An Introduction To Mathematical Modelling Fowkes Pdf

Successfully reported this slideshow.

Introduction to mathematical modelling

1

Like this presentation? Why not share!

• Email
•  

• 33 Likes
• Statistics
• Notes

Introduction to mathematical modelling

1. 1. INTRODUCTION TO MATHEMATICAL MODELLING Prof. (Dr) Samir Kumar Das Defence Institute of Advanced Technology (DIAT) Girinagar, Pune 411025
2. 2. OUTLINES OF THE PRESENTATION DIMENSIONAL ANALYSIS WHAT IS MATHEMATICAL MODELLING? WHY MATHEMATICAL MODEL IS NECESSARY? USE OF MATHEMATICAL MODEL TYPES OF MATHEMATICAL MODEL MATHEMATICAL MODELLING PROCESS
3. 3. OUTLINES OF THE PRESENTATION DISCRETE APPROACH OF COMBAT MODELS BUCKINGHAM PI THEOREM PREY-PREDATOR MODEL (LOTKA-VOLTERRA MODEL) (DISCRETE MODEL , EQUILIBRIUM CONDITIONS, STABILITY ANALYSIS) COMBAT MODELLING (LANCHESTER LAWS) (SQUARE LAW, LINEAR LAW, PARABOLIC LAW, LOG LAW) POPULATION MODEL ( LOGISTIC MODEL )
4. 4. WHAT IS MATHEMATICAL MODELLING? Representation of real world problem in mathematical form with some simplified assumptions which helps to understand in fundamental and quantitative way. It is complement to theory and experiments and often to integrate them. Having widespread applications in all branches of Science and Engineering & Technology, Biology, Medicine and several other interdisciplinary areas. 2 3 1
5. 5. WHY MATHEMATICAL MODEL IS NECESSARY? To perform experiments and to solve real world problems which may be risky and expensive or time consuming. Emerged as a powerful, indispensable tool for studying a variety of problems in scientific research, product and process development and manufacturing. Improves the quality of work and reduced changes, errors and rework However, mathematical model is only a complement but does not replace theory and experimentation in scientific research. 1 2 3
6. 6. USE OF MATHEMATICAL MODEL Solves the real world problems and has become wide spread due to increasing computation power and computing methods. Facilitated to handle large scale and complicated problems. Some areas where mathematical models are highly used are : Climate modeling, Aerospace Science, Space Technology, Manufacturing and Design, Seismology, Environment, Economics, Material Research, Water Resource, Drug Design, Populations Dynamics, Combat and War related problems, Medicine, Biology etc. 1 2 3
7. 7. TYPES OF MATHEMATICAL MODEL EMPIRICAL MODELS THEORETICAL MODELS EXPERIMENTS OBSERVATIONS STATISTICAL MATHEMATICAL COMPUTATIONAL
8. 8. TYPES OF MATHEMATICAL PROCESS REAL WORLD PROBLEM WORKING MODEL MATHEMATICAL MODELRESULT / CONCLUSIONS COMPUTATIONAL MODEL SIMPLIFY REPRESENT TRANSLATESIMULATE INTERPRET
9. 9. FORMULATION PROBLEM MATHEMATICAL PRARAMETERS START SOLUTION EVALUTION SATISFIED STOP NO YES
10. 10. TYPES OF MODELS QUALITATIVE AND QUANTITATIVE STATIC OR DYNAMIC DISCRETE OR CONTINUOUS DETERMINISTIC OR PROBABILISTIC LINEAR OR NONLINEAR EXPLICIT OR IMPLICIT 1 2 3 4 5 6
11. 11. STATIC OR DYNAMIC MODEL STATIC MODEL A static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. A static model cannot be changed, and one cannot enter edit mode when static model is open for detail view. DYNAMIC MODEL A dynamic model accounts for time- dependent changes in the state of the system. Dynamic models are typically represented by differential equations.
12. 12. DISCRETE OR CONTINUOUS MODEL DISCRETE MODEL A discrete model treats objects as discrete, such as the particles in a molecular model. A clock is an example of discrete model because the clock skips to the next event start time as the simulation proceeds. CONTINUOUS MODEL A continuous model represents the objects in a continuous manner, such as the velocity field of fluid in pipe or channels, temperatures and electric field.
13. 13. DETERMINISTIC OR PROBABILISTIC (STOCHASTIC) MODEL DETERMINISTIC MODEL A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Deterministic models describe behaviour on the basis of some physical law. PROBABILISTIC (STOCHASTIC) MODEL A probabilistic / stochastic model is one where exact prediction is not possible and randomness is present, and variable states are not described by unique values, but rather by probability distributions.
14. 14. LINEAR OR NONLINEAR MODEL LINEAR MODEL If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A linear model uses parameters that are constant and do not vary throughout a simulation. NONLINEAR MODEL A nonlinear model introduces dependent parameters that are allowed to vary throughout the course of a simulation run, and its use becomes necessary where interdependencies between parameters cannot be considered.
15. 15. EXPLICIT OR IMPLICIT MODEL EXPLICIT MODEL Calculate the state of a system at a using the past time from the state of the system at the current time. IMPLICIT MODEL Solution is obtained by solving an equation involving both the current state of the system and the later one which require extra computation and could be harder to solve.
16. 16. QUALITATIVE OR QUANTITATIVE MODEL QUALITATIVE MODEL It is basically a conceptual model that display visually of the important components of an ecosystem and linkages between them. It is a simplification of a complex system. The humans are good at common sense with qualitative reasoning. QUANTITATIVE MODEL Models are mathematically focused and many times are based on complex formulas. In addition quantitative models generally through an input-output matrix. Quantitative modelling and simulation give precise numerical answers.
17. 17. DEDUCTIVE MODEL A deductive model is a logical structure based on theory. A single conditional statement is made and a hypothesis (P) is stated. The conclusion (Q) is then deduced from the statement and hypothesis. (What this model represents ?) P Q (Conditional statement) P (Hypothesis stated) | Q (Conclusion deducted) Example #1 All men are mortal, Ram is man, Therefore, Ram is mortal.1 2 3 Example #2 If an angle satisfies 900<A<1800, then A is an obtuse angle, A=1200, Therefore, A is an obtuse angle. 1 2 3
18. 18. DEDUCTIVE MODEL An inductive model arises from empirical findings and generalizations from them. This is known as "Bottom-up" approach (Qualitative). Focus on generating new theory which is used to form hypothesis. THEORY HYPOTHESIS OBSERVATION CONFIRMATION Deductive model is more narrow in nature and is concerned with confirmation of hypothesis.
19. 19. DEDUCTIVE MODEL Deductive model is a "Top-down" approach (Quantitative). It focus on existing theory and usually begins with hypothesis. OBSERVATION PATTERN TENTATIVE HYPOTHESIS Inductive model is open ended and explanatory, specially at the beginning. THEORY
20. 20. REAL WORLD PROBLEM FALLS IN WHICH CATEGORY? This is based on how much priori information is available on the system. There are two type of models : BLACK BOX MODEL and WHITE BOX MODEL. BLACK BOX MODEL is a system of which there is no priori information available. WHITE BOX MODEL is a system where all necessary information is available.
21. 21. DIMENSIONAL ANALYSIS A method with which non-dimensional can be formed from the physical quantities occurring in any physical problem is known as dimensional analysis. This is a practice of checking relations among physical quantities by identifying their dimensions. The dimension analysis is based on the fact that a physical law must be independent of units used to measure the physical variables. 2 3 1
22. 22. DIMENSIONAL ANALYSIS The practical consequence is that any model equations must have same dimensions on the left and right sides. One must check before developing any mathematical model. 4
23. 23. DIMENSIONAL ANALYSIS EXAMPLE Let us take an example of heat transfer problem. We start with the Fourier's law of heat transfer. Rate of heat transfer Temperature gradient 2 2 K t x       Let us consider a uniform rod of length l with non-uniform temp. Lying on the x-axis form x=0 to x=l. The density of the rod ( ), specific heat (c), thermal conductivity (K) and cross-sectional area (A) are all constant.  (1)
24. 24. DIMENSIONAL ANALYSIS EXAMPLE Change of heat energy of the segment in time ( ) = Heat in from the left side – Heat out from the right side After rearranging (2) t ( , ) ( , ) x x x c A x x t t c A x x t A t K K x x                                 ( , ) ( , ) x x x K c x xx t t x t t x                           (3) After taking the limit 2 2 k t x       where K k c  (4)
25. 25. DIMENSIONAL ANALYSIS EXAMPLE (5) (6) (7) 2 2 1 TLc    3 ML  , 3 1 K MLT   , L x x , 3 1 2 0 0 2 2 1 3 00 0 MLT L O TL T ML K c k                  0    , 0T T T  , , , 0 0 Tt T       0 Lx x       2 2 0 2 2 2 Lx x       2 0 0 2 2 0 0 2 2 2 , L T T L k t Tx x              2 2T x       ,
26. 26. DIMENSIONAL ANALYSIS ASSIGNMENT #1 2 2c c D t x      2 2uu t x      , where D and are diffusion coefficient and coefficient of kinematic viscosity respectively.  CALCULATE FOR 1-D DIFFUSION EQUATION AND 1-D FLUID EQUATION IN DIMENSIONLESS FORM AS : 2 2 T x       THERE ARE GENERALLY THREE ACCEPTED METHODS OF DIMENSIONAL ANALYSIS : RAYLEIGH METHOD (1904): Conceptual method expressed as a functional relationship of some variable | BUCKINGHAM METHOD (1914): The use of Buckingham Pi ( ) theorem as the dimensional parameters was introduced by the Physicist Edger Buckingham in his classical paper | P. W. BRIDGMAN METHOD (1946): Developed on pressure physics) 
27. 27. BUCKINGHAM PI THEOREM If there are m fundamental units and n physical quantities lead to system of m linear algebraic equations with n unknowns of the form (10) This can be written in the matrix form as Ay b (11) 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 n n n n m m mn n m a y a y a y b a y a y a y b a y a y a y b             L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L
28. 28. BUCKINGHAM PI THEOREM (13) Here A is referred as coefficient matrix of order mxn, y and b are order nx1 and mx1 respectively. If any matrix C has at least one determinant of order r, then the matrix C is said to be of rank r. (14) 11 21 1 21 22 2 1 2 .............. ............. .............................. ............. n n m m mn a a a a a a A a a a             (12)  T 1 2......... ny y yy   T 1 2........ mb b bb  ( )R c r
29. 29. BUCKINGHAM PI THEOREM In order to determine the condition for the linear system equation (10), it is convenient to define the rank of the augmented matrix B. For the solution of the linear system equation (10) three possible cases arise: 11 21 1 1 21 22 2 2 1 2 .............. ............. ................................. ............ n n m m mn m b b b a a a a a a B a a a             (15) ( ) ( )R A R B ( ) ( )R A R B r n   ( ) ( )R A R B r n   CASE I : CASE II: CASE III: (16) CASE I : In this case, no solution exists. CASE-II: In this case, a unique solution exists CASE-III: In this case, an infinite number of solution with n-r arbitrary unknown exists
30. 30. BUCKINGHAM PI THEOREM The procedure of dimensional analysis makes use of the following assumptions: It is possible to select m independent units (Example: m=3, ie, length, time and mass ) There exist n quantities involved in a phenomenon whose dimensional formulae can be expressed in terms of m fundamental units The dimensional quantity Q0 can be related to the independent dimensional quantities by1 2 1, ,......... nQ Q Q     11 2 1 2 1 1 2 10 , ,......... , ,......... nyy y n nQ Q Q Q Q QQ F K     Where K is a non-dimensional constant and are integer exponent 1 2 1, ......... ny y y  (17)
31. 31. BUCKINGHAM PI THEOREM Equation (17) is independent of the type of units chosen and is dimensionally Homogeneous , ie, the quantities occurring on both sides of the equations must have same dimensions.
32. 32. BUCKINGHAM PI THEOREM EXAMPLE Consider the problem of freely falling body near the surface of the Earth. If x, w, g and t represent the distance measured from the initial height , the weight of the body, the gravitational acceleration and the time respectively, obtain a relation of x as a function of w, g and t. SOLUTION Using the fundamental units of force F, length L and time T and four physical quantities, , , and , involve three fundamental units. Hence m=3 and n=4 as mentioned in assumptions one and two. 0Q x 1Q w 2Q g 3Q t
33. 33. BUCKINGHAM PI THEOREM SOLUTION By using the assumption three, assume a relation of the form   31 2 , , yy y w g tx F K w g t  (18) where K is an arbitrary non-dimensional quantity. Let [ ] denote dimensions of a quantity. Then the relation can be written (using the assumption four) as        1 2 3y y y x w g t       2 1 3 2 31 2 220 1 0 yy y y yy y LTF LT F T F L T     1: 0F y 2: 1L y 2 3: 0 2T y y   3 22 2y y  0 1 2 x K w g t 2 x K g t (19) (20) (21) (22) (23) The constant in this case is ½ which can not obtained form dimensional analysis
34. 34. BUCKINGHAM PI THEOREM ASSIGNMENT #2 Consider the problem of drag force on a body moving through a fluid. Let D, and V be the drag force, density of the fluid, viscosity effect of the fluid, reference length and velocity. Using fundamental units obtain a relation of drag force as function of density, viscosity, length and velocity. HINTS , ,l    31 2 4 , , , yy y y l vD F K l v     1 0 0 D F L T 4 2 FL T   2 FL T  , ,
35. 35. POPULATION MODEL (LOGISTIC MODEL) Logistic model was developed by Belgian Mathematician Pierre Verhulst (1838) who suggested that the rate may be limited, ie, it may depend on population density. P kP t    1 P k r K        , where (1) At low population (P<<0), the population growth rate is maximum and equal To r. Parameter r can be interpreted as population growth rate in the absence of intra-species competition. 0 r K
36. 36. POPULATION MODEL (LOGISTIC MODEL) ... the population growth rate declines with the population number P and reaches zero when P=K, parameter K is the upper limit of population growth and it is called carrying capacity. It is usually expressed as the amount of resources available. If the population number exceeds K, then population growth rate becomes negative and population numbers decline. (2) There are three possible model out comes: 1. Population increases and reaches a plateau (Po<K) 2. Population decreases and reaches a plateau (Po>K) 3. Population does not change (Po=K or Po=0) 0 r K 1 P P r t K         
37. 37. DISCRETE DYNAMICAL MODEL Behavior of the Discrete Dynamical Model near an Equilibrium If f '(Pe) > 1, then the solutions of the discrete dynamical model grow away from the equilibrium (monotonically). Thus, the equilibrium is unstable. 2. If 0 < f '(Pe) < 1, then the solutions of the discrete dynamical model approach the equilibrium (monotonically). Thus, the equilibrium is stable. 3. If -1 < f '(Pe) < 0, then the solutions of the discrete dynamical model oscillate about the equilibrium and approach it. Thus, the equilibrium is stable. 4. If f '(Pe) < -1, then the solutions of the discrete dynamical model oscillate but move away from the equilibrium. Again, the equilibrium is unstable. 1 1 n n n n P P P rP K         
38. 38. PREY-PREDATOR MODEL Prey-Predator model is known as Lotka-Vloterra model. This model was developed independently by Alfraid J. Lotka and Vito Voterra in 1920's which characterized by oscillation in the population size of both prey-predator. The prey- predator dynamics can be written in the simplified form by using pair of differential equations. This describes the relation between herbivore-plant, parasitoid-host, lions-gazelles, birds-insects, shark-fish etc. ALFRED LOTKA (1880-1949) VITO VOLTERRA (1860-1940)
39. 39. PREY-PREDATOR MODEL This model makes several simplifying assumptions: The population will grow exponentially when predator is absent1 2 3 The predator population will starve in the absence of prey population Predator can consume infinite number of prey Both the populations can move randomly through a homogeneous environment The prey has unlimited food supply 4 5
40. 40. PREY-PREDATOR MODEL If there is no predator, the first assumption would imply that prey grows exponentially. Let us consider N is prey, using Pierre Verhulst model this can be written as N rN t    0IC : (0)N N(4) (5) The solution of equation (4) can be expressed as 0( ) rt N t N e Where is initial populations and r is the growth rate. Here the number of pray would increase exponentially. Since there are predators, which Must account for negative component of growth rate. (6) N rN aPN t     Where P is number of predators and a is the attack rate. The term shows the losses from prey population due to predation. (7) 0N aPN
41. 41. PREY-PREDATOR MODEL Now we consider predator population. If there are no food supply, the population would die out at rate proportional to its size. (8) Predator mortality rate In the presence of prey, this decline is opposed by the predator birth rate Where the term is the birth rate and c a constant conversion rate of eaten prey into new predator in abundance. Combining equation (7) and equation (9), coupled model can be obtained which is known as Lotka-Volterra model or prey-predator model. (9) P qp t     q  P qP caNP t      caNP
42. 42. PREY-PREDATOR MODEL Prey Model: (10) The Lotka-Volterra model consist of a system of linked differential equations that can be separated from each other and can not be solved in closed form. Assuming N>0 and P>0 N rN aPN t     Prey Model: P qP caNP t      EQUILIBRIUM CONDITIONS: 0 N t    ,   0r aP N  , r P a   0 P t    ,   0caN q P  q N ca   (11) (12)
43. 43. PREY-PREDATOR MODEL GRAPHICAL EQUILIBRIUM Prey pop size Predator Pop Size r/a dN/dt =0 Prey (H) equilibrium (dN/dt=0) is determined by predator population size. If the predator population size is large the prey population will go extinct. If the predator population is small the prey population size increases.
44. 44. PREY-PREDATOR MODEL GRAPHICAL EQUILIBRIUM Prey pop size Predator Pop Size Predator (P) equilibrium (dP/dt=0) is determined by prey population size. If the prey population size is large the predator population will increase. If the prey population is small the predator population goes extinct. q/ca dP/dt =0
45. 45. PREY-PREDATOR MODEL PREY-PREDATOR INTERACTION Prey pop size Predator Pop Size The stable dynamic of predators and prey is a cycle. CASE – I : When Prey population is above equilibrium and Predator population below the equilibrium CASE – II : When both Prey and Predator populations are below equilibrium CASE – III : When Predator population size is above the equilibrium and Prey below equilibrium CASE – IV : When both Prey and Predator populations are above the equilibrium q/ca dN/dt =0 r/a Case - II Case - I Case - III Case - IV dP/dt =0
46. 46. PREY-PREDATOR MODEL Three possible outcomes of interactions The oscillations are stable The oscillations are damped (convergent oscillation) The oscillations are divergent and can lead to extinction This model predicts cyclical Relationship between predator (P) and Prey (N) Due to increase of consumption rate , decrease of prey takes place and therefore aPN decreases .
47. 47. PREY-PREDATOR MODEL LOWINTIMACYHIGH LOW LETHALITY HIGH PARASITOIDSPARASITE GRAZER PREDATOR
48. 48. STABILITY ANALYSIS 0 N t      1 1 1 , f f df N P dN dp N P       0 P t      2 2 2 , f f df N P dN dp N P       1 1 2 2 ( ) ( ) ( ) ( ) N PN P f f r aP N r aP N N P N P J f f caN q P caN q P N PN P                                         Since and P N Prey Zero Growth Isocline PredatorZeroGrowth Isocline
49. 49. STABILITY ANALYSIS   r q r a a r aP aN a ca J r qcaP caN q ca ca q a ca                                            0 0 q J c cr         The trace of J is equal to zero which is not meeting stability conditions. Therefore, equilibrium condition is not stable.   0 0 0 , det 0 0 q q A Ac c cr cr                       2 2 0, ,qr qr i qr       
50. 50. STABILITY ANALYSIS The British theorist F. W. Lanchester (1914) developed this theory based on World War-I, aircraft engagement to explain why concentration of forces was useful in modern warfare. Both the models work on the basis of attrition : HOMOGENEOUS 1. A single scalar represents a unit's combat power. 2. Both sides have the same weapon effectiveness. HETEROGENEOUS Attrition is assured by weapon type and target type and other variability factors.
51. 51. STABILITY ANALYSIS HOMOGENEOUS 1. Homogeneous Model is an "Academic Model". 2. Useful for review the ancient battles. 3. Not proper model for modern warfare. HETEROGENEOUS More appropriate for "Modern Battlefield". The following battlefield functions are sometimes combined and can be modeled by separate algorithm/theory. Direct Fire | Indirect Fire | Air to Ground Fire | Ground to Air Fire | Air to Air Fire | Minefield Attrition
52. 52. COMBAT MODELLING (LANCHESTER LAWS) 1. Forces are homogeneous. 2. Similar Weapons Systems. 3. Which can accomplish the same effects. Lanchester's Combat model shows force-on-force interaction. Blue Force (B) acts upon a Red Force (R) in accordance with same effect ( ). Red Force (R) acts upon a Blue Force (B) in accordance with same effect ( ). Blue Force Red Force B(t) R(t)    
53. 53. COMBAT MODELLING (LANCHESTER LAWS) This and are attrition between forces. The variables which represents these effects are and , are called attrition rates as they represent the rates at which reds kill blues and blues kill reds.     Attrition of Blue forces dB R dt   , for B>0 dR B dt   , for R>0 IC: at t=0 0(0)B B 0(0)R R (1) (2) Attrition of Red forces
54. 54. COMBAT MODELLING (LANCHESTER LAWS) (3) (5) ; dB dR dt dt R B      , dB dR BdB RdR R B         0 0 B R B R BdB RdR   0 0 2 2 2 2 B R B R B R                     2 2 2 2 0 0B B R R    (6) (7) (4)
55. 55. SQUARE LAW Since the strengths of the opposing forces appear with exponents of two, the name "Square Law" is given to that law which the set of equations describes. It is assumed that combat continues until one side's unit count reaches to zero. If this is true, then at the end of the combat , we define the number of units remaining as Bf and Rf respectively, we can write
56. 56. SQUARE LAW 0fB  2 2 0 0fR R B           2 2 0 0B R  2 2 0 0fB B R           2 2 0 0R B  0fR  then then And If : If :
57. 57. SQUARE LAW OTHER ASSUMPTIONS 1. The forces are within weapons range of one another. 2. The effects of weapons rounds are independent. 3. Fire is uniformly distributed over the enemy targets. 4. Attrition coefficient are constant and known. 5. All of the forces are committed at the beginning and there are no reinforcements. These assumptions appear to restrict the applicability of Lanchester's Model. Combat on today's battlefield is very complex and very much from the type proposed by Lanchester's Model. Some of these are not longer seen to be appropriate in modelling what we term combat under modern conditions. Although, it is not necessary that all the assumptions fit the experimental model perfectly, some deviation is quite possible.
58. 58. SQUARE LAW ASSIGNMENT#3 Initial Blue Force = 2000 ( men ) =B0 Initial Red Force = 1000 ( men ) =R0 Blue Effect = = 0.002( 1/hour ) Red Effect = = 0.001( 1/hour )   HINTS To determine who will win, each must have victory conditions. Battle termination model, assuming both side fight to annihilation.    0 and 0f fR t B t     0 and 0f fB t R t  0 0 B R    It can be shown that a square-law battle will be won by the blue (B) If and only if Who will win the battle after how many hours? Red Wins : Blue Wins :
59. 59. EXTENSION OF LANCHESTER LAWS 1 w dB B R dt R           1 w dB B R dt R           1 w dR R B dt B           0=> T/Tw 1/2=> FT/FTw 1=> F/Fw
60. 60. Write to me at samirkdas@diat.ac.in hod_am@diat.ac.in samirkumar_d@yahoo.com THANK YOU Download http://www.slideshare.net/slideshow/embed_code/42588379

• 

  RamVijayYadav1

  Sep. 23, 2021

• 

  chandanareddy75

  May. 9, 2021

• 

  SowmyaTeku

  Mar. 17, 2021

• 

  VaishaliWadhwa6

  Oct. 22, 2020

• 

  GajananWaghmare3

  Oct. 3, 2020

• 

  mubashiralighaffar

  Aug. 12, 2020

• 

  mubashiralighaffar

  Aug. 12, 2020

• 

  mastercoin

  Aug. 9, 2020

• 

  reeGowrav

  Jul. 8, 2020

• 

  RuchaDesai10

  May. 31, 2020

• 

  ZankrutiPatel

  May. 22, 2020

• 

  NimishaKhaneja

  Apr. 1, 2020

• 

  SivakumarSivakumar20

  Jan. 12, 2020

• 

  AnukshanGhosh

  Nov. 3, 2019

• 

  SonalSaroha

  Mar. 22, 2019

• 

  manan9641

  Mar. 15, 2019

• 

  KarunyaGraceAmalados

  Jan. 6, 2019

• 

  JosephOcen

  Jan. 2, 2019

• 

  sonujha41

  Nov. 4, 2018

• 

  FathollahPourfayaz

  Sep. 30, 2018

Indicon 2014, Pune

An Introduction To Mathematical Modelling Fowkes Pdf

Source: https://www.slideshare.net/arupparia/introduction-to-mathematical-modelling-42588379

Posted by: grahamprinag1964.blogspot.com

Share this post