Linear Inhomogeneous Differential Equation Research Articles

Solvability of Inhomogeneous Autonomous Differential Equation With Aftereffect on the Negative Semi-Axis

We obtain a necessary and sufficient solvability condition for a linear autonomous inhomogeneous functional differential equation with aftereffect and find the representation of all solutions in the special spaces of integrable functions with exponential weight. The obtained results are applied for study of two inhomogeneous differential equations with aftereffect (the first equation is with concentrated aftereffect, and the second one is with distributed aftereffect). We give the description of solutions spaces for these equations in terms of parameters of the initial problem.

Об общем решении линейного однородного дифференциального уравнения в банаховом пространстве в случае комплексных характеристических операторов

A linear inhomogeneous differential equation (LIDE) of the n th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the n -parametric family of solutions to LHDE is indicated. Operator functions eAt ; sinBt ; cosBt of real argument t ∈ [0;∞) are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied

Fast and Parallel Runge--Kutta Approximation of Fractional Evolution Equations

We consider a linear inhomogeneous fractional evolution equation which is obtained from a Cauchy problem by replacing its first-order time derivative with Caputo's fractional derivative. The operator in the fractional evolution equation is assumed to be sectorial. By using the inverse Laplace transform a solution to the fractional evolution equation is obtained which can be written as a convolution. Based on $L$-stable Runge--Kutta methods a convolution quadrature is derived which allows a stable approximation of the solution. Here, the convolution quadrature weights are represented as contour integrals. On discretizing these integrals, we are able to give an algorithm which computes the solution after $N$ time steps with step size $h$ up to an arbitrary accuracy $\varepsilon$. For this purpose the algorithm only requires $\mathcal{O}(N)$ Runge--Kutta steps for a large number of scalar linear inhomogeneous ordinary differential equations and the solutions of $\mathcal{O}(\log(N)\log(\frac{1}{\varepsilon}))$ linear systems which can be done in parallel. In numerical examples we illustrate the algorithm's performance.

Representation of solution of initial value problem for fuzzy linear multi-term fractional differential equation with continuous variable coefficient

We consider the representation of solutions of the initial value problems of fuzzy linear multi-term in-homogeneous fractional differential equations with continuous variable coefficients.

The Method of Determined Coefficients

The method of undetermined coefficients is a well-established procedure for finding a particular solution to any inhomogeneous linear differential equation with constant coefficients, in which the ...

An infinite sequence of conserved quantities for the cubic Gross–Pitaevskii hierarchy on ℝ

We consider the cubic Gross–Pitaevskii (GP) hierarchy on R \mathbb {R} , which is an infinite hierarchy of coupled linear inhomogeneous partial differential equations appearing in the derivation of the cubic nonlinear Schrödinger equation from quantum many-particle systems. In this work, we identify an infinite sequence of operators which generate infinitely many conserved quantities for solutions of the GP hierarchy.

Numerical solution of linear inhomogeneous fuzzy delay differential equations

We investigate inhomogeneous fuzzy delay differential equation (FDDE) in which initial function and source function are fuzzy. We assume these functions be in a special form, which we call triangular fuzzy function. We define solution as a fuzzy bunch of real functions such that each real function satisfies the equation with certain membership degree. We develop an algorithm to find the solution, and we provide the existence and uniqueness results for the considered FDDE. We also present an example to show the applicability of the proposed algorithm.

Indefinite integrals of special functions from inhomogeneous differential equations

ABSTRACTA method is presented for deriving integrals of special functions which obey inhomogeneous second-order linear differential equations. Inhomogeneous equations are readily derived for functions satisfying second-order homogeneous equations. Sample results are derived for Bessel functions, parabolic cylinder functions, Gauss hypergeometric functions and the six classical orthogonal polynomials. For the orthogonal polynomials the method gives indefinite integrals which reduce to the usual orthogonality conditions on the usual orthogonality intervals. These indefinite integrals for the orthogonal polynomials appear to be new. All results have been checked with Mathematica.

The existeness and uniqueness result of solutions to initial value problems of nonlinear diffusion equations involving with the conformable variable derivative

In this paper, based on the conformable derivative, we introduce the concept of conformable variable order derivative. The conformable derivative is a local operator, it has many properties like usual integer derivative’s ones. Similar to the conformable derivative, we study some properties of the conformable variable order derivative. We investigate the fundamental solutions to initial value problem for linear homogeneous and inhomogeneous diffusion differential equations with the conformable variable order derivative. In addition, using upper and lower solutions and monotone iterative method, we consider the existence and uniqueness of solutions to an initial value problem for nonlinear diffusion differential equations involving with the conformable variable order derivative.

New special function recurrences giving new indefinite integrals

ABSTRACTSequences of new recurrence relations are presented for Bessel functions, parabolic cylinder functions and associated Legendre functions. The sequences correspond to values of an integer variable r and are generalizations of each conventional recurrence relation, which correspond to r=1. The sequences can be extended indefinitely, though the relations become progressively more intricate as r increases. These relations all have the form of a first-order linear inhomogeneous differential equation, which can be solved by an integrating factor. This gives a very general indefinite integral for each recurrence. The method can be applied to other special functions which have conventional recurrence relations. All results have been checked numerically using Mathematica.

A problem in non-linear Diophantine approximation

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.

On Lower Estimates of Solutions and Their Derivatives to a Fourth-Order Linear Integrodifferential Volterra Equation

We examine solutions of the problem on sufficient conditions that guarantee a lower estimate and tending to infinity of solutions and their derivatives up to the third order to a fourth-order linear integrodifferential Volterra equation. For this purpose, we develop a method based on the nonstandard reduction method (S. Iskandarov), the Volterra transformation method, the method of shearing functions (S. Iskandarov), the method of integral inequalities (Yu. A. Ved’ and Z. Pakhyrov), the method of a priori estimates (N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, and P. M. Simonov, 1991, 2001), the Lagrange method for integral representations of solutions to first-order linear inhomogeneous differential equations, and the method of lower estimate of solutions (Yu. A. Ved’ and L. N. Kitaeva).

A diagonal recurrence relation for the Stirling numbers of the first kind

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces

The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used.

The Hurwitz Product, p-Adic Topology on ℤ, and Fundamental Solution to Linear Differential Equation in the Ring ℤ[[x

We study the linear inhomogeneous first order differential equation by′ + f(x) = y in the ring of formal power series with integer coefficients. Using the p-adic topology on the ring of integers, we construct a counterpart of the Hurwitz product of the Euler series and an arbitrary formal power series with integer coefficients. It is shown that the Euler series can be interpreted as the fundamental solution to the equation under consideration. Bibliography: 8 titles.

Boundary-Value Problem for a System of Linear Inhomogeneous Differential Equations of the First Order with Rectangular Matrices

Boundary-Value Problem for a System of Linear Inhomogeneous Differential Equations of the First Order with Rectangular Matrices

Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision

We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

On one method for the analysis of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation

A sequence converging to the solution of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation is constructed. This sequence is also asymptotic in the sense that the deviation (in the norm of the space of continuous functions) of its nth element from the solution of the problem is proportional to the (n + 1)th power of the perturbation parameter. A similar sequence is constructed for the case of an inhomogeneous first-order linear equation, on the example of which the application of such a sequence to the justification of the asymptotics obtained by the method of boundary functions is demonstrated.

Linear differential-algebraic equations perturbed by Volterra integral operators

We study linear inhomogeneous vector ordinary differential equations of arbitrary order in which the matrix multiplying the highest derivative of the unknown vector function is singular in the domain where the equations are defined. We also study perturbations (not necessarily small) of such equations, which are linear integro-differential equations with a Volterra operator. We obtain sufficient conditions for the solvability of such equations and give representations of their general solutions; solvability and uniqueness conditions are also given for initial value problems for such equations. The influence of small perturbations of the free term and the initial data on the solution is considered. A numerical method is suggested. The results of numerical experiments are given.

Hyers–Ulam stability of the time independent Schrödinger equations

In Jung and Roh (2017), we investigated some properties of approximate solutions of the second-order inhomogeneous linear differential equations. In this paper as an application of above paper we will study the stability of the time independent Schrödinger equation with a potential box of finite walls.