Solutions of non-homogeneous linear complex differential equations in some function spaces

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.

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.

In this paper, we investigate analytic solutions of higher order linear non-homogeneous directional differential equations whose coefficients are analytic functions in the unit ball. We use methods of theory of analytic functions in the unit ball having bounded L-index. Our proofs are based on application of inequalities from analog of Hayman’s theorem for analytic functions in the unit ball. There are presented growth estimates of their solutions which contain parameters depending on the coefficients of the equations. Also, we obtained sufficient conditions that every analytic solution of the equation has bounded L-index in the direction. The deduced results are also new in one-dimensional case, i.e. for functions analytic in the unit disc. Keywords: Analytic function, analytic solution, slice function, unit ball, directional differential equation, growth estimate, bounded L-index in direction.

Variation of parameters and the method of Kryloff and Bogoliuboff

(1971). On the solutions of certain linear nonhomogeneous second-order differential equations. Applicable Analysis: Vol. 1, No. 1, pp. 57-63.

I. Third Order Linear Homogeneous Differential Equations in Normal Form.- 1. Fundamental Properties of Solutions of the Third Order Linear Homogeneous Differential Equation.- 1. The Normal Form of a Third Order Linear Homogeneous Differential Equation.- 2. Adjoint and Self-adjoint Third Order Linear Differential Equations.- 3. Fundamental Properties of Solutions.- 4. Relationship between Solutions of the Differential Equations (a) and (b).- 5. Integral Identities.- 6. Notion of a Band of Solutions of the First, Second and Third Kinds.- 7. Further Properties of Solutions of the Differential Equation (a) Implied by Properties of Bands.- 8. Weakening of Property (v) for the Laguerre Invariant.- 2. Oscillatory Properties of Solutions of the Differential Equation (a).- 1. Basic Definitions.- 2. Sufficient Conditions for the Differential Equation (a) to Be Disconjugate.- 3. Sufficient Conditions for Oscillatoricity of Solutions of the Differencial Equation (a).- 4. Further Conditions Concerning Oscillatoricity or Non-oscillatoricity of Solutions of the Differential Equation (a).- 5. Relation between Solutions without Zeros and Oscillatoricity of the Differential Equation (a).- 6. Sufficient Conditions for Oscillatoricity of Solutions of the Differential Equation (a) in the Case A(x) ? 0, x ? (a, ?).- 7. Conjugate Points, Principal Solutions and the Relationship between the Adjoint Differential Equations (a) and (b).- 8. Criteria for Oscillatoricity of the Differential Equations (a) and (b) Implied by Properties of Conjugate Points.- 9. Further Criteria for Oscillatoricity of the Differential Equation (b).- 10. The Number of Oscillatory Solutions in a Fundamental System of Solutions of the Differential Equation (a).- 11. Criteria for Oscillatoricity of Solutions of the Differential Equation (a) in the Case that the Laguerre Invariant Does Not Satisfy Condition (v).- 12. The Case, When the Laguerre Invariant Is an Oscillatory Function of x.- 13. The Differential Equation (a) Having All Solutions Oscillatory in a Given Interval.- 3. Asymptotic Properties of Solutions of the Differential Equations (a) and (b).- 1. Asymptotic Properties of Solutions without Zeros of the Differential Equations (a) and (b).- 2. Asymptotic Properties of Oscillatory Solutions of the Differential Equation (b).- 3. Asymptotic Properties of All Solutions of the Differential Equation (a).- 4. Boundary Value Problems.- 1. The Green Function and Its Applications.- 2. Further Applications of Integral Equations to the Solution of Boundary-value Problems.- 3. Generalized Sturm Theory for Third Order Boundary-value Problems.- 4. Special Boundary-value Problems.- II. Third Order Linear Homogeneous Differential Equations with Continuous Coefficients.- 5. Principal Properties of Solutions of Linear Homogeneous Third Order Differential Equations with Continuous Coefficients.- 1. Principal Properties of Solutions of the Differential Equation (A).- 2. Bands of Solutions of the Differential Equation (A).- 3. Application of Bands to Solving a Three-point Boundary-value Problem.- 6. Conditions for Disconjugateness, Non-oscillatoricity and Oscillatoricity of Solutions of the Differential Equation (A).- 1. Conditions for Disconjugateness of Solutions of the Differential Equation (A).- 2. Solutions without Zeros and Their Relation to Oscillatoricity of Solutions of the Differential Equation (A).- 3. Conditions for the Existence of Oscillatory Solutions of the Differential Equation (A).- 4. On Uniqueness of Solutions without Zeros of the Differential Equation (A).- 5. Some Properties of Solutions of the Differential Equation (A) with r(x) ? 0.- 7. Comparison Theorems for Differential Equations of Type (A) and Their Applications.- 1. Comparison Theorems.- 2. A Simple Application of Comparison Theorems.- 3. Remark on Asymptotic Properties of Solutions of the Differential Equation (A).- III. Concluding Remarks.- 1. Special Forms of Third Order Differential Equations.- 2. Remark on Mutual Transformation of Solutions of Third Order Differential Equations.- IV. Applications of Third Order Linear Differential Equation Theory.- 8. Some Applications of Linear Third Order Differential Equation Theory to Non-linear Third Order Problems.- 1. Application of Quasi-linearization to Certain Problems Involving Ordinary Third Order Differential Equations.- 2. Three-point Boundary-value Problems for Third Order Non-linear Ordinary Differential Equations.- 3. On Properties of Solutions of a Certain Non-linear Third Order Differential Equation.- 9. Physical and Engineering Applications of Third Order Differential Equations.- 1. On Deflection of a Curved Beam.- 2. Three-layer Beam.- 3. Survey of Some Other Applications of Third Order Differential Equations.- References.

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.

Entire Solutions of Nonhomogeneous Linear Differential Equations

In management science research, a large number of cases analysis often requier from qualitative analysis to quantitative analysis, differential equation and differential equations are often used in quantitative analysis. The core of quantitative analysis of a case is to build a corresponding mathematical model. The differential equation models are widely used mathematical models. In particular, linear differential equation models with constant coefficients are often used mathematical models in the quantitative analysis of cases in management science research. In practice, the particular solution of linear differential equation model with constant coefficients often been found by Laplace transformation. Follow this way, a method to get the general solutions of linear differential equation with constant coefficients based on Laplace transformation be given. In this method, it is easy to find the general solution of homogeneous linear differential equation, of non-homogeneous linear differential equation and of linear differential equations with constant coefficients based on Laplace transformation. Key words : Management; Mathematical model; Linear differential equation; Laplace transformation

This paper deals with the fixed point and hyper order of solutions of nonhomogeneous higher order linear differential equations with meromorphic function coefficent, and gets two results of fixed point and hyper order of homogeneous higher order linear differenliad equations.Moreover,we generalize the related results of some authors.

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.

In this paper, we investigate the iterated order of solutions of higher order homogeneous linear differential equations with entire coef- ficients. We improve and extend some results of Bela¨odi and Hamouda by using the concept of the iterated order. We also consider nonhomogeneous linear differential equations.

1. The basic idea of the application of integral operators to the Weierstrass-Hadamard direction. In order to generate and investigate solutions of differential equations, operators p (defined as the integral operators of the first kind) have been introduced in [2; 6](2). p transforms analytic functions of one and two variables into solutions of linear elliptic differential equations of two and three variables, respectively. It has been shown in the abovementioned papers that p (as well as some other operators connected with p) preserves many properties of the functions to which the operator is applied. This situation permits us to use theorems in the theory of functions to obtain theorems not merely on harmonic functions in two variables, but on solutions of other linear differential equations as well(3). In the present paper the above-mentioned method is used to prove connections between the properties in the large of solutions i1 of certain linear differential equations, see (1.1) and (1.3), on one side and the structure of certain subsequences of the coefficients of the series development of V/ at the origin on the other. Let us formulate these procedures in a somewhat more concrete manner, at first for equations in two variables. Let i1 be a (real) solution of the differential equation

A solution of First Order Nonhomogeneous Linear Fuzzy Differential Equation with initial condition as Triangular Fuzzy Number is discussed in this paper and to obtained a general solution, we have used a method of interval arithmetic on α-cut interval. Here, we have proposed a solution Non-Homogeneous Fuzzy Differential Equation by considering four different possible cases of real valued functions involved in differential equations. At the last, an example of first order nonhomogeneous linear fuzzy differential equation under fuzzy initial condition is being solved to verify the result.

In this paper, we have considered second order non-homogeneous linear differential equations whose coefficients are entire functions. We have established conditions ensuring the non-existence of finite order solution of such type of differential equations. Moreover, we have also extended our results to higher order non-homogeneous differential equations.