scholarly journals Study of composite fractional relaxation differential equation using fractional operators with and without singular kernels and special functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Azhar Ali Zafar ◽  
Jan Awrejcewicz ◽  
Olga Mazur ◽  
Muhammad Bilal Riaz
Keyword(s):  
Differential Equation ◽  
Special Functions ◽  
Order Derivative ◽  
Laplace Transform Method ◽  
Derivative Operator ◽  
Transform Method ◽  
Integer Order ◽  
The Laplace Transform ◽  
Singular Kernels ◽  
Fractional Relaxation

AbstractOur aim in this article is to solve the composite fractional relaxation differential equation by using different definitions of the non-integer order derivative operator $D_{t}^{\alpha }$ D t α , more specifically we employ the definitions of Caputo, Caputo–Fabrizio and Atangana–Baleanu of non-integer order derivative operators. We apply the Laplace transform method to solve the problem and express our solutions in terms of Lorenzo and Hartley’s generalised G function. Furthermore, the effects of the parameters involved in the model are graphically highlighted.

scholarly journals Exact solution of time-fractional partial differential equations using Laplace transform

2016 ◽  
Vol 5 (1) ◽  
pp. 86
Author(s):  
Naser Al-Qutaifi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Partial Differential Equations ◽  
Laplace Transform Method ◽  
Partial Differential ◽  
Transform Method ◽  
Integer Order ◽  
First Derivative ◽  
The Laplace Transform

<p>The idea of replacing the first derivative in time by a fractional derivative of order , where , leads to a fractional generalization of any partial differential equations of integer order. In this paper, we obtain a relationship between the solution of the integer order equation and the solution of its fractional extension by using the Laplace transform method.</p>

scholarly journals A SIMPLE METHOD OF SOLVING ILKOVIC'S DIFFERENTIAL EQUATION FOR THE TRANSFER OF DEPOLARIZER TO THE DROPPING MERCURY ELECTRODE

1962 ◽  
Vol 40 (2) ◽  
pp. 296-300 ◽  
Author(s):  
R. S. Subrahmanya
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Mercury Electrode ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Simple Method ◽  
Dropping Mercury Electrode ◽  
The Laplace Transform ◽  
New Variables

The method developed by Ilkovic for solving the differential equation[Formula: see text]which governs the diffusion of the depolarizer to the surface of the dropping mercury electrode is very difficult. In the present work a simple method is presented. The above equation is transformed into ∂C/∂T = ∂2C/∂s2 by introducing the two new variables s = xt2/3/2. √((3/7)D) and T = (1/4)t/7/3. The boundary conditions for the transformed differential equation are formulated and the equation is solved by the Laplace transform method.

Analytical Solution of Transient Response of Gas-to-Gas Parallel and Counterflow Heat Exchangers

1987 ◽  
Vol 109 (4) ◽  
pp. 848-855 ◽  
Author(s):  
D. D. Gvozdenac
Keyword(s):  
Transient Response ◽  
Heat Exchangers ◽  
Special Functions ◽  
Dynamic Responses ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Independent Variables ◽  
The Laplace Transform

This paper shows how the transient response of gas-to-gas parallel and counterflow heat exchangers may be calculated by an analytical method. Making the usual idealizations for analysis of dynamic responses of heat exchangers, the problem of finding the temperature distributions of both fluids and the separating wall as well as the outlet temperatures of fluids is reduced to the solution of an integral equation. This equation contains an unknown function depending on two independent variables, space and time. The solution is found by using the method of successive approximations, the Laplace transform method, and special functions defined in this paper.

Dynamic Rate Effects on Timoshenko Beam Response

1985 ◽  
Vol 52 (2) ◽  
pp. 439-445 ◽  
Author(s):  
T. J. Ross
Keyword(s):  
Timoshenko Beam ◽  
Bending Moment ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Rate Effects ◽  
The Laplace Transform ◽  
Step Loading ◽  
Fixed End ◽  
Constitutive Formulation ◽  
Viscoelastic Timoshenko Beam

The problem of a viscoelastic Timoshenko beam subjected to a transversely applied step-loading is solved using the Laplace transform method. It is established that the support shear force is amplified more than the support bending moment for a fixed-end beam when strain rate influences are accounted for implicitly in the viscoelastic constitutive formulation.

scholarly journals Mathematical modeling and algorithm for calculation of thermocatalytic process of producing nanomaterial

2021 ◽  
Vol 23 (3) ◽  
pp. 1590
Author(s):  
Bakhtiyar Ismailov ◽  
Zhanat Umarova ◽  
Khairulla Ismailov ◽  
Aibarsha Dosmakanbetova ◽  
Saule Meldebekova
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Solid Phase ◽  
Nonlinear Effects ◽  
Operating Characteristics ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Wide Range ◽  
The Laplace Transform ◽  
Complex Mathematical Model

<p>At present, when constructing a mathematical description of the pyrolysis reactor, partial differential equations for the components of the gas phase and the catalyst phase are used. In the well-known works on modeling pyrolysis, the obtained models are applicable only for a narrow range of changes in the process parameters, the geometric dimensions are considered constant. The article poses the task of creating a complex mathematical model with additional terms, taking into account nonlinear effects, where the geometric dimensions of the apparatus and operating characteristics vary over a wide range. An analytical method has been developed for the implementation of a mathematical model of catalytic pyrolysis of methane for the production of nanomaterials in a continuous mode. The differential equation for gaseous components with initial and boundary conditions of the third type is reduced to a dimensionless form with a small value of the peclet criterion with a form factor. It is shown that the laplace transform method is mainly suitable for this case, which is applicable both for differential equations for solid-phase components and calculation in a periodic mode. The adequacy of the model results with the known experimental data is checked.</p>

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