scholarly journals Solution of a system of differential equations with constant coefficients using in-verse moments problem techniques

2019 ◽  
Vol 7 (3) ◽  
pp. 71
Author(s):  
Dra. María B. Pintarelli
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Initial Conditions ◽  
Analytical Form ◽  
Inverse Transformation ◽  
General Integral ◽  
Inverse Transform ◽  
Moments Problem ◽  
Constant Coefficients ◽  
Linear Differential

It is known that given a system of simultaneous linear differential equations with constant coefficients you can apply the Laplace method to solve it. The Laplace transforms are found and the problem is reduced to the resolution of an algebraic system of equations of the determining functions, and applying the inverse transformation the generating functions are determined, solutions of the given system. This implies the need to know the analytical form of the inverse transform of the function. In this case the initial conditions consist in knowing the value that the generating function and its derivatives takes at zero. A generalization of this method is proposed in this work, which is to define a more general integral operator than the Laplace transform, the initial conditions consist of Cauchy conditions in the contour. And finally, we find a numerical approximation of the inverse transformation of the generating functions, using the techniques of inverse moment problems, without being necessary to know the analytical form of the inverse transform of the function.

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

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Ravshan Ashurov ◽  
Alberto Cabada ◽  
Batirkhan Turmetov
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Operational Calculus ◽  
Operator Method ◽  
Caputo Fractional Derivatives ◽  
Operator Representation ◽  
Integer Order ◽  
Representation Of Solutions ◽  
Constant Coefficients ◽  
Linear Differential

AbstractOne 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.

scholarly journals V. On the calculus of symbols, with applications to the theory of differential equations

1861 ◽  
Vol 151 ◽  
pp. 69-82 ◽  
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
New Methods ◽  
Methods Of Calculation ◽  
Great Utility ◽  
Long Time ◽  
Constant Coefficients ◽  
Linear Differential

The calculus of generating functions, discovered by Laplace, was, as is well known, highly instrumental in calling the attention of mathematicians to the analogy which exists between differentials and powers. This analogy was perceived at length to involve an essential identity, and several analysts devoted themselves to the improvement of the new methods of calculation which were thus called into existence. For a long time the modes of combination assumed to exist between different classes of symbols were those of ordinary algebra; and this sufficed for investigations respecting functions of differential coefficients and constants, and consequently for the integration of linear differential equations, with constant coefficients. The laws of combination of ordinary algebraical symbols may be divided into the commutative and distributive laws; and the number of symbols in the higher branches of mathematics, which are commutative with respect to one another, is very small. It became then necessary to invent an algebra of non-commutative symbols. This important step was effected by Professor Boole, for certain classes of symbols, in his well-known and beautiful memoir published in the Transactions of this Society for the year 1844, and the object of the paper which I have now the honour to lay before the Society is to perfect and develope the methods there employed. For this purpose I have constructed systems of multiplication and division for functions of non-commutative symbols, subject to the same laws of combination as those assumed in Professor Boole’s memoir, and I thus arrive at equations of great utility in the integration of linear differential equations with variable coefficients.

scholarly journals On the generalized superstability of nth-order linear differential equations with initial conditions

2015 ◽  
Vol 98 (112) ◽  
pp. 243-249 ◽  
Author(s):  
Jinghao Huang ◽  
Qusuay Alqifiary ◽  
Yongjin Li
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Constant Coefficients ◽  
Linear Differential

We establish the generalized superstability of differential equations of nth-order with initial conditions and investigate the generalized superstability of differential equations of second order in the form of y??(x) + p(x)y?(x)+q(x)y(x) = 0 and the superstability of linear differential equations with constant coefficients with initial conditions.

scholarly journals I. On the calculus of symbols, with applications to the theory of differential equations

1862 ◽  
Vol 11 ◽  
pp. 84-85
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Linear Differential Equations ◽  
New Methods ◽  
Methods Of Calculation ◽  
Long Time ◽  
Constant Coefficients ◽  
Linear Differential

The calculus of generating functions, discovered by Laplace, was, as is well known, highly instrumental in calling the attention of mathematicians to the analogy which exists between differentials and powers. This analogy was perceived at length to involve an essential identity, and several analysts devoted themselves to the improvement of the new methods of calculation which were thus called into existence. For a long time the modes of combination assumed to exist between different classes of symbols were those of ordinary algebra; and this sufficed for investigations respecting functions of differential coefficients and constants, and consequently for the integration of linear differential equations, with constant coefficients.

scholarly journals The variational iteration method for solving n-th order fuzzy differential equations

Open Physics ◽  
2012 ◽  
Vol 10 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Mohammad Saeidy ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Analytical Method ◽  
Iteration Method ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Linear Differential Equations ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
Linear Differential

AbstractThe variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

scholarly journals A comparative analysis of methods: mimetics, finite differences and finite elements for 1-dimensional stationary problems

2021 ◽  
Vol 8 (1) ◽  
pp. 1-11
Author(s):  
Abdul Abner Lugo Jiménez ◽  
Guelvis Enrique Mata Díaz ◽  
Bladismir Ruiz
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Stationary Regime ◽  
Heat Transfer Fluid ◽  
Mimetic Finite Difference Method ◽  
Linear Differential

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

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