On a Numerical Method for Solution of the Mathieu-Hill Type Equations

1980 ◽  
Vol 102 (3) ◽  
pp. 619-626 ◽  
Author(s):  
A. Midha ◽  
M. L. Badlani
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Initial Conditions ◽  
Linear Equations ◽  
Solution Process ◽  
Constraint Equations ◽  
Step Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Simultaneous Linear Equations

This paper presents a computer-programmable numerical method for the solution of a class of linear, second order differential equations with periodic coefficients of the Mathieu-Hill type. The method is applicable only when the initial conditions are prescribed and the solution is not requiried to be periodic. The solution is facilitated by representing the coefficient functions as a sum of step functions over corresponding sub-intervals of the fundamental interval. During each sub-interval, the solution form is assumed to be that of the differential equations with “constant” coefficients. Constraint equations are derived from imposing the conditions of “compatibility” of response at the end nodes of the intermediate sub-intervals. This set of simultaneous linear equations is expressed in matrix form. The matrix of coefficients may be represented as a triangular one. This form greatly simplifies the solution process for simultaneous equations. The method is illustrated by its application to some specific problems.

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 USING OF MULTILAYER NEURAL NETWORKS FOR THE SOLVING SYSTEMS OF DIFFERENTIAL EQUATIONS

2021 ◽  
pp. 81-88
Author(s):  
Natalia Marchenko ◽  
Ganna Sydorenko ◽  
Roman Rudenko
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Loss Function ◽  
Initial Conditions ◽  
Solution Process ◽  
Analytical Form ◽  
Individual Values ◽  
Systems Of Differential Equations ◽  
Solving Equations ◽  
Multilayer Neural Networks

The article considers the study of methods for numerical solution of systems of differential equations using neural networks. To achieve this goal, thefollowing interdependent tasks were solved: an overview of industries that need to solve systems of differential equations, as well as implemented amethod of solving systems of differential equations using neural networks. It is shown that different types of systems of differential equations can besolved by a single method, which requires only the problem of loss function for optimization, which is directly created from differential equations anddoes not require solving equations for the highest derivative. The solution of differential equations’ system using a multilayer neural networks is thefunctions given in analytical form, which can be differentiated or integrated analytically. In the course of this work, an improved form of constructionof a test solution of systems of differential equations was found, which satisfies the initial conditions for construction, but has less impact on thesolution error at a distance from the initial conditions compared to the form of such solution. The way has also been found to modify the calculation ofthe loss function for cases when the solution process stops at the local minimum, which will be caused by the high dependence of the subsequentvalues of the functions on the accuracy of finding the previous values. Among the results, it can be noted that the solution of differential equations’system using artificial neural networks may be more accurate than classical numerical methods for solving differential equations, but usually takesmuch longer to achieve similar results on small problems. The main advantage of using neural networks to solve differential equations` system is thatthe solution is in analytical form and can be found not only for individual values of parameters of equations, but also for all values of parameters in alimited range of values.

The DOMAIN family of propagation operations for intervals on simultaneous linear equations

1995 ◽  
Vol 9 (3) ◽  
pp. 197-210 ◽  
Author(s):  
R. Chen ◽  
A.C. Ward
Keyword(s):  
Linear Equations ◽  
Constraint Equation ◽  
Linear Constraint ◽  
The Other ◽  
Interval Matrix ◽  
Domain Family ◽  
Output Vector ◽  
Constraint Equations ◽  
Simultaneous Linear Equations ◽  
Mathematical Operations

AbstractThis paper defines, develops algorithms for, and illustrates the utility in design of a class of mathematical operations. These accept as inputs a system of linear constraint equations, Ax = b, an interval matrix of values for the coefficients, A, and an interval vector of values for either x or b. They return a set of values for the “domain” of the other vector, in the sense that all combinations of the output vector values set and values for A, when inserted into the constraint equation, correspond to values for the input vector that lie within the input interval. These operations have been mostly overlooked by the interval matrix arithmetic community, but are mathematically interesting and useful in the design, for example, of structures.

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.

Improvements in a method for solving fractional integral equations with some links with fractional differential equations

2018 ◽  
Vol 21 (1) ◽  
pp. 174-189 ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Source Term ◽  
Initial Conditions ◽  
Standard Procedure ◽  
Relaxation Equation ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure. In this procedure, there were two major obstacles that did not allow to obtain a full result. These were the assumptions over the smoothness of the source term and the assumption about the rationality of the orders. So, one of the main topics of this document is to describe a modification of the procedure presented in [4], when the source term is not smooth enough to differentiate the required amount of times. Furthermore, we will also study the fractional integral equations with non-rational orders by a limit process of fractional integral equations with rational orders. Finally, we will connect the previous material with some fractional differential equations with Caputo derivatives described in [7]. For instance, we will deal with the fractional oscillation equation, the fractional relaxation equation and, specially, its particular case of the Basset problem. We also expose how to compute these solutions for the Riemann-Liouville case.

scholarly journals Efficient homotopy perturbation method for fractional non-linear equations using Sumudu transform

2015 ◽  
Vol 19 (4) ◽  
pp. 1167-1171 ◽  
Author(s):  
Ming-Feng Zhang ◽  
Yan-Qin Liu ◽  
Xiao-Shuang Zhou
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Linear Equations ◽  
Solution Process ◽  
Homotopy Perturbation ◽  
Non Linear ◽  
Sumudu Transform ◽  
Non Linear Equations

In this paper, we propose an efficient modification of the homotopy perturbation method for solving fractional non-linear equations with fractional initial conditions. Sumudu transform is adopted to simplify the solution process. An example is given to illustrate the solution process and effectiveness of the method.

scholarly journals REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS AND NONPERMUTABLE VARIABLE COEFFICIENTS

2020 ◽  
Vol 25 (2) ◽  
pp. 303-322
Author(s):  
Michal Pospíšil
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Multiple Delays ◽  
Matrix Coefficients ◽  
Representation Of Solutions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Delay Differential

Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications.Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.

The RANGE family of propagation operations for intervals on simultaneous linear equations

1995 ◽  
Vol 9 (3) ◽  
pp. 183-196 ◽  
Author(s):  
R. Chen ◽  
A.C. Ward
Keyword(s):  
Interval Arithmetic ◽  
Linear Equations ◽  
Interval Matrix ◽  
Design Problems ◽  
The Matrix ◽  
Simultaneous Linear Equations ◽  
Systems Of Linear Equations ◽  
Mathematics Community

AbstractInterval arithmetic has been extensively applied to systems of linear equations by the interval matrix arithmetic community. This paper demonstrates through simple examples that some of this work can be viewed as particular instantiations of an abstract “design operation,” the RANGE operation of the Labeled Interval Calculus formalism for inference about sets of possibilities in design. These particular operations promise to solve a variety of design problems that lay beyond the reach of the original Labeled Interval Calculus. However, the abstract view also leads to a new operation, apparently overlooked by the matrix mathematics community, that should also be useful in design; the paper provides an algorithm for computing it.

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