Particular Solution of Linear Sequential Fractional Differential equation with Constant Coefficients by Inverse Fractional Differential Operators

2017 ◽  
Vol 25 (3) ◽  
pp. 373-383 ◽  
Author(s):  
Sanaa L. Khalaf ◽  
Ayad R. Khudair
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Differential Operators ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equation ◽  
Fractional Differential Operators

scholarly journals Dynamic Keynesian Model of Economic Growth with Memory and Lag

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 178 ◽  
Author(s):  
Vasily Tarasov ◽  
Valentina Tarasova
Keyword(s):  
Differential Equation ◽  
Economic Growth ◽  
Fractional Differential Equation ◽  
Differential Operators ◽  
Continuous Distribution ◽  
National Income ◽  
Fading Memory ◽  
Kummer Functions ◽  
Distributed Lag ◽  
Fractional Differential

A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the Abel-type integral and integro-differential operators with the confluent hypergeometric Kummer function in the kernel. These operators allow us to propose an economic accelerator, in which the memory and lag are taken into account. The fractional differential equation, which describes the dynamics of national income in this generalized model, is suggested. The solution of this fractional differential equation is obtained in the form of series of the confluent hypergeometric Kummer functions. The asymptotic behavior of national income, which is described by this solution, is considered.

scholarly journals Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1944
Author(s):  
Tohru Morita ◽  
Ken-ichi Sato
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Green’S Function ◽  
Fractional Differential Equation ◽  
Green's Function ◽  
Nonstandard Analysis ◽  
Polynomial Coefficients ◽  
Particular Solution ◽  
Fractional Differential ◽  
Detailed Derivation

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green’s function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation.

scholarly journals Fractional Differential Equation-Based Instantaneous Frequency Estimation for Signal Reconstruction

2021 ◽  
Vol 5 (3) ◽  
pp. 83
Author(s):  
Bilgi Görkem Yazgaç ◽  
Mürvet Kırcı
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Frequency Estimation ◽  
Signal Reconstruction ◽  
Reconstruction Method ◽  
Reconstruction Algorithms ◽  
Instantaneous Frequency Estimation ◽  
Proposed Model ◽  
Fractional Differential ◽  
Instantaneous Frequencies

In this paper, we propose a fractional differential equation (FDE)-based approach for the estimation of instantaneous frequencies for windowed signals as a part of signal reconstruction. This approach is based on modeling bandpass filter results around the peaks of a windowed signal as fractional differential equations and linking differ-integrator parameters, thereby determining the long-range dependence on estimated instantaneous frequencies. We investigated the performance of the proposed approach with two evaluation measures and compared it to a benchmark noniterative signal reconstruction method (SPSI). The comparison was provided with different overlap parameters to investigate the performance of the proposed model concerning resolution. An additional comparison was provided by applying the proposed method and benchmark method outputs to iterative signal reconstruction algorithms. The proposed FDE method received better evaluation results in high resolution for the noniterative case and comparable results with SPSI with an increasing iteration number of iterative methods, regardless of the overlap parameter.

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