Particular Solutions of Linear Differential and (q-) Difference Systems with Hypergeometric Right-hand Sides

2019 ◽  
Vol 45 (5) ◽  
pp. 298-302
Author(s):  
A. A. Ryabenko
Keyword(s):  
Particular Solutions ◽  
Right Hand ◽  
Difference Systems ◽  
Linear Differential

A Method for the Numerical Solution of a Boundary Value Problem for a Linear Differential Equation with Interval Parameters

2019 ◽  
Vol 16 (07) ◽  
pp. 1850115 ◽  
Author(s):  
Nizami A. Gasilov ◽  
Müjdat Kaya
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Numerical Method ◽  
Boundary Value ◽  
Real Life ◽  
Boundary Values ◽  
Right Hand ◽  
Linear Differential ◽  
The Right

In many real life applications, the behavior of the system is modeled by a boundary value problem (BVP) for a linear differential equation. If the uncertainties in the boundary values, the right-hand side function and the coefficient functions are to be taken into account, then in many cases an interval boundary value problem (IBVP) arises. In this study, for such an IBVP, we propose a different approach than the ones in common use. In the investigated IBVP, the boundary values are intervals. In addition, we model the right-hand side and coefficient functions as bunches of real functions. Then, we seek the solution of the problem as a bunch of functions. We interpret the IBVP as a set of classical BVPs. Such a classical BVP is constructed by taking a real number from each boundary interval, and a real function from each bunch. We define the bunch consisting of the solutions of all the classical BVPs to be the solution of the IBVP. In this context, we develop a numerical method to obtain the solution. We reduce the complexity of the method from [Formula: see text] to [Formula: see text] through our analysis. We demonstrate the effectiveness of the proposed approach and the numerical method by test examples.

scholarly journals II. On the solution of linear differential equations

1848 ◽  
Vol 138 ◽  
pp. 31-54 ◽  
Keyword(s):  
Differential Equations ◽  
Differential Expression ◽  
Linear Differential Equations ◽  
Right Hand ◽  
Independent Variable ◽  
Linear Differential ◽  
The Right

If the operation of differentiation with regard to the independent variable x be denoted by the symbol D, and if ϕ (D) represent any function of D composed of integral powers positive or negative, or both positive and negative, it may easily be shown, that ϕ (D){ψ x. u } = ψ x. ϕ (D) u + ψ' x. ϕ' (D) u + ½ψ" x. ϕ" (D) u + 1/2.3 ψ"' x. ϕ"' (D) u + . . . (1.) and that ϕx .ψ(D) u = ψ(D){ ϕx. u } - ψ'(D){ ϕ'x. u } + ½ψ"(D){ ϕ"x. u } - 1/2.3ψ"'(D){ ϕ"'x. u } + . . (2.) and these general theorems are expressions of the laws under which the operations of differentiation, direct and inverse, combine with those operations which are de­noted by factors, functions of the independent variable. It will be perceived that the right-hand side of each of these equations is a linear differential expression; and whenever an expression assumes or can be made to assume either of these forms, its solution is determined; for the equations ϕ (D){ψ x. u } = P and ϕx . ψ(D) u = P are respectively equivalent to u = (ψ x ) -1 { ϕ (D)} -1 P and u = {ψ(D)} -1 (( ϕx ) -1 P).

On the asymptotic behaviour of the solutions of a class of non-linear differential equations

1950 ◽  
Vol 46 (3) ◽  
pp. 406-418
Author(s):  
F. G. Friedlander
Keyword(s):  
Existence And Uniqueness ◽  
Lipschitz Constant ◽  
Asymptotic Properties ◽  
Physical Significance ◽  
Right Hand ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Mechanics ◽  
Image Position ◽  
The Right

1. This paper is concerned with certain asymptotic properties of the solutions of the differential equationwhere dots indicate differentiation with respect to t, k is a small parameter, and f(x, ẋ, t) satisfies certain conditions which will be formulated below. Equations of this type occur frequently in non-linear mechanics; for k = 0 a system satisfying (1·1) behaves as a harmonic oscillator. To ensure the existence and uniqueness of the solutions of (1·1) it must be assumed that the right-hand side is bounded and satisfies a Lipschitz condition, at least for finite x, ẋ and say all t ≥ 0. The parameter k may be considered as a measure of the ‘smallness’ of the upper bound, and of the Lipschitz constant, of the right-hand side, and need not have any intrinsic physical significance.

scholarly journals APPLICATIONS OF LIE SYSTEMS IN DISSIPATIVE MILNE–PINNEY EQUATIONS

2009 ◽  
Vol 06 (04) ◽  
pp. 683-699 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
JAVIER DE LUCAS
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Geometric Approach ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Particular Solutions ◽  
Systems Of Differential Equations ◽  
Lie Systems ◽  
Linear Differential

We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express the general solution of a dissipative Milne–Pinney equation in terms of particular solutions of a system of second-order linear differential equations and a set of constants.

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