Fractional Differential Equations

Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed.

https://www.mdpi.com/2227-7390/7/6/509/pdf

On History of Mathematical Economics: Application of Fractional Calculus

(An) order level inventory model for deteriorating items = 부패하는 제품의 재고 모형

A generalization of the economic model of natural growth, which takes into account the power-law memory effect, is suggested. The memory effect means the dependence of the process not only on the current state of the process, but also on the history of changes of this process in the past. For the mathematical description of the economic process with power-law memory we used the theory of derivatives of non-integer order and fractional-order differential equation. We propose equations take into account the effects of memory with one-parameter power-law damping. Solutions of these fractional differential equations are suggested. We proved that the growth and downturn of output depend on the memory effects. We demonstrate that the memory effect can lead to decrease of output instead of its growth, which is described by model without memory effect. Memory effect can lead to increase of output, rather than decrease, which is described by model without memory effect.

Fractional Dynamics of Natural Growth and Memory Effect in Economics

In this paper, a fractional-order SIRD mathematical model is presented with Caputo derivative for the transmission of COVID-19 between humans. We calculate the steady-states of the system and discuss their stability. We also discuss the existence and uniqueness of a non-negative solution for the system under study. Additionally, we obtain an approximate response by implementing the fractional Euler method. Next, we investigate the first and the second waves of the disease in Iran and Japan; then we give a prediction concerning the second wave of the disease. We display the numerical simulations for different derivative orders in order to evaluate the efficacy of the fractional concept on the system behaviors. We also calculate the optimal control of the system and display its numerical simulations.

On the fractional SIRD mathematical model and control for the transmission of COVID-19: The first and the second waves of the disease in Iran and Japan.

In this paper, we have studied some memory dependent Economic order quantity models as the memory effect has an important role to handle the business policy of the inventory system. Demand of a company product depends on many factors like behavior of staff, environment of the shop, product quality etc which are the main reasons of memory effect on the system. Once the customers gain some poor experience, further they will never purchase products from those companies or shop. So inclusion of memory effect in the inventory model is necessary to handle practical business policy. One of the best way of inclusion of memory effect in the EOQ model is the use of fractional calculus as fractional derivative is defined in terms of integration where the limits of integration are the initial state and current state. Three fractional order models have been developed considering (i) only the rate of change of the inventory level of fractional order  (ii) demand rate as a fractional polynomial of degree 2 ,  where  is the rate of change of the inventory level (iii) demand rate as a fractional polynomial of degree 2m , where m may be different from the order of the rate of change of the inventory level. Here fractional order is physically treated as an index of memory. To develop the models here Caputo type fractional derivative has been applied. Due to solve those problems, we have used primal geometric programming method and finally some numerical examples are cited to establish the memory effect. Our investigation establishes the existence of memory effect on inventory management through fractional formulation of EOQ models which can never be obtained from classical calculus.

/pdf/study-of-memory-effects-in-an-inventory-model-using-1968f3k1qq.pdf

Study of memory effects in an inventory model using fractional calculus

Study of memory effect in an inventory model for deteriorating items with partial backlogging

Our purpose of the paper is to propose an inventory modelwit h memory effect.Generally, in deriving the solution of the class ical inventory model, first order differential equation is considered. The integer orde r differential equation is not able to incorporate the memory or past experience of the inventory system. But in case of real life inventory problems, the past exper ience or memory effect has great impact. We also want to give importance on sa lvage value of an inventory system. A comparisonalso has beenbuilt with our pr evious developed paper where salvage value is not considered. The paper aims to show the result of long memory effect and short memory effect on the min i ized total average cost and the optimal ordering interval. The numerical exampl e is presented toillustrate the purpose of the paper.

/pdf/study-of-memory-effect-in-an-inventory-model-with-quadratic-1cwcowuqsn.pdf

Study of memory effect in an inventory model with quadratic type demand rate and salvage value

The purpose of this paper is to establish the memory effect in an inventory model. In this model, price dependent demand is considered during the shortage period. Primal geometric programming is introduced to solve the minimized total average cost and optimal ordering interval. And finally we have taken a numerical example to justify the memory effect of this type inventory system. From the result it is clear that the model is suitable for short memory affected business i.e. newly started business.

/pdf/application-of-memory-effect-in-an-inventory-model-with-48jqwraudl.pdf

Application of Memory Effect in an Inventory Model with Price Dependent Demand Rate during Shortage

The study of memory effect in an economic order quantity model has a great impact on the inventory system. Although business policy almost depends on the past experiences of the system, usually the classical inventory model does not include the past experience or memory effect, i.e. one important part of the system is ignored. Our purpose is to include memory or past experience in the inventory model. The purpose of this paper is to incorporate the existence of dynamic memory in an inventory model with shortage via fractional calculus. To derive the memory dependent inventory model associated with inventory holding cost, shortage cost has been developed. Analytical solution of the proposed inventory model has been solved via primal geometric programming method. Numerically long memory effect or short memory effect of the inventory system has been established. In this paper, an effort has also been made to compare the memory effect on the minimized total average cost and the optimal ordering interval using different numerical examples.

/pdf/study-of-memory-effect-in-an-economic-order-quantity-model-1h0thz92s5.pdf

Rituparna Pakhira

Papers

Study of memory effects in an inventory model using fractional calculus

Study of memory effect in an inventory model for deteriorating items with partial backlogging

Study of memory effect in an inventory model with quadratic type demand rate and salvage value

Application of Memory Effect in an Inventory Model with Price Dependent Demand Rate during Shortage

Study of memory effect in an economic order quantity model with quadratic type demand rate