variation from those calculated by the analytical solution of linear differential equation which is based on the constant material property assumption. These differences in temperature distribution affect the accuracy of the predicted power and thus the predicted performance of thermoelectric modules or devices.

On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point

Assume that the coefficients and solutions of the equation f (n)+p n−1(z)f (n−1)+. . .+p s+1(z)f (s+1)+ . . . + p 0(z)f = 0 have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients p j , j = s+1, . . . ,n−1, increase slower (in terms of the Nevanlinna characteristics) than p s (z). It is proved that this equation has at most s linearly independent solutions of finite order.

Solution of linear differential equations in chemical kinetics through flow graph theory approach

A flow graph theory is a method for finding the analytical solution of linear differential equations which arise in chemical kinetics through Cramer's method of determinants. This article presents the applicability of flow graph theory for deriving the analytical solution of kinetic equations which arise in modeling of complex reaction system such as hydrocracking of heavy oils. A discrete lumped model for hydrocracking of heavy oils was developed and analytical solution for the governing model equations was derived using Laplace transforms earlier. In this work, a new method involving flow graph theory was used instead of Laplace transforms. The kinetic equations which describe the performance of a hydrocracker are governed by linear differential equations and a general analytical solution was successfully derived using flow graph theory. The analytical solution obtained through flow graph theory is similar with the reported solution using Laplace transforms for the kinetic equations of hydrocracking of heavy oils. Furthermore, the relative errors between the experimental data and model calculations using analytical solution of the three lump hydrocracker model are reasonable except for few data points.

Exponential Dichotomy and Bounded Solutions of Differential Equations in the Fréchet Space

UDC 517.9 We establish necessary and sufficient conditions for the existence of bounded solutions of linear differential equations in the Fr´ echet space. The solutions are constructed with the use of a strong generalized inverse operator.

DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

Detecting Nonexistence of Rational Solutions of Linear Difference Equations in Early Stages of Computation

We discuss some changes of the traditional scheme for finding rational solutions of linear difference equations (homogeneous or inhomogeneous) with rational coefficients. These changes in some cases allow one to detect the absence of rational solutions in an early stage of computation and to stop the work. The corresponding strategy may be useful for the algorithms based on finding rational solutions of equations of considered form, e.g., for some symbolic summation algorithms.