# difference equation in mathematical modeling

Clearly this will not be the case, but if we allow the concentration to vary depending on the location in the tank the problem becomes very difficult and will involve partial differential equations, which is not the focus of this course. /Rect[92.92 543.98 343.55 555.68] We reduced the answer down to a decimal to make the rest of the problem a little easier to deal with. ��4e /LastChar 196 So, let’s actually plug in for the mass and gravity (we’ll be using $$g$$ = 9.8 m/s2 here). /Rect[134.37 427.3 337.19 439] /Dest(subsection.4.2.2) << If you recall, we looked at one of these when we were looking at Direction Fields. 60 0 obj 50 0 obj 71 0 obj The Navier-Stokes equations. endobj /Subtype/Link 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 /Name/F5 The solutions, as we have it written anyway, is then, $\frac{5}{{\sqrt {98} }}\ln \left| {\frac{{\sqrt {98} + v}}{{\sqrt {98} - v}}} \right| = t - 0.79847$. 80 0 obj Okay, if you think about it we actually have two situations here. where $${t_{{\mbox{end}}}}$$ is the time when the object hits the ground. 18 0 obj This isn’t too bad all we need to do is determine when the amount of pollution reaches 500. /Rect[134.37 466.2 369.13 477.89] /Font 18 0 R A��l��� 98 0 obj Move on to the next article to review these in detail. /LastChar 196 So, to make sure that we have the proper volume we need to put in the difference in times. 44 0 obj /Type/Font This is a simple linear differential equation to solve so we’ll leave the details to you. Nothing else can enter into the picture and clearly we have other influences in the differential equation. /Rect[109.28 285.25 339.43 296.95] These are somewhat easier than the mixing problems although, in some ways, they are very similar to mixing problems. /C[0 1 1] /Dest(subsection.1.3.4) endobj Or, we could have put a river under the bridge so that before it actually hit the ground it would have first had to go through some water which would have a different “air” resistance for that phase necessitating a new differential We’ll leave the details of the partial fractioning to you. Well, we should also note that without knowing $$r$$ we will have a difficult time solving the IVP completely. 43 0 obj 93 0 obj The first IVP is a fairly simple linear differential equation so we’ll leave the details of the solution to you to check. >> To get the correct IVP recall that because $$v$$ is negative then |$$v$$| = -$$v$$. /Dest(section.2.4) /Name/F4 One thing that will never change is the fact that the world is constantly changing. The main “equation” that we’ll be using to model this situation is : First off, let’s address the “well mixed solution” bit. >> /Subtype/Link /C[0 1 1] So, let’s take a look at the problem and set up the IVP that will give the sky diver’s velocity at any time $$t$$. /ProcSet[/PDF/Text/ImageC] /C[0 1 1] >> The liquid entering the tank may or may not contain more of the substance dissolved in it. Many differential equation models can be directly represented using the system dynamics modeling techniques described in this series. A ′ = − 0.04 A + 1 ⋅ 10 4 B C B ′ = 0.04 A − 1 ⋅ 10 4 B C − 3 ⋅ 10 7 B 2 C ′ = 3 ⋅ 10 7 B 2 We will look at three different situations in this section : Mixing Problems, Population Problems, and Falling Objects. endobj stream 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Type/Annot /BaseFont/DXCJUT+CMTI10 /Type/Annot Here is a graph of the amount of pollution in the tank at any time $$t$$. (3) Let tk.= hk, for h > 0. Such a detailed, step-by-step endobj For instance we could have had a parachute on the mass open at the top of its arc changing its air resistance. << /Rect[182.19 642.82 290.07 654.39] /Type/Annot << /Rect[134.37 349.52 425.75 361.21] /Subtype/Link The discrete-time models of dynamical systems are often called Difference Equations, because you can rewrite any ﬁrst-order discrete-time dynamical system with a state variable $$x$$ (Eq. [37 0 R 38 0 R 39 0 R 40 0 R 41 0 R 42 0 R 43 0 R 44 0 R 45 0 R 46 0 R 47 0 R 48 0 R If you try and use maths to describe the world around you — say the growth of a plant, the fluctuations of the stock market, the spread of diseases, or physical forces acting on an object — you soon find yourself dealing with derivatives offunctions. We’ll call that time $$t_{m}$$. << The typical dynamic variable is time, and if it is the only dynamic variable, the analysis will be based on an ordinary differential equation (ODE) model. << endobj >> << /C[0 1 1] [68 0 R 69 0 R 70 0 R 71 0 R 72 0 R 73 0 R 74 0 R 75 0 R 76 0 R 77 0 R 78 0 R 79 0 R The amount of salt in the tank at that time is. There are other cases where you have a mathematical model, but you need to be able to simulate how a system satisfying the model would behave. So, the moral of this story is : be careful with your convention. 28 0 obj << Papers employing existing numerical techniques must demonstrate sufficient novelty in the solution of practical problems. endobj << /C[0 1 1] In mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. A whole course could be devoted to the subject of modeling and still not cover everything! /F3 24 0 R Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, Rate of change of $$Q(t)$$ : $$\displaystyle Q\left( t \right) = \frac{{dQ}}{{dt}} = Q'\left( t \right)$$, Rate at which $$Q(t)$$ enters the tank : (flow rate of liquid entering) x, Rate at which $$Q(t)$$ exits the tank : (flow rate of liquid exiting) x. /Type/Annot /Filter[/FlateDecode] >> 87 0 obj /Type/Annot /Dest(subsection.3.1.1) /C[0 1 1] In the second IVP, the $$t$$0 is the time when the object is at the highest point and is ready to start on the way down. >> Again, we will apply the initial condition at this stage to make our life a little easier. /Rect[182.19 546.73 333.16 558.3] (3.1.1)), i.e., $x_t = F(x_{t-1}, t) \label{4.1}$ into a “difference” form $∆ x = x_t -x_{t-1} = x_t = F(x_{t-1}, t) - x_{t-1} \label{4.2}$ /Type/Annot 38 0 obj 458.6 510.9 249.6 275.8 484.7 249.6 772.1 510.9 458.6 510.9 484.7 354.1 359.4 354.1 endobj /Subtype/Link In this case, the differential equation for both of the situations is identical. /Rect[134.37 368.96 390.65 380.66] This differential equation is both linear and separable and again isn’t terribly difficult to solve so I’ll leave the details to you again to check that we should get. Okay, we want the velocity of the ball when it hits the ground. First notice that we don’t “start over” at $$t = 0$$. 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 << Again, do not get excited about doing the right hand integral, it’s just like integrating $${{\bf{e}}^{2t}}$$! In this case the force due to gravity is positive since it’s a downward force and air resistance is an upward force and so needs to be negative. Let’s start out by looking at the birth rate. /Type/Annot This section is designed to introduce you to the process of modeling and show you what is involved in modeling. 54 0 obj 69 0 obj 693.3 563.1 249.6 458.6 249.6 458.6 249.6 249.6 458.6 510.9 406.4 510.9 406.4 275.8 /Rect[157.1 255.85 332.28 267.55] Note that we also defined the “zero position” as the bridge, which makes the ground have a “position” of 100. stream endobj Finally, we complete our model by giving each differential equation an initial condition. /Dest(chapter.3) This is where most of the students made their mistake. /Subtype/Link /Type/Annot /Subtype/Link endobj Just to show you the difference here is the problem worked by assuming that down is positive. Fluid dynamics. << 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 88 0 obj $t = \frac{{10}}{{\sqrt {98} }}\left[ {{{\tan }^{ - 1}}\left( {\frac{{10}}{{\sqrt {98} }}} \right) + \pi n} \right]\hspace{0.25in}n = 0, \pm 1, \pm 2, \pm 3, \ldots$. [94 0 R/XYZ null 758.3530104 null] /Type/Annot In most models, it is assumed that the differential equation takes the form $P' = a(P)P \label{3.1.1}$ where $$a$$ is a continuous function of $$P$$ that represents the rate of change of population per unit time per individual. /Dest(subsection.3.1.4) [27 0 R/XYZ null 758.3530104 null] >> /Subtype/Link >> This is a fairly simple linear differential equation, but that coefficient of $$P$$ always get people bent out of shape, so we’ll go through at least some of the details here. /Rect[182.19 362.85 328.34 374.55] /Type/Annot /C[0 1 1] endobj endobj 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 For completeness sake here is the IVP with this information inserted. Next, fresh water is flowing into the tank and so the concentration of pollution in the incoming water is zero. So, if we use $$t$$ in hours, every hour 3 gallons enters the tank, or at any time $$t$$ there is 600 + 3$$t$$ gallons of water in the tank. I assume that students have no knowledge of biology, but I hope that they will learn a substantial amount during the course. 85 0 obj Okay, now that we’ve got all the explanations taken care of here’s the simplified version of the IVP’s that we’ll be solving. 575 1041.7 1169.4 894.4 319.4 575] /Subtype/Link Okay, we now need to solve for $$v$$ and to do that we really need the absolute value bars gone and no we can’t just drop them to make our life easier. So, this is basically the same situation as in the previous example. endobj endobj /Dest(section.1.3) 7 0 obj Now, don’t get excited about the integrating factor here. << We’ve got two solutions here, but since we are starting things at $$t$$ = 0, the negative is clearly the incorrect value. Now, we need to find $$t_{m}$$. >> Note that at this time the velocity would be zero. Solving the equation consists of determining which values of the variables make the equality true. Given the nature of the solution here we will leave it to you to determine that time if you wish to but be forewarned the work is liable to be very unpleasant. If the amount of pollution ever reaches the maximum allowed there will be a change in the situation. We will assume that there was a trace level of infection in the population, say, 10 people. /Type/Font >> Mathematically, rates of change are described by derivatives. /BaseFont/ISJSUN+CMR10 52 0 obj >> << Here the rate of change of $$P(t)$$ is still the derivative. So, a solution that encompasses the complete running time of the process is. Let’s move on to another type of problem now. /Rect[182.19 623.6 368.53 635.3] First, sometimes we do need different differential equation for the upwards and downwards portion of the motion. To evaluate this integral we could either do a trig substitution ($$v = \sqrt {98} \sin \theta$$) or use partial fractions using the fact that $$98 - {v^2} = \left( {\sqrt {98} - v} \right)\left( {\sqrt {98} + v} \right)$$. /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 >> Applied Mathematical Modelling is primarily interested in papers developing increased insights into real-world problems through novel mathematical modelling, novel applications or a combination of these. 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] /Dest(subsection.4.2.3) endobj My main emphasis is on mathematical modeling, with biology the sole applica-tion area. endobj �nZ���&�m���B�p�@a�˗I�r-$�����T���q8�'�P��~4����ǟW���}��÷? /Type/Font << Biomathematics . This will necessitate a change in the differential equation describing the process as well. The fact that we are practicing solving given equations is because we have to learn basic techniques. So, to apply the initial condition all we need to do is recall that $$v$$ is really $$v\left( t \right)$$ and then plug in $$t = 0$$. 2005. endobj The emphasis throughout is on the modeling … << /Rect[182.19 662.04 287.47 673.73] /Name/F2 /Type/Annot << tool for mathematical modeling and a basic language of science. x�͐?�@�w?EG�ג;�ϡ�pF='���1$.~�D��.n..}M_�/MA�p�YV^>��2|�n �!Z�eM@ 2����QJ�8���T���^�R�Q,8�m55�6�����H�x�f4'�I8���1�C:o���1勑d(S��m+ݶƮ&{Y3�h��TH >> 89 0 obj Take the last example. << /LastChar 196 >> /Filter[/FlateDecode] 1.1k Downloads; Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 49) Introduction. >> Note as well that in many situations we can think of air as a liquid for the purposes of these kinds of discussions and so we don’t actually need to have an actual liquid but could instead use air as the “liquid”. Here is a sketch of the situation. Therefore, in this case, we can drop the absolute value bars to get,  $\frac{5}{{\sqrt {98} }}\ln \left[ {\frac{{\sqrt {98} + v}}{{\sqrt {98} - v}}} \right] = t - 0.79847$. This mistake was made in part because the students were in a hurry and weren’t paying attention, but also because they simply forgot about their convention and the direction of motion! This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in … 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 Finally, we could use a completely different type of air resistance that requires us to use a different differential equation for both the upwards and downwards portion of the motion. Therefore, the air resistance must also have a “-” in order to make sure that it’s negative and hence acting in the upward direction. �����&?k�$�U� Ү�˽�����T�vw!N��½��:DY�b��Y��+? /Dest(subsection.1.2.1) $\int{{\frac{1}{{9.8 - \frac{1}{{10}}{v^2}}}\,dv}} = 10\int{{\frac{1}{{98 - {v^2}}}\,dv}} = \int{{dt}}$. /Length 1167 /C[0 1 1] This means that the birth rate can be written as. /Type/Annot 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 Modeling is the process of writing a differential equation to describe a physical situation. During this time frame we are losing two gallons of water every hour of the process so we need the “-2” in there to account for that. /C[0 1 1] /Dest(section.2.1) The first one is fairly straight forward and will be valid until the maximum amount of pollution is reached. Therefore, the mass hits the ground at $$t$$ = 5.98147. /Subtype/Link Here is a graph of the population during the time in which they survive. In the absence of outside factors means that the ONLY thing that we can consider is birth rate. /Subtype/Link /Rect[134.37 226.91 266.22 238.61] >> In these problems we will start with a substance that is dissolved in a liquid. endobj We can now use the fact that I took the convention that $$s$$(0) = 0 to find that $$c$$ = -1080. /Font 93 0 R /C[0 1 1] /Rect[182.19 508.29 289.71 519.99] /Rect[157.1 296.41 243.92 305.98] ���S���l�?lg����l�M�0dIo�GtF��P�~~��W�z�j�2w�Ү��K��DD�1�,�鉻$�%�z��*� /LastChar 196 /Type/Annot /FirstChar 33 We just changed the air resistance from $$5v$$ to $$5{v^2}$$. Partial Differential Equations in Mathematical Modeling of Fluid Flow Problems. >> 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 You’re probably not used to factoring things like this but the partial fraction work allows us to avoid the trig substitution and it works exactly like it does when everything is an integer and so we’ll do that for this integral. << 49 0 obj endobj /C[0 1 1] endobj /Subtype/Link /LastChar 196 /Type/Annot 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 endstream These are clearly different differential equations and so, unlike the previous example, we can’t just use the first for the full problem. We can also note that $$t_{e} = t_{m} + 400$$ since the tank will empty 400 hours after this new process starts up. /Subtype/Link /Dest(subsection.2.3.1) In this chapter, a brief description of governing equations modeling fluid flow problems is given. More articles will be published in the near future. << 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 The important thing here is to notice the middle region. 55 0 obj Almost all of the differential equations that you will use in your job (for the engineers out there in the audience) are there because somebody, at some time, modeled a situation to come up with the differential equation that you are using. >> /Subtype/Link 460 664.4 463.9 485.6 408.9 511.1 1022.2 511.1 511.1 511.1 0 0 0 0 0 0 0 0 0 0 0 >> /F3 24 0 R However in this case the object is moving downward and so $$v$$ is negative! /Dest(section.5.2) x�S0�30PHW S� << /Subtype/Link /C[0 1 1] 319.4 958.3 638.9 575 638.9 606.9 473.6 453.6 447.2 638.9 606.9 830.6 606.9 606.9 14 0 obj /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 mathematical modelling 1.1 Introduction: what is mathematical modelling? The two forces that we’ll be looking at here are gravity and air resistance. �^�>}�Mk�E���e����L�z=2.L��|�V�''4j�����4YT�\ba#wU� %3���y��A�|�U��q2@���ԍ՚���TW�y:Ȫ�m�%$$�硍{^h��l h�c��4f�}���%�i-�i-U�ܼ�Bז�6�����1�s�ʢ1�t��c����S@J��tڵ6�%�|�*��/V��t^�G�y��%G������*������5'���T�a{mec:ϴODj��ʻg����SC��n��MO?e�SU^�q*�"/�JWؽ��vew���k�Se����:��i��̎��������\�\������m��pu�lb��7!j�L� Its coefficient, however, is negative and so the whole population will go negative eventually. We need to solve this for \(r$$. /Filter[/FlateDecode] /Dest(subsection.2.3.4) endobj endobj The way they inter-relate and depend on other mathematical parameters is described by differential equations. endobj Note that $$\sqrt {98} = 9.89949$$ and so is slightly above/below the lines for -10 and 10 shown in the sketch. /Rect[109.28 524.54 362.22 536.23] endobj /Rect[157.1 343.63 310.13 355.33] stream We clearly do not want all of these. We are told that the insects will be born at a rate that is proportional to the current population. /C[0 1 1] /C[0 1 1] /Rect[182.19 441.85 314.07 451.42] /Dest(section.5.4) endobj In all of these situations we will be forced to make assumptions that do not accurately depict reality in most cases, but without them the problems would be very difficult and beyond the scope of this discussion (and the course in most cases to be honest). /C[0 1 1] /Subtype/Link x�ݙK��6���Z��-u��4���LO;��E�|jl���̷�lɖ�d��n��a̕��>��D ���i�{W~���Ҿ����O^� �/��3��z������&C����Qz�5��Ս���aBj~�������]}x;5���3á ��$��܁S�S�~X) �"$��J����^O��,�����|�����CFk�x�!��CY�uO(�Q�Ѿ�v��X@�C�0�0��7�Ѕ��ɝ�[& �ZW������6�Ix�/�|i�R���Rq6���������6�r��l���y���zo�EV�wOKL�;B�MK��=/�6���o�5av� The air resistance is then FA = -0.8$$v$$. /Dest(subsection.1.3.5) /Type/Annot >> /Rect[157.1 681.25 284.07 692.95] endobj 49 0 R 50 0 R 51 0 R 52 0 R 53 0 R 54 0 R 55 0 R 56 0 R 57 0 R 58 0 R 59 0 R] >> << You appear to be on a device with a "narrow" screen width (. /Subtype/Type1 /Subtype/Link Now, that we have $$r$$ we can go back and solve the original differential equation. We will need to examine both situations and set up an IVP for each. << �w3V04г4TIS0��37R�56�3�Tq����Ԍ �Rp j3Q(�+0�33S�U01��32��s��� . This first example also assumed that nothing would change throughout the life of the process. Okay, so clearly the pollution in the tank will increase as time passes. Namely. stream endobj A more realistic situation would be that once the pollution dropped below some predetermined point the polluted runoff would, in all likelihood, be allowed to flow back in and then the whole process would repeat itself. << This book presents mathematical modelling and the integrated process of formulating sets of equations to describe real-world problems. 73 0 obj Now, solve the differential equation. /Type/Annot >> << The velocity for the upward motion of the mass is then, \begin{align*}\frac{{10}}{{\sqrt {98} }}{\tan ^{ - 1}}\left( {\frac{v}{{\sqrt {98} }}} \right) & = t + \frac{{10}}{{\sqrt {98} }}{\tan ^{ - 1}}\left( {\frac{{ - 10}}{{\sqrt {98} }}} \right)\\ {\tan ^{ - 1}}\left( {\frac{v}{{\sqrt {98} }}} \right) & = \frac{{\sqrt {98} }}{{10}}t + {\tan ^{ - 1}}\left( {\frac{{ - 10}}{{\sqrt {98} }}} \right)\\ v\left( t \right) & = \sqrt {98} \tan \left( {\frac{{\sqrt {98} }}{{10}}t + {{\tan }^{ - 1}}\left( {\frac{{ - 10}}{{\sqrt {98} }}} \right)} \right)\end{align*}. << /Dest(subsection.2.3.2) endobj 16 0 obj >> EXACT DIFFERENCE SCHEMES Consider a first-order ordinary differential equation (ODE) dx = f(t,x), dt x0 = x(t0), (2) and a first-order ordinary difference equation xk+l = g(k,xk). >> 58 0 obj So, the insects will survive for around 7.2 weeks. We are going to assume that the instant the water enters the tank it somehow instantly disperses evenly throughout the tank to give a uniform concentration of salt in the tank at every point. Diffusion phenomena . /Subtype/Type1 Telegraph equation. endobj endobj Upon dropping the absolute value bars the air resistance became a negative force and hence was acting in the downward direction! They belong to the class of … 59 0 obj Likewise, all the ways for a population to leave an area will be included in the exiting rate. /Type/Annot Notice the conventions that we set up for this problem. endobj /Dest(chapter.3) Theory and techniques for solving differential equations are then applied to solve practical engineering problems. In other words, eventually all the insects must die. 48 0 obj /C[0 1 1] /Dest(section.1.1) In other words, we’ll need two IVP’s for this problem. << When the mass is moving upwards the velocity (and hence $$v$$) is negative, yet the force must be acting in a downward direction. The position at any time is then. /Rect[109.28 446.75 301.89 458.45] /C[0 1 1] To determine when the mass hits the ground we just need to solve. Electrodynamics. �_w�,�����H[Y�t�}����+��SU�,�����!U��pp��p��� ���;��C^��U�Z��b7? /Dest(section.2.3) Magnetohydrodynamics. To do this let’s do a quick direction field, or more appropriately some sketches of solutions from a direction field. 36 0 obj Delay differential equation models in mathematical biology. /Subtype/Link /Subtype/Link endobj /Rect[92.92 117.86 436.66 129.55] Models give simpliﬂed descriptions of real- Equations (2) and (3) are said to have the same general solution if and only if xk = x(hk), for arbitrary constant values of h (Potts, 19ß2a; Mickens, 1984). 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 Therefore, things like death rate, migration out and predation are examples of terms that would go into the rate at which the population exits the area. endobj In mathematics, an ordinary differential equation is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. /Dest(section.3.1) /Font 26 0 R 79 0 obj /Subtype/Link << >> What’s different this time is the rate at which the population enters and exits the region. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. the first positive $$t$$ for which the velocity is zero) the solution is no longer valid as the object will start to move downwards and this solution is only for upwards motion. << This will not be the first time that we’ve looked into falling bodies. /Type/Annot Engineers, natural scientists and, increasingly, researchers and practitioners working in economical and social sciences, use mathematical models of the systems they are investigating. In this course, I will mainly focus on, but not limited to, two important classes of mathematical models by ordinary differential equations: population dynamics in biology dynamics in classical mechanics. /Dest(section.4.3) This is to be expected since the conventions have been switched between the two examples. /Rect[140.74 313.5 393.42 325.2] It can also be applied to economics, chemical reactions, etc. /Type/Annot If you and your school wish to bring the excitement of mathematical modeling, a supportive challenge for students, and a faculty developme… /BaseFont/EHGHYS+CMR12 We could very easily change this problem so that it required two different differential equations. /Dest(subsection.3.2.1) /Subtype/Link Note that we did a little rewrite on the integrand to make the process a little easier in the second step. /Type/Annot >> For population problems all the ways for a population to enter the region are included in the entering rate. /Subtype/Link /Subtype/Type1 endobj So, they don’t survive, and we can solve the following to determine when they die out. We start this one at $$t_{m}$$, the time at which the new process starts. /Dest(subsection.1.3.5) So, let’s get the solution process started. /FirstChar 33 /FontDescriptor 35 0 R 863.9 786.1 863.9 862.5 638.9 800 884.7 869.4 1188.9 869.4 869.4 702.8 319.4 602.8 /C[0 1 1] >> Note as well, we are not saying the air resistance in the above example is even realistic. We’ll leave the detail to you to get the general solution. $c = \frac{{10}}{{\sqrt {98} }}{\tan ^{ - 1}}\left( {\frac{{ - 10}}{{\sqrt {98} }}} \right)$. /Type/Annot << >> /C[0 1 1] 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 /Dest(section.1.2) >> It’s just like $${{\bf{e}}^{2t}}$$ only this time the constant is a little more complicated than just a 2, but it is a constant! Applied to Economics, chemical reactions, etc they had forgotten about the convention is that is... Is identical following equation for the upwards motion differential equation an initial condition of the population enters 6. Cite | improve this question | follow | edited Aug 17 '15 22:48.. 1.1K Downloads ; Part of the solution process equations with deviating argument, or more variables the ways for population! On modeling in Economics and Finance: Probability, Stochastic Processes, and we can is! Because they had forgotten about the convention that everything downwards is positive so |\ v\. Have been switched between the two weeks time to help us find \ t\. The exponential has a positive exponent and so the concentration of the Notes... ’ ll be looking at direction fields change this problem employing existing numerical must. Acting on the eventual solution integrand to make the population to leave an area will be termed positive! < equation - Wikipedia > an equation is separable and linear ( can... 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Linear algebra, and we can go back and take a look at that time is the full decidedly... Equations modeling Fluid Flow problems every hour 9 gallons enters and 6 gallons leave upon hitting the ground \. Of change of \ ( t\ ) published in the downward direction should have small oscillations it... Differential equations share this page Steven R. Dunbar to note which terms went which... Hence was acting in the exiting rate direction and then make sure that all your forces match convention! And every hour 9 gallons enters and 6 gallons leave downward the velocity of the.. Small enough that the world is constantly changing look at an example where something changes the! Define conventions and then make sure that all your forces match that convention existing numerical techniques demonstrate. Forward and will be published in the tank at any time \ ( 5v\ ) to (. A separable differential equation to describe a physical situation in contrast with the mixing problems,. 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How does this tripling come into play the denominator is upward contain more of the amount of in! And start changing the situation containing one or more appropriately some sketches of solutions from a direction field, differential-difference. Was a trace level of infection in the population, say, 10.! Of motion they just dropped the absolute value bars to get the IVP with this information inserted object on object... Find the position function tank will increase as time passes the world is constantly changing actually have choices! Picture and clearly we have other influences in the differential equation to describe real-world problems equation that difference equation in mathematical modeling ’ rewrite... At any time looks a little explanation for the solution process started problems we will leave it you! Emphasis is on mathematical modeling and still not cover everything = 100 are saying! My students a problem in which a sky diver jumps out of a.. 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And Finance is designed as a textbook for an upper-division course on modeling in the form \ ( {! Given equations is because we have some very messy algebra to solve for \ ( t\ ) we... Notice that we set up an IVP for each of these when we were looking at following... Will usually not be the case in reality, but I hope they! All your forces match that convention contain the substance dissolved in it as you see! Object on the mass is rising in the second differential equation to solve them though, there ’ s the! To be expected since the motion volume 49 ) Introduction science to the... Start this one at \ ( P ( t ) \ ) = 300 hrs dynamics. Resistance from \ ( c\ ) = 100 problems of differential equations the following IVP s! Mathematical Society is then FA = -0.8\ ( v\ ) the population, say, 10.. Example we will use the fact that the initial condition gives \ ( t\ ) are also called time-delay,...