This is an exam that I will get at 9am UK time. So I will send the questions then. The attached files are the past paper and it's solution
1 Faculty of Science, Engineering and Computing Undergraduate Regulations April/May Examinations or Assessments 2020/2021 Level 6 MODULE: MA6500: Advanced Mathematical Methods and Models DURATION:Three Hours Instructions to Candidates This paper contains FOUR questions Answer THREE questions only All questions carry equal marks Candidates are reminded that the major steps in all calculations are to be set out clearly. Number of Pages: 1 – 7 General instructions 1. This examination/assessment was designed to be completed within the duration specified on the front cover. However, Canvas will remain open for a further two hours to allow additional time for any technical difficulties. 2. If you experience technical difficulties, eg. access and upload issues, or identify a potential error in a question please email the module leader who will be available throughout the exam. 3. You must not collaborate with anyone on this exam/assessment, it should be wholly your own work. Your work will be checked for evidence of plagiarism and/or collusion using Turnitin. Submission Guidance 1. You should submit your answers as a single word document via Canvas. Add your ID to the top of each page and indicate the questions you have answered on the first page of your document. 2. If you include graphics in your answer, these should be embedded into the word document (eg. a photograph of a hand-drawn graphic). The source of any copied and pasted figures should be cited. 3. Please make sure to save your work regularly and leave plenty of time to upload your work before the deadline. Late submissions will not be marked. Continued… 1.On a remote island, environmentalists are studying the population numbers of a rare species of mammals whose natural birth rate and natural death rate are such that . This investigation has been prompted by the outbreak amongst the animals there of a disease associated with the Perona virus. The number of the mammals with the disease at any time is denoted by . Some of the animals in the group may be symptomless and their natural lifespan is unaffected. However, the disease can be fatal in other cases, resulting in the death rate and the scientists have estimated that . If represents the number of the animals free from the disease (healthy) at any time, and is a positive constant, then the system of ODEs modelling the scenario is : (a)Draw a compartmental diagram representing this system for the two classes of the population, and , ensuring that your flow rates arrows have correctly directed heads and are labelled with the appropriate flow rates of animals. (3 marks) (b)Find the co-ordinates of the two critical points of the above system (explain your major steps and number important equations for clarity). Show that one of these points is in the positive quadrant of the plane. (8 marks) (c)Linearise the system by obtaining the Jacobian, in terms of the model’s variables and parameters. (4 marks) (d)Classify these two critical points in terms of type and stability. Note that for one of these points it will not be possible to state conclusively the nature of the point from the given information, so be sure to find and to state what the alternative types are for this point, along with the relationship between the parameters needed to be satisfied for these to be definitely identified. (See overleaf.) (16 marks) Question 1 continued on page 3… Continuation of Question 1: [For part (d) you may wish to refer to refer to the diagram opposite, as presented in the lecture slides, where and ] (e)Assuming that suitable units had been chosen when this model was formulated, what can you deduce about the nature and location in the plane of the critical points given that: State also what represents in the model (2 marks) Continued… 2.(a)Assuming that and are all positive quantities independent of , use the definition of the Laplace transform of a function , to show that where is the unit step function. (6 marks) (b)Consider the boundary problem for the function , (i)Take the Laplace transform of the PDE to show that where . Note that you may quote results from the KU Tables for this and future transforms in this question, meaning there is no need to use the integral definition. (6 marks) (ii)Solve the ODE (1) above for and use the transform of the boundary condition to determine the constant of integration. (14 marks) (iii)Finally, using KU Tables, take the inverse transform of and rewrite the unit step function it involves to show that the solution to the PDE is (7 marks) Continued… 3.(a)(i)Let Show (include details of all major steps in your working) that the (half-range) Fourier Sine Series representation of this function is given by: (6 marks) (ii)Suppose that a full-range Fourier Series representing is required. Without attempting to evaluate the series state very briefly how this series would differ qualitatively from the series given above in part (i). (2 marks) (b)Now consider the following partial differential equation: with: Use the method of separation of variables to obtain a series solution for the above problem, giving details of all the steps in your calculations. You may assume that the solution consists of a steady state solution and a transient solution , where and its partial derivatives tend to zero as . You may also assume that in any separation of variables the separation constant is negative. You may quote results from part (a) if/where appropriate. (22 marks) By considering appropriate terms in your solution obtain the value of accurate to five significant figures (3 marks) Continued… 4.(a)Consider the first order (quasi-)linear partial differential equation (PDE) given by With boundary condition, . (i) Write down the differential equations that govern the characteristics of the equation above. (2 marks) (ii)Carefully obtain the general solution to the original PDE. (5 marks) (iii)Show that the general solution satisfies the PDE. (4 marks) (iv)Obtain the particular solution to the PDE for the given boundary condition above. (2 marks) (b)Consider the following (quasi-)linear partial differential equation (i)Show that the above equation has only one set of characteristics: where is an arbitrary constant. (3 marks) (ii)Taking and show that the canonical form of the equation is: (where and are constants that you should determine). (7 marks) (iii)Hence show that the general solution is: where and are two arbitrary functions. (5 marks) Question 4 continued on page 7… Continuation of Question 4: (iv)If you are also given that: and find and which satisfy the above conditions. (5 marks) END OF EXAMINATION PAPER 2 Q1 (Solution) (a) [3] (b) We seek the simultaneous solution to ???? ???? = ???? ???? = 0, meaning [1] ??(?? + ??) − ???? − ?????? = 0 ... (1) ?????? − (?? + ??)?? = 0 ... (2) From (2), ??(???? − ?? − ??) = 0 so Either ?? = 0 ⇒ ??(?? − ??) = 0 in (1) [1] But, as we are given that the birth rate exceed the natural death rate (?? > ??) then ?? = 0 Hence, one critical point is (0,0) [1] Or ?? ≠ 0 and so from (2), ???? − ?? − ?? = 0 ⇒ ?? = ??+?? ?? ... (3) [1] From (1) ??(?? − ????) = (?? − ??)?? so ?? = (??−??)?? ??−???? and using the value of H from (3) yields ?? = (??−??)??+???? ??−????+???? = (??−??)(??+??) ????−??(??+??) or ?? = (??−??)(??+??) ??(??+??−??) [3] Meaning the second critical point is at �??+?? ?? , (??−??)(??+??) ??(??+??−??) � and we note that as (?? − ??)(?? + ??) > 0 and ?? > ?? − ?? then this point is interior to the first quadrant of the (??,??) plane. [1] (c) Setting ?? = ??(?? + ??) − ???? − ?????? and ?? = ?????? − (?? + ??)?? the Jacobian is ?? = �???? ???????? ???? � [1] ??(?? + ??) ??(??) (?? + ??)?? ???? ?????? ??(??) That is, ?? = �?? − ?? − ???? ?? − ???????? ???? − ?? − ??� [3] (d) At the critical point (0,0) we have ?? = �?? − ?? ??0 −?? − ??� [1] In the notation of the classification diagram, ?? = |??| = −(?? − ??)(?? + ??) < 0="" since="" all="" parameters="" are="" positive="" and="" we="" are="" given="" that=""> ??, then (0,0) is a saddle point (unstable). [2] Equally, we could note that the eigenvalues on the diagonal of this triangular matrix are of opposite sign to come to the same conclusion. At the critical point ?? = ??+?? ?? ,?? = (??−??)(??+??) ??(??+??−??) we have ???? = ?? − ?? − ???? = ?? − ?? − (??−??)(??+??) (??+??−??) = (?? − ??) �1 − (??+??)(??+??−??)� = ??(??−??)(??+??−??) [1] ???? = ?? − ???? = ?? − ?? − ?? [1] ???? = ???? = (??−??)(??+??) (??+??−??) [1] ???? = ???? − ?? − ?? = ?? + ?? − ?? − ?? = 0 [1] So, at this second critical point, ?? = � ??(??−??) (??+??−??) ?? − ?? − ?? (??−??)(??+??) (??+??−??) 0 � [1] Here ?? = |??| = (?? − ??)(?? + ??) > 0 so the point is not a saddle, but its stability