Chernoff Approximations as a Method for Finding the Resolvent of a Linear Operator and Solving a Linear ODE with Variable Coefficients

The goal of the work. Proposals for methods of solving systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients that defined in interval form and intended for modeling exchange processes in multicomponent environments. Research subject: systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients defined in interval form. Research method: interval analysis. The obtained results. Systems of linear homogeneous and non-homogeneous differential equations, which are used in modeling exchange processes in multicomponent environments, are considered. Such systems can be considered, for example, in problems of chemical kinetics, materials science, and the theory of Markov processes. To obtain the solution of these equations, specialized calculators of analytical transformations were used and tested. The Matlab system (ode15s solver) was used for numerical analysis of systems of differential equations. It is shown that the application of interval methods of numerical analysis at the initial stage of system modeling has some advantages over probabilistic methods because they do not require knowledge of the laws of distribution of the results of the system state parameter measurements and their errors. It is shown that existing methods of solving systems of linear differential equations can be divided into two groups. Common to these groups is the use of interval expansion of classical methods for solving differential equations given in interval form. The difference between these two groups of methods is as follows. The methods of the first group can be used for all types of differential equations but require the creation of special software. The peculiarity of the methods of the second group is that they can be used to solve equations analytically or using numerical analysis packages. The application of the methods of the second group is shown on the example of solving a system of differential equations, the coefficients of which are determined in interval form. The system of these equations is intended for modeling the processes of exchange with the external environment of the elements of the model of a specific physical system. In the case when the coefficients of these equations are variables, their piecewise-constant approximation is applied and a criterion that determines the possibility of its application is given. The technique proposed in the paper can be applied to solve systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients if they are given by slowly varying functions. In the case when the coefficients of the equations are determined in the interval form, the technique allows obtaining their solution also in the interval form and does not require the creation of special software.

Использование многочлена Тейлора второй степени при аппроксимации производных конечными разностями приводит ко второму порядку аппроксимации традиционного метода сеток при численном интегрировании краевых задач для неоднородных линейных обыкновенных дифференциальных уравнений второго порядка с переменными коэффициентами. В работе при исследовании краевых задач для неоднородных линейных обыкновенных дифференциальных уравнений четвертого порядка с переменными коэффициентами рассмотрен предложенный ранее метод численного интегрирования, использующий средства матричного исчисления, в котором аппроксимация производных конечными разностями не использовалась. Согласно указанному методу, при составлении системы разностных уравнений может быть выбрана произвольная степень многочлена Тейлора в разложении искомого решения задачи в ряд Тейлора. В работе возможные граничные условия дифференциальной краевой задачи записаны как в виде производных степеней от нуля до трех, так и в виде линейных комбинаций этих степеней. Краевая задача названа симметричной, если количества граничных условий в левой и правой границах совпадают и равны двум; в противном случае задача названа несимметричной. Для дифференциальной краевой задачи составлена ее аппроксимирующая разностная краевая задача в виде двух подсистем: в первую подсистему вошли уравнения, при построении которых не были использованы граничные условия краевой задачи; во вторую подсистему вошли четыре уравнения, при построении которых были использованы граничные условия задачи. Теоретически выявлены закономерности между порядком аппроксимации разностной краевой задачи и степенью используемого многочлена Тейлора. Установлено следующее: а) порядок аппроксимации первой и второй подсистем пропорционален степени используемого многочлена Тейлора; б) порядок аппроксимации первой подсистемы меньше степени многочлена Тейлора на две единицы при ее четном значении и меньше на три единицы при ее нечетном значении; в) порядок аппроксимации второй подсистемы меньше степени многочлена Тейлора на три единицы независимо как от четности или нечетности этой степени, так и от степени старшей производной в граничных условиях краевой задачи. Вычислен порядок аппроксимации разностной краевой задачи со всеми возможными комбинациями граничных условий. Теоретические выводы подтверждены численными экспериментами.

We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential operators that are polynomials in the variables and their partial derivatives. The output is a quantum state whose wavefunction is proportional to a specific solution of the non-homogeneous differential equation, which can be measured to reveal features of the solution. The algorithm consists of three stages: preparing fixed resource states in ancillary systems, performing Hamiltonian simulation, and measuring the ancilla systems. The algorithm can be carried out using standard methods for gate decompositions, but we improve this in two ways. First, we show that for a wide class of differential operators, it is possible to derive exact decompositions for the gates employed in Hamiltonian simulation. This avoids the need for costly commutator approximations, reducing gate counts by orders of magnitude. Additionally, we employ methods from machine learning to find explicit circuits that prepare the required resource states. We conclude by studying an example application of the algorithm: solving Poisson's equation in electrostatics.

We consider a nonhomogeneous linear mixed type differential equation with variable coefficients and establish an asymptotic result for the solutions. Our result is obtained by the use of a solution of the so-called generalized characteristic equation of the corresponding homogeneous linear mixed type differential equation.

In this manuscript, we discuss the Tarig transform for homogeneous and non-homogeneous linear differential equations. Using this Tarig integral transform, we resolve higher-order linear differential equations, and we produce the conditions required for Hyers–Ulam stability. This is the first attempt to use the Tarig transform to show that linear and nonlinear differential equations are stable. This study also demonstrates that the Tarig transform method is more effective for analyzing the stability issue for differential equations with constant coefficients. A discussion of applications follows, to illustrate our approach. This research also presents a novel approach to studying the stability of differential equations. Furthermore, this study demonstrates that Tarig transform analysis is more practical for examining stability issues in linear differential equations with constant coefficients. In addition, we examine some applications of linear, nonlinear, and fractional differential equations, by using the Tarig integral transform.

Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives correspond to the given equations, respectively. In this new analytic method, an integrating factor function formula for second-order ODEs has been carefully formulated and derived, making every second-order ODE with variable coefficients reducible to its lower-order form, specifically first-order ODEs. To ensure the accuracy of the new method, two well-known classes of second-order linear ODEs, namely the Whittaker second-order linear ODE and the Modified Bessel equation, were applied. The results demonstrated that the new analytic method effectively solves these equations, producing exact analytical solutions. To validate the effectiveness and efficiency of the new analytic method, a comparative analysis was conducted using illustrative examples, followed by graphical representations of the solution results.

In this paper we describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type Fractional Derivative, and describe this method developed by us, to find out Particular Integrals, for several types of forcing functions. The solutions are obtained in terms of Mittag-Leffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to non-homogeneous FDE. We have demonstrated these developed methods with few examples of FDE, and also applied in fractional damped forced differential equation. This method proposed by us is useful as it is having conjugation with the classical methods of solving non-homogeneous linear differential equations, and also useful in understanding physical systems described by FDE.

In this study, we consider a linear nonhomogeneous differential equation with variable coefficients and variable delays and present a novel matrix-collocation method based on Morgan–Voyce polynomials to obtain the approximate solutions under the initial conditions. The method reduces the equation with variable delays to a matrix equation with unknown Morgan–Voyce coefficients. Thereby, the solution is obtained in terms of Morgan–Voyce polynomials. In addition, two test problems together with error analysis are performed to illustrate the accuracy and applicability of the method; the obtained results are scrutinized and interpreted by means of tables and figures.

We make use of linear operators to derive the formulae for the general solution of elementary linear scalar ordinary differential equations of order n. The key lies in the factorization of the linear operators in terms of first-order operators. These first-order operators are then integrated by applying their corresponding integral operators. This leads to the solution formulae for both homogeneous- and nonhomogeneous linear differential equations in a natural way without the need for any ansatz (or educated guess). For second-order linear equations with nonconstant coefficients, the condition of the factorization is given in terms of Riccati equations.

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

In this paper sufficient conditions have been obtained for non-oscillation of non-homogeneous canonical linear differential equations of third order. Some of these results have been extended to non-linear equations.

<p style='text-indent:20px;'>Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper.</p>

Solutions of Linear Difference Equations with Variable Coefficients

In this paper, it is shown how non-homogeneous linear differential equations, especially those of the second order, are solved by means of GeoGebra applets, This is done by indeterminate coefficient methods and variation of parameters, in the course of differential equations for engineering students of the University of Antofagasta in Chile. The use of free software GeoGebra has been increasing in the teaching of mathematics, mainly in non-homogeneous linear differential equations, because it facilitates the teaching and learning process

The solution of a nonhomogeneous linear Caputo fractional differential equation of order nq,(n−1)<nq<n with Caputo fractional initial conditions can be expressed using suitable Mittag–Leffler functions. In order to extend this result to such a nonhomogeneous linear Caputo fractional differential equation of order nq,(n−1)<nq<n, that also includes lower order fractional derivative terms, we can reduce such a problem to an n-system of Caputo fractional differential equations of order q,0<q<1, with corresponding initial conditions. In this work, we use an approximation method to solve the resulting system of Caputo fractional differential equations of order q with initial conditions, using the fundamental matrix solutions involving the matrix Mittag–Leffler functions. Furthermore, we compute the fundamental matrix solution using the standard eigenvalue method. This fundamental matrix solution then allows us to express the component-wise solutions of the system using initial conditions, similar to the scalar case. As a consequence, we obtain solutions to linear nonhomogeneous Caputo fractional differential equations of order nq,(n−1)<nq<n, with Caputo fractional initial conditions having lower-order Caputo derivative terms. We illustrate the method with several examples for two and three system, considering cases where the eigenvalues are real and distinct, real and repeated, or complex conjugates.