Operational calculus techniques for solving differential equations

2005 ◽  
pp. 23-34 ◽  
Author(s):  
Nikolaos Glinos ◽  
B. David Saunders
Keyword(s):  
Differential Equations ◽  
Operational Calculus

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 On the solutions of some fractional q-differential equations with the Riemann-Liouville fractional q-derivative

2021 ◽  
Vol 104 (4) ◽  
pp. 130-141
Author(s):  
S. Shaimardan ◽  
◽  
N.S. Tokmagambetov ◽  
◽  
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Numerical Solutions ◽  
Operational Calculus ◽  
Explicit Solutions ◽  
Cauchy Type ◽  
Numerical Treatment ◽  
Type Problem ◽  
Real Numbers ◽  
Cauchy Type Problem

This paper is devoted to explicit and numerical solutions to linear fractional q-difference equations and the Cauchy type problem associated with the Riemann-Liouville fractional q-derivative in q-calculus. The approaches based on the reduction to Volterra q-integral equations, on compositional relations, and on operational calculus are presented to give explicit solutions to linear q-difference equations. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. Finally, some examples are provided to illustrate our main results in each subsection.

scholarly journals Solution of Some Types of Differential Equations: Operational Calculus and Inverse Differential Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
K. Zhukovsky
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Laguerre Polynomial ◽  
Operational Calculus ◽  
Integral Transforms ◽  
Operational Technique ◽  
Mathematical Problems ◽  
Equations Of Mathematical Physics ◽  
Transport Problems ◽  
General Method

We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.

scholarly journals Interval Version of the Operational Calculation Method for Solving Ordinary Differential Equations

2021 ◽  
Vol 04 (10) ◽  
Author(s):  
Oybek Zhumaboyevich Khudayberdiyev ◽  
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Calculation Method ◽  
Initial Conditions ◽  
Operational Calculus ◽  
Real Solution ◽  
Proved Theorem ◽  
The Cauchy Problem

This article discusses the interval variant of solving ordinary differential equations with given initial conditions, i.e. the Cauchy problem, by the method of operational calculus. This is where the interval version of the operational calculus is motivated and built. As a result, on the basis of the proved theorem in this article, an analytic interval set of solutions is obtained that is guaranteed to contain a real solution to the problem.

scholarly journals Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications

2021 ◽  
Vol 24 (2) ◽  
pp. 518-540
Author(s):  
Hafiz Muhammad Fahad ◽  
Arran Fernandez
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
General Setting ◽  
Fractional Differentiation ◽  
Fractional Calculus Operators ◽  
Liouville Fractional Derivative ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński’s operational calculus approach is used to obtain exact solutions of fractional differential equations with constant coefficients and with this type of fractional derivatives. These solutions can be expressed in terms of Mittag-Leffler type functions.

scholarly journals On the solution of the problem of synthesis of the control system for the process of dosed feed of electrode wire for arc welding equipment

2021 ◽  
pp. 20-26
Author(s):  
V.A. Lebedev
Keyword(s):  
Mathematical Model ◽  
Control System ◽  
Differential Equations ◽  
Electric Drive ◽  
Electric Motor ◽  
Mathematical Description ◽  
Operational Calculus ◽  
Feedback Circuit ◽  
Electric Drive System ◽  
Electrode Wire

Goal. Refinement of the methodology for the development of an effective control system for an electric drive with controlled relay-type regulators for organizing a metered feed of an electrode wire using the parameters of the arc process with the possibility of using it in design practice and practice of technological application. Methodology. The proposed method for the mathematical description (mathematical model) of the system of the developed structure electric drive - arc process with current feedback of welding with a variable structure device is based on the theory of automatic control as applied to nonlinear elements, the application of the theory of operational calculus. At the same time, a selection and description of a nonlinear node in the feedback circuit in the form of a relay element with a certain structure and subsequent linearization of this element was made. As an electric motor of the electrode wire feeder, a new development of a specialized valve electric motor is used, which is used in the system with a microprocessor controller. Results. Due to the presence of a substantially nonlinear link, the calculation of the valve electric drive system – the arc process can be found on the basis of a system of nonlinear differential equations, which is practically impossible for practical application. In this work, these complications are overcome on the basis of a rational choice of the description of the nonlinear link, its harmonic linearization and obtaining on this basis a mathematical description of the system, from which, using the methodology of operational calculus, the relations necessary for calculating the parameters of the system are determined in analytical form. Originality. The problem of calculating a rather complex problem of mathematical description of the valve electric drive system – a technological link in the form of an arc process with a substantially nonlinear link in the feedback circuit in the work is solved with the effective use of a set of methodological methods, which include as a means of representing individual links, including nonlinear links selected simplifications and solutions of the obtained differential equations using original methods of operational calculus. The proposed method (mathematical model) is tested in two directions – oscillography of a real system, as well as system simulation. Practical significance. Using the developed methods for describing the control system, it is possible to calculate its characteristics and, on their basis, select the parameters for setting the electric drive controller, which allows, without additional experimental research, to obtain the necessary character of the transfer of electrode metal, and, consequently, the quality of the result of the arc process.

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