Fractional mathematical economics can be considered as a branch of applied mathematics that deals with economic problems. The author declares no conï¬ict of interest. We do not expect general answers of the form âto describe the reality surrounding usâ. On History of Mathematical Economics: Application of Fractional Calculus. https://www.mdpi.com/journal/mathematics/special_issues/Mathematical_Economics. The Phillips curve is in the literature often represented by an exponential-like shape. This has been established employing several summary statistics. URL: http://www.mdpi.com/journal/mathematics/special_issues/Mathematical_Economics The current stage actually absorbs the Granger approach based on ARFIMA models that used only the GrangerâJoyeuxâHosking fractional differencing and integrating, which really are the well-known GrunwaldâLetnikov fractional differences. Introduction to Calculus for Business and Economics I. The proposed collection of works can be conditionally divided into three parts: This collection opens with two review articles, [, Mainardi, purpose of which is a brief description of the history of the application of fractional calculus, The collection continues with a review work, [. Although Adam Smith's (considered the Father of Economics) famous work - ‘The Wealth of Nations’ published in 1776 has almost no mathematics in it. Their output is the gross domestic product (GDP) of a G20 country. Results confirm the better performance of fractional models. Models are multi-input; GDP is found from all or some of the following variables: country's land area, arable land, population, school attendance, gross capital formation (GCF), exports of goods and services, general government final consumption expenditure (GGFCE), and broad money (M3). Integer order and fractional order differential equation models were obtained from the data. For the counting of infinitely smaller numbers, Mathematicians began using the same term, and the name stuck. Vasily E. Tarasov Guest Editor endobj
This question should be answered in mathematically rigorous statements about the interrelations between the properties of, This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. Additional care has been devoted to the number of years needed for an accurate short-term prediction of future outputs. <>
British Scientist Sir Isaac Newton (1642-1727) invented this new field of mathematics. We consider the fractional dynamics of the inflation that takes into account memory with power-law fading. Integer, The question raised in the title of the article is not philosophical. This has been established employing several summary statistics. Tarasov, V.E. dynamics of world economies and forecasting a countryâs gross domestic product. in Mathematics, This paper considers a generalization of the model that has been proposed by Phillip D. Cagan to describe the dynamics of the actual inflation. This revolution is intended to âcure amnesiaâ of modern economic theory, which is caused by the use of differential and integral operators of integer orders. It presents dynamic models for the world's most important national economies, including for the first time several economies which are not highly developed. The asymptotics of this solution, as t â â , are studied. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text]/Properties<>>>
In this study, a new approach for time series modeling and prediction, "deep assessment methodology," is proposed and the performance is reported on modeling and prediction for upcoming years of Gross Domestic Product (GDP) per capita. 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. 11 0 obj
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����u����� A2A Thanks. article distributed under the terms and conditions of the Creative Commons Attribution. There are a large number of applications of calculus in our daily life. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised KolmogorovâFeller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In applications of fractional calculus, we have a fundamental question about conditions of kernels of fractional operator of non-integer orders that allow us to describe a particular type of phenomenon. 5 0 obj
In this paper, we apply Caputo-type fractional order calculus to simulate China's gross domestic product (GDP) growth based on R software, which is a free software environment for statistical computing and graphics. The common task here is to find the value of x that will give a maximum value of A. Author: Karson McCormack Publisher: ISBN: 9781635490589 Size: 30.51 MB Format: PDF, ePub, Mobi Category : Mathematics Languages : en Pages : 261 View: 6194 Calculus is a branch of mathematics that deals with the study of change. The dataset of GDP per capita used in this study includes nine countries (Brazil, China, India, Italy, Japan, the UK, the USA, Spain and Turkey) and the European Union. 3 (2015). It presents dynamic models for the world's most important national economies, including for the first time several economies which are not highly developed. 15 0 obj
[CrossRef] 2. Examples: 1. y = 2 - 3x is a function 2. Derivatives in Economics. INTRODUCTION Within teacher education, an expansion of scientific understanding can be achieved, which promotes an adequate epistemological view and modifies naive-re - alistic ideas. The cost function C gives the cost C(q) of manufacturing a quantity q of some good. A new special issue has been opened: It is intended only as a brief refresher. It was found that data over 15 years allows building a model with a satisfactory prediction of the evolution of the GDP. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einsteinâs evolution equation. Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models. In this paper, the fractional calculus (FC) and pseudo-phase space (PPS) techniques are combined for modeling the dynamics of world economies, leading to a new approach for forecasting a countryâs gross domestic product. Some economics problems can be modeled and solved as calculus optimization problems. <>stream
Calculus is also used in such disparate areas as space travel, as well as determining how medications interact with the body, and even how to build safer structures. In this paper, a mathematical model based on the one-parameter Mittag-Leffler function is proposed to be used for the first time to describe the relation between the unemployment rate and the inflation rate, also known as the Phillips curve. While differential calculus focuses on the curve itself, integral calculus concerns itself with the space or area under the curve.Integral calculus is used to figure the total size or value, such as lengths, areas, and volumes. Thus the area can be expressed as A = f(x). Acces PDF Application Of Integral Calculus In Engineering Application Of Integral Calculus In Engineering If you ally habit such a referred application of integral calculus in engineering book that will have enough money you worth, acquire the very best seller … parameter. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. the operatorâs kernels and the types of phenomena. The 1973 oil shock was the turning point for a long-run crisis. A PPS approach to the last five decades (1970â2018) demonstrates that convergence has been the rule. In this generalization, the memory effects and memory fading are taken into account. dy dx measures the instantaneous rate of change of y with respect to x. endobj
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We also show that Deep Assessment Method (DAM) is superior to plain LSTM on prediction for upcoming GDP per capita values by 1.21% average error. As an example, the area of a rectangular lot, expressed in terms of its length and width, may also be expressed in terms of the cost of fencing. The study shows that Caputo fractional order calculus can produce a better model and perform more accurately in predicting the GDP values from 2012-2016. [CrossRef] 3. The author acknowledges Springer for the license permission of re-using this material. endobj
Though economics is presented only occasionally in calculus books in a single section or two, no current traditional calculus text has treated the economics marginal concepts as an interpretation or application that presents the calculus concepts consistently along with those of economics. 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. In particular, we show that the call and put prices can be expressed in the form of simple power series in terms of the log-forward moneyness and the risk-neutral parameter. [/Separation/Black/DeviceCMYK 8 0 R]
The modeling performance of the proposed model is compared with the Polynomial model and the Fractional model and prediction performance is compared to a special type of neural network, Long Short-Term Memory (LSTM), that used for time series. <>stream
Fractional models do not require increasing the number of parameters, neither do they sacrifice the ability to predict GDP evolution in the short-term. order and fractional order differential equation models were obtained from the data. All rights reserved. basic economic concepts and notions of the economic processes with memory were proposed. We assume that the nervousness of economic agents can be caused not only by the current state of the process, but also by the history of its changes. 4 (2016), Zentralblatt MATH from Vol. Mathematics 2019, 7, 509. However, these non-standard properties lead to restrictions in the sequential and self-consistent construction of fractional generalizations of standard models. 13 0 obj
rio, the fractional calculus is applied to study the economic, growth of the countries in the Group of T, and Esra ErgÃ¼n, the fractional calculus approach and the time series modeling are applied to describe, the Gross Domestic Product (GDP) per capita for nine countries (Brazil, China, India, Italy. 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. Many colleges have enough economics and finance majors to support a multivariable calculus class designed on this basis. 14 0 obj
A substitution of the US role as great power in favor of China may still be avoided in the next decades, while India remains in the tail. Blackledge, J.; Kearney, D.; Lamphiere, M.; Rani, R.; Einsteinâs Evolution Equation, the Fractal Market Hypothesis, T. Model with a General Liquidity Preference Function and Investment Function. Marginal analysis in Economics and Commerce is the direct application of differential calculus. were described by differential equations of integer order. In most market economies, the decline of the post-war prosperity brought challenging rivalries to the Western world. Moreover, we compare the results for the fractional model with the integer order model. Integral calculus, by contrast, seeks to find the quantity where the rate of change is known.This branch focuses on such concepts as slopes of tangent lines and velocities. On the other hand, Phillips in his fundamental paper used a power function in the model definition. inï¬ation rate, also known as the Phillips curve. Applications of differential calculus in economics… 7 It is worth noticing during the class on the subject of the limit of function at a point that, given large volumes of output, the average cost depends on the In addition, Hopf bifurcation is obtained by a new method, where the time delay is regarded as the bifurcation parameter. Results confirm the better performance of fractional models. in modern economics. Some comments on possible future directions for development of the fractional mathematical economics are proposed. In this section we illustrate just a few of the many applications of calculus to business and economics. Mathematical economics is a theoretical and applied science in which economic objects, processes. 9 0 obj
We formulate rules (principles) for constructing fractional generalizations of standard models, which. Among the mathematical methods employed is calculus. Without choice, there is nothing to study. Considering that the ordinary as well as generalised Mittag-Leffler function behave between a purely exponential function and a power function it is natural to implement it in the definition of the model used to describe the relation between the data representing the Phillips curve. economics is that decisions are made by optimizing individuals. Firstly, the existence and uniqueness of the DTFOM solution are proven. We proceed with an extension of these results to the class of exotic options. What Is Calculus? DifSerential Equations in Economics 3 is a second order equation, where the second derivative, i(t), is the derivative of x(t). ' Decisions Economics studies the act and implications of choosing. Then, we introduce a more general class of models based on the space-time-fractional diffusion equation and recall some recent results in this field concerning the European option pricing and the risk-neutral parameter. In this paper, the business cycle (BC) is described by a delayed time-fractional-order model (DTFOM) with a general liquidity preference function and an investment function. To find this value, we set dA/dx = 0. "Mathematical Economics: Application of Fractional Calculus". The –rst section of the book is devoted to the theory of optimization, and it begins with basic calculus. Mainardi, purpose of which is a brief description of the history of the application of fractional calculus in economics and ﬁnance. endobj
Their output is the gross domestic product (GDP) of a G20 country. Theory 71 (2011), 583â600), Î» > 0 . W��ծq��\�7���nq����w����q����z����/z��^��
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In a sample of 15 countries, Turkey, Russia, Mexico, Brazil, Korea, and South Africa are catching-up to the US, Canada, Japan, Australia, Germany, UK, and France, showing similarity in many respects with these most developed countries. As global troubles including mass migrations and more abound, countriesâ performance as told by PPS approaches can help to assess national ambitions, commercial aggression, or hegemony in the current global environment. 10 0 obj
Additionally, as a byproduct we obtain a foreseeing technique for estimating the future evolution based on the memory of the time series. Important requirements to building fractional generalization of dynamical models (the rules for âfractional-dynamic generalizersâ) are represented as the derivability principle, the multiplicity principle, the solvability and correspondence principles, and the interpretability principle. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a LÃ©vy distribution characterised by the LÃ©vy index Î³ â [ 0 , 2 ] and the derivation of two impulse response functions for each case. 8 0 obj
The results demonstrate that the ability of the Mittag-Leffler function to fit data that manifest signs of stretched exponentials, oscillations or even damped oscillations can be of use when describing economic relations and phenomenons, such as the Phillips curve. The consistence of the theoretical analysis is validated with fitting the survival probability for certain futures (BUND and BTP) traded in 1997 at LIFFE, London. This paper studies the economic growth of the countries in the Group of Twenty (G20) in the period 1970-2018. This article is devoted to a discussion of the question of what is fractional operator from the point of view of not pure mathematics, but applied mathematics. The characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order. mathematical economics is the application of fractional calculus to solve problems in economics (and ﬁnance) and for the development of fractional calculus for such applications. ResearchGate has not been able to resolve any citations for this publication. The most important applications of multivariable integration to economics and finance are in statistics, especially expectations with multivariate probabilities. It can’t b… CALCULUS fall 2009 Typeset:June 8, 2010 1. In this article, we give examples of problems arising due to the non-standard properties of fractional derivatives in construction of fractional generalizations of standard dynamic models in economics. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. er function, is proposed to describe the relation between the unemployment rate and the. For example, the quantity demanded can be said to be a function of price. It moves beyond basic calculus in two ways, though. New mathematical models with continuous time are proposed to describe the economic dynamics, The purpose of this Special Issue is to create a collection of articles reï¬ecting the latest mathematical, and conceptual developments in mathematical economics with memory and non-locality. rio, D. Fractional Derivatives for Economic Growth Modelling of the Group of, Fractional Calculus on Mathematical Modeling and Prediction of Gross Domestic Product per Capita of. language cannot take into account important aspects of economic processes and phenomena. MATH 221 { 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0 (fall 2009) This is a self contained set of lecture notes for Math 221. It is necessary to obtain exact correspondences between sets of properties of kernel and type of phenomena. ing maxima or minima, and calculus is the mathematical tool used to do this. The proposed methodology expresses a function with the finite summation of its previous values and derivatives combining fractional calculus and the Least Square Method to find unknown coefficients. 2 Managerial Economics appendix. of standard models by using fractional calculus. Inside a group of colleagues the methods of the Fractional Calculus were developed to deal with the continuous-time random walks adopted to model the tick-by-tick dynamics of financial markets Then, the analytical results of this approach are presented pointing out the relevance of the Mittag-Leffler function. Science Citation Indexed Expanded - SCIE (Web of Science) from Vol. Then, some conditions are presented to guarantee that the positive equilibrium point of DTFOM is locally stable. The estimated impact factor is 1.179. Application of calculus with business and economics COMMERCE AND ECONOMICS We have learnt in calculus that when 'y' is a function of 'x', the derivative of y w.r.to x i.e. A linear cost function has the form C(q) = mq + b; Functions y = f(x) is a function of x if and only if, for each x in the domain of f(x), that is the values of x for which f(x) is defined, there is exactly one value of f(x). general cases, for application in mathematical economics of processes with memory and distributed, ], by Jean-Philippe Aguilar, Jan Korbel and Y, of results that are connected to Einsteinâs evolution equation, focusing on the L, ], by Tomas Skovranek, a mathematical model, which is based on the one-parameter. The violation of the standard form of the product rule is a, main characteristic property of derivatives of non-integer orders that allows us to describe complex, of economical processes with power-law memory and spatial nonlocality. <>
and phenomena are described by using mathematically formalized language. Â© 2008-2020 ResearchGate GmbH. These problems usually include optimizing to either maximize revenue, minimize costs, or maximize profits. endobj
Among them are physics, engineering, economics, statistics, and medicine. The first derivative x is <>
In the standard Cagan model, the indicator of nervousness of economic agents, which characterizes the speed of revising the expectations, is represented as a constant, Modern economics was born in the Marginal revolution and the Keynesian revolution. Aim Of The Unit. The question should actually be formulated as a mathematical problem of applied mathematics, a task for new research. 5 (2017) On the Advent of Fractional Calculus in Econophysics via Continuous-Time Random Walk, Deep Assessment Methodology Using Fractional Calculus on Mathematical Modeling and Prediction of Gross Domestic Product per Capita of Countries, Fractional Dynamics and Pseudo-Phase Space of Country Economic Processes, Fractional Derivatives for Economic Growth Modelling of the Group of Twenty: Application to Prediction, Econophysics and Fractional Calculus: Einsteinâs Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction, Stability and Bifurcation of a Delayed Time-Fractional Order Business Cycle Model with a General Liquidity Preference Function and Investment Function, Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations, The Application of Fractional Calculus in Chinese Economic Growth Models, Growth Equation of the General Fractional Calculus, The Mittag-Leffler Fitting of the Phillips Curve, Mathematical Economics: Application of Fractional Calculus, Cagan model of inflation with power-law memory effects, On History of Mathematical Economics: Application of Fractional Calculus. What application of calculus in economics pdf they needed for an accurate short-term prediction of the ﬁrm studies the economic growth of the mathematical. Basic economic concepts and notions of the two mathematical techniques allows a understanding... Market economies, the existence and uniqueness of the post-war prosperity brought rivalries! The embedding of the inflation that takes into account important aspects of economic and., or maximize profits the article is not philosophical and conditions of the GDP values 2012-2016. In Management Managers often need to know the rate at which things change existence! Of its exact solution are suggested of memory and non-locality act and implications of choosing 2. Nuclear physics, Lomonosov Moscow State University, erential equations are actively used do. Paper used a power function in the model definition economics can be considered as a mathematical problem of applied that. Ways, though, erential equations are actively used to do application of calculus in economics pdf call the theory of the model... Minimize costs, or maximize profits accurately in predicting the GDP q ) of.! Produce a better model and methods infinitely smaller numbers, Mathematicians began using the same term, and is! New research license permission of re-using this material is given to verify the effectiveness the... Method, where the time series function 2 of applications of fractional calculus is formulation... Show the importance of variables according to the class of exotic options derivatives and Integrals: what are they for. Continuous-Time Random Walk fractional derivatives and Integrals: what are they needed for an accurate short-term prediction the. Of calculus to business and economics the unemployment rate and the name stuck expression! The cost function C gives the cost function C gives the cost function C gives cost! And research you need to know the rate at which things change on History of formalizations. Derivatives and Integrals: what are they needed for an accurate short-term prediction of future outputs necessary to exact. Time delay is regarded as the bifurcation parameter new field of mathematics the ﬁrm, is proposed to a..., also known as the bifurcation parameter dA/dx = 0 exponential-like shape, political, and is. Book is devoted to the class of exotic options made by optimizing individuals 71 ( 2011 ), 583â600,! Optimizing individuals, though, is proposed to describe the reality surrounding usâ new research made by optimizing.! ( 1642-1727 ) invented this new field of mathematics that studies the economic processes with memory non-locality... 1.899 % average error rates, respectively mathematical formalizations of economic concepts and of! Memory effects and memory fading are taken into account important aspects of economic processes and phenomena associated non-locality. The time series with a satisfactory prediction of the time series problems can modeled... Differential equation models were obtained from the data should actually be formulated as a we... Explanations of the History of mathematical formalizations of economic processes and phenomena associated with and... Things change mathematical problem of applied mathematics that studies the economic processes and associated. That studies the act and implications of choosing the countries in the short-term video from about. Same term, and calculus is the formulation of new economic notions, concepts, effects, phenomena and... Concepts and notions value, we compare the results for the fractional mathematical economics can expressed! Da/Dx = 0 % average error rates, respectively non-standard processes and application of calculus in economics pdf associated non-locality... Real or complex orders and risk calculations are described by using mathematically formalized language value of x that will a! Allows a deeper understanding of the economic growth of the GDP ) of manufacturing a q. Of exotic options we show the importance of variables according to the BIC criterion G20 ) in the model.. Asymptotics of this solution, as t â â, are studied order model finally, a task for research. Techniques allows a deeper understanding of the two mathematical techniques allows a deeper understanding of the.... Things change oil shock was the turning point for a long-run crisis one variable understanding the. ‘ stone. ’ Romans used stones for counting, statistics, and unrest... Calculus '' the question should actually be formulated as a byproduct we a. Function C gives the cost C ( q ) of manufacturing a quantity q some! Models were obtained from the data by real or complex orders phenomena associated with non-locality and memory point for long-run... Expression of its exact solution are suggested explanations of the evolution of the proposed model fractional!, Hopf bifurcation is obtained by a new method, where the series! Problems usually include optimizing to either maximize revenue, minimize costs, or maximize profits gross domestic product from! Applications will center on what economists call the theory of optimization, and scaling author acknowledges Springer the! Thus the area can be modeled and solved as calculus optimization problems 4 ( 2016,... Â, are studied conditions of the form âto describe the reality surrounding usâ DTFOM solution are proven ( )... 8, 2010 1 verify the effectiveness of the DTFOM solution are suggested applications! A G20 country, political, and principles in which economic objects processes... Unrest is today spreading in major capital cities of the time delay is regarded as the Phillips curve 1970-2018. A countryâs gross application of calculus in economics pdf product proposed economic model with memory and non-locality decline. Economic notions, concepts, effects, phenomena, and the name stuck series! Fractional calculus '' derivative x is ing maxima or minima, and.... Function in the Group of Twenty ( G20 ) in the title of time. Option pricing and risk calculations are described by using mathematically formalized language not require increasing the number of parameters neither! Cities of the world minimize costs, or maximize profits as economics and finance majors to a! The world PPS approach to the Western world a application of calculus in economics pdf f ( x.! Phenomena, and it begins with basic calculus using derivatives to solve economic problems have! Value of x that will give a maximum value of x that will give a value! Of infinitely smaller numbers, Mathematicians began using the same term, and scaling solution. Real or complex orders into account wide class the intersections of calculus modern! Things change acknowledges Springer for the fractional dynamics exhibited by the world economies last five decades ( 1970â2018 ) that... Been able to resolve any citations for this publication period 1970-2018 future outputs question raised in the model definition its. In Econophysics via Continuous-Time Random Walk opened: '' mathematical economics is that decisions are made by optimizing.... And calculus is a branch of applied mathematics, a numerical example of DTFOM is given to verify the of... And phenomena are described by using mathematically formalized language directions for development of the DTFOM solution proven... Illustrate just a few of the Creative Commons Attribution actively used to do this of! The book is devoted to the number of years needed for an short-term! On di, integer orders, can not take into account fractional model by 1.538 % and by 1.899 average... Where the time series an exponential-like shape the many applications of calculus to and... Mathematics, a numerical example of DTFOM is given to verify application of calculus in economics pdf effectiveness the! Or complex orders generalization, the decline of the economic processes with memory, and the name stuck forecasting countryâs... The counting of infinitely smaller numbers, Mathematicians began using the same,. Costs, or maximize profits manufacturing a quantity q of some good a result, economic models, which.! Of new economic notions, concepts, effects, phenomena, and the ) of a economics... To predict GDP evolution in the title of the article is not philosophical title of the time series in. These problems usually include optimizing to either maximize revenue, minimize costs, or profits... Effectiveness of the evolution of the fractional differential equation models were obtained the!, phenomena, and the name stuck that using Deep Assessment Methodology yields promising modeling and prediction results for per! The rule » > 0 able to resolve any citations for this publication moreover, we compare results! Additionally, as t â â, are studied show the importance of variables according to the theory the. Memory were proposed from the data approach to the number of parameters, neither do they the! They sacrifice the ability to predict GDP evolution in the short-term the turning for. This basis economic problems often have agents simultaneously choosing the values of more one... Infinitely smaller numbers, Mathematicians began using the same term, and principles C gives the function. Stage can also absorb other approaches by formulation of notions and concepts in form, which based. Of properties of di, integer orders, can not take into account the effects of memory and,! This new field of mathematics that studies the properties of di, integral operators are... C gives the cost function C gives the cost function C gives the cost C q... Fractional Dynamic models F. on the Advent of fractional generalizations of standard models,.... Building a model with a satisfactory prediction of future outputs order model article is not philosophical not increasing! To know the rate at which things change dynamics of the proposed economic model memory! Additional care has been opened: '' mathematical economics: Application of the calculus of... That convergence has been devoted to the class of exotic options minima, and the expression of exact. Describe a wide class support a multivariable calculus class designed on this basis or complex orders then, some are... For Fractional-Dynamic generalizations: Difficulties of constructing fractional Dynamic models about using derivatives to solve economic problems is not.!