Modelling by Differential Equations from Properties of Phenomenon to its Investigation

Transcription

1 Modelling by Differential Eqations from Properties of Phenomenon to its Investigation V. Kleiza and O. Prvinis Kanas University of Technology, Lithania Abstract The Panevezys camps of Kanas University of Technology comprises the Bsiness and Administration Faclty and the Faclty of Technology. They host stdents of bsiness administration, civil engineering, electrical engineering, management and mechanical technology. Stdents of all specialities need mathematical knowledge for the soltion of practical problems and for research from specific domains, sch as mechanics, control, techniqes, physics, economics etc. The paper discsses methodological isses of teaching ordinary differential eqations to the bachelor and master stdents from the Faclty of Technology and bachelor stdents from the Bsiness and Administration Faclty. Or experience of many years shows that stdents nderstand the theory of ordinary differential eqations better if it is illstrated with applications. For bilding p models we se mechanical and economical problems and solve the inverse problems for ordinary differential eqations as well. Or opinion is that inverse problems promote better nderstanding of the idea of modelling idea. Introdction The Panevėžys camps of Kanas University of Technology consists of the Faclty of Technology and the Bsiness and Administration Faclty. General higher mathematics topics are covered dring two terms, 3 hors for theory and or 3 hors for ttorials per week. It is possible to introdce first and second order ordinary differential eqations (ODE). The level of proficiency of different mathematical topics slightly differs among different specialities. Engineers of all specialties will need mathematical knowledge for the soltion of practical problems and for research in the fields of mechanics, control, techniqes, and physics. Stdents of bsiness and economics shold master skills that will be sefl in economics, statistics and econometrics. For stdents of both specialities we point ot that differential eqations are tools for solving real-world problems that have been discssed in lectres of other sbjects. Dring ttorials we firstly master finding soltions of particlar ODEs. After that it is easier to draw attention to the applications of the ODE. Generally, modelling provides a chance to develop practical skills of applying mathematics in particlar domains (Gershenfeld, 1998).

2 To bild a model, stdent mst make many decisions. Therefore it is convenient to divide the whole modelling process into separate phases (Shier & Wallenis, 1999). We sally follow five steps when demonstrating investigations of applied problems by means of differential eqations. 1. Analysis of the real-world problem. Usally it is a problem that stdents have already considered in engineering or economics lectres. This involves evalation of the major known properties of phenomenon and the formlation of what properties shold be investigated by modelling sing an ODE as a tool.. Set p a mathematical model, i.e. write down a differential eqation of the model. Or opinion is that in this step it is important to point ot how the differential eqation describes the known properties discssed in the first step and what fndamental laws of natre and economics shold be taken into accont (Самарский, 001). Attention shold be drawn to the fact that the ODE is an expression that incldes first or higher order derivatives. Therefore it represents the rate of change of the properties of the system or phenomenon nder discssion. This is a good point to review the mechanical and economic meaning of the first and second order derivatives. 3. Solve the ODE. Here one can often bt not always focs solely mathematical knowledge. 4. Analysis of the soltion of the ODE. This step generally remains mathematically oriented and it incldes analysis of the soltion s closed-form expression if it is available. The analysis of the soltion may inclde a discssion of the expression, evalation its analytical properties, pointing ot how the soltion depends on bondary or initial vales, plotting the graph etc. 5. Reformlate the mathematically discovered characteristics of the soltion into engineering or economical langage. This is a good place to compare the new properties and knowledge abot the problem revealed by modelling with already known facts, to predict the behavior of the device or economical process and to check if it is what the engineers expected (Dym, 004). This step also contribtes to the validation of the model. Practical examples For example we consider the determination of the expression for demand for some good, depending on income, when the elasticity is constant. 1. Analysis of the problem. This problem comes from the microeconomics. Constant elasticity fnctions are often discssed in economic theory and they are considered as simple fnctions. Bt the analytical expression is not always given. So it is interesting to find this expression and to investigate properties of it from the mathematical viewpoint.. The definition of elasticity of demand (D) with respect to income x is D x D and the reqirement to be constant yields ( D x D ) = 0.

3 3. This is a second order ODE and there are no difficlties in showing c1 its soltion is D= cx with arbitrary constants c 1 and c. 4. When performing the mathematical analysis of the fnction D, it is D x D = c. Therefore graphs of the demand fnction D essential to point ot that ( ) 1 with varios elasticity c 1 vales inclding c 1 >1 and c 1 <1 shold be plotted. 5. The main reslt is that the demand of constant elasticity is a power fnction. The demand D has netral elasticity, i. e, when c 1 =1, only when D=c x and in this case the demand is proportional to income x. Bt the linear fnction of general form y=kx+b can not have the constant elasticity property. Instrctors of applied mathematics are typically more concerned with direct problems. In real life, the problem of interest to engineers and scientists is often an inverse problem. For instance, in many practical sitations it is necessary to find coefficients of the ODE, parameters of the sorce term or parameters of the bondary vales. The engineer seeks to determine these nknown parameters from collected measrements or other accessible knowledge abot the soltion of the given ODE. Or experience shows that stdents nderstand the theory of ODE better if this theory is also illstrated with inverse ODE problems. In addition, considering inverse problems also promotes better comprehension of the essence of ODE models. The theory of inverse ODE problems nowadays is well developed, and has wide applications de to the accessibility of powerfl compters. When the edcator introdces sch problems, the major difficlty that stdents face is that of case and effect, i.e. the direct problem and the inverse problem switch places with one another. Therefore the edcator mst pay special attention to this pecliarity and explain it. As an example, we consider one problem that reveals methodological aspects of soltion of the inverse problem. We start from the bondary vale problem that describes the transversal deflection ( x ) of the heterogeneos beam when it is spported at points x= 0 and x= L. d M = ( ), F x dx M (0) = M ( L) = 0, d D( x) = M ( x), dx (0) = ( L) = 0. (1) where F( x ) is a distribted load, M ( x ) is bending moment acting in the beam cross section, D( x) = E( x) I( x) is flexral rigidity of beam, E( x ) is Yong s modls of the beam material, I( x ) is the second moment of the cross section area. The first two eqations of (1) have analytical soltion

4 x v x L v, () M ( x) = F( t) dtdv x F( t) dtdv and provided that D( x ) > 0, we obtain the soltion of problem (1) x v x L v M ( t) M ( t) ( x)= dtdv x dtdv D( t) D( t). (3) Hence we have redced the problem (1) into bondary-vale problem for the linear non-homogeneos differential eqation d P( x), (0) ( L) 0 dx = = = (4) where P( x) = M ( x) D( x). Assming ( x ) and 1 ( ) x are linearly independent soltions of the homogeneos problem. Then the Green s fnction of the problem (4) is given by expression where W ( s) ( x) ( s) ( s) ( x) [ ] W ( s) W ( s) G 1 1 ( x, s ) = 1 H ( x s) + H ( x s) (5) = 1 is the Wronskian and ( ) 1 Then the soltion of the problem (1) is H x is Heaviside s step fnction. L ( x) = G( x, s) P( s) ds (6) 0 Now we shall solve the inverse problem, i.e. we seek P( x ) for given ( x), F( x), x (0, L). Notice that fnction M ( x ) becomes known when the problem () is solved. Hence one can derive the bending stiffness M ( x) D( x) = (7) P( x) and then, assming that the second moment of the cross section area I( x ) is known, one can also derive Yong s modls distribtion D( x) E( x) =. (8) I( x)

5 The problem (6) can be solved by applying nmerical qadratre formla. The interval (0, L) is divided by points s = il N, i= 0, N into N eqal sbranges S = ( s, s ), i= 0, N 1 and then i i i+ 1 where ( ) L N 1 si+ 1 N 1 si+ 1 0 i i= 0 s i= 0 i ( j) G( x, s) p( s) ds= G( x, s) P( s) ds P s G( s j, s) ds, si = si+ s i + 1 are middle points of intervals S i. If i = ( si ) are measrements then the soltion of the inverse problem (6) is the soltion of linear system si where si+ 1 A G( s, s) ds ij =, si j AP T U 1 N = (,,..., ), = U, (10) T P p1 p pn pi p si = (,,..., ), = ( ). Now the edcator can consider qite important real-world engineering problems. Beams made of materials like concrete, show degradation of flexral stiffness dring their service life de to mechanical and environmental loadings. Using the method otlined above, we can investigate how the material parameters (Yong s modls and flexral stiffness) change. Finally, we note that, from the expression of the ODE it is sometimes possible to draw conclsions abot which real-world problems and systems this ODE can serve as a model. For instance, when covering the first order differential eqations, we consider the eqation y =k 1 y-k y. Edcators can draw attention that the eqation models general process changes that are inflenced by two factors corresponding to terms k 1 y and k y. The first term increases the changes while the second contribtes to the decrease. Conclsions 1. Modelling skills shold be taght by practically bilding models in concrete domains inclding data acqisition from these domains. Models become more meaningfl when stdents collect their own data.. The backgrond of sccessfl modelling with differential eqations is the knowledge of the meaning of derivatives. 3. The consideration of inverse ODE problems contribtes to better nderstanding of the essence of modelling by differential eqations.

6 4. When a particlar differential eqation is given it may be sefl to discss what processes and what relationships between properties of the process it can describe. 5. Modelling teaches stdents to apply mathematical concepts to solve real problems. References Dym, Clive L. (004) Principles of mathematical modeling. Academic Press. Gershenfeld, Neil A. (1998) The natre of mathematical modeling. Cambridge University Press. Shier, Doglas R. and Wallenis K. T. (1999) Applied mathematical modeling: a mltidisciplinary approach. CRC. Самарский А. А., Михайлов А. П. (001) Математическое моделирование: Идеи. Методы. Примеры. Физматлит.