Browsing by Author "Karunathilake, N.G.A."

Now showing 1 - 7 of 7

Convergence properties in numerical algorithms of option price calculations
(University of Kelaniya, 2011) de Silva, T.M.M.; Karunathilake, N.G.A.
Financial derivatives can be used to minimize losses caused by price fluctuations of the underlying assets where derivative means financial instrument whose value depends on the price of some other financial assets or some underlying factors. Options are widely used on markets and exchanges. Options can be used, for instance, to hedge assets and portfolios in order to control the risk due to movements in the share price. The famous Black-Scholes model is a convenient way to calculate the price of an option. And also the Black–Scholes model [1] is a mathematical description of financial markets and derivative investment instruments. In an idealized financial market the price of the European option can be obtained as the solution of the celebrated Black-Scholes equation. This equation also provides a hedging portfolio that replicates the contingent claim. However, the Black- Scholes equation has been derived under quite restrictive assumptions (for instance, frictionless, liquid, complete markets). In recent years, some of these restrictive assumptions have been relaxed in order to model, for instance, the presence of transaction costs, imperfect replication and investor‟s preferences, introduction of a given stock-trading strategy of a large trader, and risk from unprotected portfolio. These models lead to a generalized Black-Scholes equation for the price of an option in which the volatility need not be necessarily constant and it may depend on the asset price as well as the option price. Again, if transaction costs are taken into account then the classical Black-Scholes theory is no longer applicable. In order to maintain the delta hedge one has to make frequent portfolio adjustments yielding thus a substantial increase in transaction costs. However, these modifications convert the Black-Scholes equation into non-linear form and it is very difficult to solve the equation analytically. In the absence of analytical methods, various numerical discretization methods can be used in order to approximate the solution of the Black- Scholes equation. In our conditions we investigate the numerical solution of the Black-Scholes equation for the European call option,     xx x x xx n i u  S u  u  1 D u  u  where   xx x n i sign u u t K S              2 , 2 2  r D  , K is strike price, is volatility, r is interest rate.
Coupling Shallow Water Equation with Navier-Stokes Equations: A viscous shallow water model
(Faculty of Graduate Studies, University of Kelaniya, 2008) Karunathilake, N.G.A.
The general characteristic of shallow water flows is that the vertical characteristic scale D is essentially smaller than and typical horizontal scale L .i.e. £ := D << 1 . L In many classical derivations, in order to obtain the shallow water approximation of the Navier-Stokes's Equations, the molecular viscosity effect is neglected and a posteriori is added into the shallow water model to represent the efficient-viscosity ( a friction term through the Chezzy formula which involes empirical constants) at the bottom topography. However, the validity of this approach has been questioned in some applications as the models lead to different Rankine-Hugoniot curves (see e.g. [1]). Therefore, it can be useful to consider the molecular viscosity effect directly in the derivation of the shallow water model. On the other hand the classical shallow water models are derived under the assumption of slowly varying bottom topographies. Hence, for the description of incompressible shallow water laminar flow in a domain with a free boundary and highly varying bottom topography, the classical Shallow Water Equations are not applicable. The remedy consist of dividing the flow domain into two sub-domains namely, near field (sub domain with the bottom boundary) and far field (sub domain with the free boundary) with a slowly varying artificial interface and employ the Navier-Stokes Equations and Viscous Shallow Water Equations in the near field and far field, respectively. In this work, we derive a two-dimensional Viscous Shallow Water model for incompressible laminar flows with free moving boundaries and slowly varying bottom topographies to employ in the far field. In this approach, the effect of the molecular viscosity is retained and thereby corrections to the velocities and the hydrostatic pressure approximations are established. Coupling modified shallow water model with NSE has been carried out in a separate work. In order to derive the viscous shallow water model the two-dimensional Incompressible Navier-Stokes equations in usual notations au + au2 + auw + ap = 􀁊 ( 2v au) + 􀁌 (v au +V 8 w) , at ax az ax ax ax az az az aw + auw + 8w2 + ap =-g+􀁋 (vau +j.law) +􀁌 ( 2vau), ---------------------------(1) at ax az az . ax az ax az az a-w+ a-w= 0. ax az are employed in the far field with the suitable boundary conditions. On the free surface, we assume that the fluid particle does not leave the free surface and we neglect the wind effect and the shear stress. On the artificial boundary we set the conditions according with the Navier-Stokes solution at the interface. On the lateral boundaries inflow and outflow conditions are employed. Rescaling the variables with the typical characteristic 151 Proceedings of the Annual Research Symposium 2008- Faculty of Graduate Studies University of Kelaniya scales L and D, the dimensionless form of the Navier-Stokes's equations for shallow water flows are obtained. Similarly, assuming that the bottom boundary is regular and the gradient of the free surface remains bounded we obtain the dimensionless boundary conditions. The second order terms with respect to & in the system are neglected and asymptotic analysis is carried out under the assumptions, the flow quantities admit linear asymptotic expansion to the second order with respect to & and the molecular viscosity of the water is very small. Then, rescaling the depth averaged first momentum equation of the resulting system and substituting the zeroth order solution for the velocity and the pressure in it the zeroth order first momentum equation which include the interface conditions is obtained. Again integrating the continuity equation of the dimensionless system from z1 to H(t, x ), a more detailed view of the vertical velocity component is established. Similarly, integrating the vertical momentum equation the dimensionless system from z1to H(t, x ) and replacing boundary conditions, the second order correction to the hydrostatic pressure distribution is derived. Then, dropping o(s2) in the system and switching to the variables with dimensions, the following results are established. Proposition: The formal second order asymp t ot ic ex p ansion of t he Navier-St okes Equat ions for t he shallow wat er laminarfl ow is given by ( z -z I ) ou . I ou 2 u(t, x, z) = u(t,x,z1) + I--- -(x, z1 ,t)(z-z1)---(x, z1 ,t)(z-z1) 2h oz 2h oz h(t,x)+z1 OU w(t,x,z)=w(t,x,z1)- f -d1] OX Z=Zt - ou ou p(t,x,z) = g(h+ z1 -z)-v-(t,x,z)-v-(x,t) ox ox wit h t he viscous shallow wat er equations ah + £(􀁍h)= ( w _ u az 1 ) , at ax ax z=Zt 􀁎 (􀁍 h) +£ {􀁍z h) + £( gh2 J = £( 4v h a􀁍J -rl ' at ax 􀁏 8x 2 ax 8x where r, 􀁐 [ p : +v: +v: -2v : : +u(u: -w) L,, and z 􀁐 z,(x, t ) is t he interface. Concluding remarks In the zeroth order expansion as well as in many classical shallow water models, the horizontal velocity does not change along with the vertical direction. In contrast, our first order correction gives a quadratic expansion to the vertical velocity components retaining more details of the flow. As many classical models we do not neglect the viscosity effect but just assume that it is very small. Also, the zeroth order hydrostatic pressure approximation has been upgraded to the first order giving a parabolic correction to the pressure distribution.
Coupling Shallow Water Equation with Navier-Stokes Equations: A viscous shallow water model
(University of Kelaniya, 2008) Karunathilake, N.G.A.
The general characteristic of shallow water flows is that the vertical characteristic scale D is essentially smaller than and typical horizontal scale L .i.e. £ := D << 1 . L In many classical derivations, in order to obtain the shallow water approximation of the Navier-Stokes's Equations, the molecular viscosity effect is neglected and a posteriori is added into the shallow water model to represent the efficient-viscosity ( a friction term through the Chezzy formula which involes empirical constants) at the bottom topography. However, the validity of this approach has been questioned in some applications as the models lead to different Rankine-Hugoniot curves (see e.g. [1]). Therefore, it can be useful to consider the molecular viscosity effect directly in the derivation of the shallow water model. On the other hand the classical shallow water models are derived under the assumption of slowly varying bottom topographies. Hence, for the description of incompressible shallow water laminar flow in a domain with a free boundary and highly varying bottom topography, the classical Shallow Water Equations are not applicable. The remedy consist of dividing the flow domain into two sub-domains namely, near field (sub domain with the bottom boundary) and far field (sub domain with the free boundary) with a slowly varying artificial interface and employ the Navier-Stokes Equations and Viscous Shallow Water Equations in the near field and far field, respectively. In this work, we derive a two-dimensional Viscous Shallow Water model for incompressible laminar flows with free moving boundaries and slowly varying bottom topographies to employ in the far field. In this approach, the effect of the molecular viscosity is retained and thereby corrections to the velocities and the hydrostatic pressure approximations are established. Coupling modified shallow water model with NSE has been carried out in a separate work. In order to derive the viscous shallow water model the two-dimensional Incompressible Navier-Stokes equations in usual notations au + au2 + auw + ap = �� ( 2v au) + �� (v au +V 8 w) , at ax az ax ax ax az az az aw + auw + 8w2 + ap =-g+�� (vau +j.law) +�� ( 2vau), ---------------------------(1) at ax az az . ax az ax az az a-w+ a-w= 0. ax az are employed in the far field with the suitable boundary conditions. On the free surface, we assume that the fluid particle does not leave the free surface and we neglect the wind effect and the shear stress. On the artificial boundary we set the conditions according with the Navier-Stokes solution at the interface. On the lateral boundaries inflow and outflow conditions are employed. Rescaling the variables with the typical characteristic scales L and D, the dimensionless form of the Navier-Stokes's equations for shallow water flows are obtained. Similarly, assuming that the bottom boundary is regular and the gradient of the free surface remains bounded we obtain the dimensionless boundary conditions. The second order terms with respect to & in the system are neglected and asymptotic analysis is carried out under the assumptions, the flow quantities admit linear asymptotic expansion to the second order with respect to & and the molecular viscosity of the water is very small. Then, rescaling the depth averaged first momentum equation of the resulting system and substituting the zeroth order solution for the velocity and the pressure in it the zeroth order first momentum equation which include the interface conditions is obtained. Again integrating the continuity equation of the dimensionless system from z1 to H(t, x ), a more detailed view of the vertical velocity component is established. Similarly, integrating the vertical momentum equation the dimensionless system from z1to H(t, x ) and replacing boundary conditions, the second order correction to the hydrostatic pressure distribution is derived. Then, dropping o(s2) in the system and switching to the variables with dimensions, the following results are established. Proposition: The formal second order asymp t ot ic ex p ansion of t he Navier-St okes Equat ions for t he shallow wat er laminarfl ow is given by ( z -z I ) ou . I ou 2 u(t, x, z) = u(t,x,z1) + I--- -(x, z1 ,t)(z-z1)---(x, z1 ,t)(z-z1) 2h oz 2h oz h(t,x)+z1 OU w(t,x,z)=w(t,x,z1)- f -d1] OX Z=Zt - ou ou p(t,x,z) = g(h+ z1 -z)-v-(t,x,z)-v-(x,t) ox ox wit h t he viscous shallow wat er equations ah + £(��h)= ( w _ u az 1 ) , at ax ax z=Zt �� (�� h) +£ {��z h) + £( gh2 J = £( 4v h a��J -rl ' at ax �� 8x 2 ax 8x where r, �� [ p : +v: +v: -2v : : +u(u: -w) L,, and z �� z,(x, t ) is t he interface. Concluding remarks In the zeroth order expansion as well as in many classical shallow water models, the horizontal velocity does not change along with the vertical direction. In contrast, our first order correction gives a quadratic expansion to the vertical velocity components retaining more details of the flow. As many classical models we do not neglect the viscosity effect but just assume that it is very small. Also, the zeroth order hydrostatic pressure approximation has been upgraded to the first order giving a parabolic correction to the pressure distribution.
New Theorem on Primitive Pythagorean Triples
(University of Kelaniya, 2005) Piyadasa, R.A.D.; Karunathilake, N.G.A.
As a result of our survey on primitive Pythagorean triples, we were able to prove the following theorem: All primitive Pythagorean triples can be generated by almost one parametera , satisfyinga > 2 +1. Furthermore, a is either an integer or of the form h a = g where g and h (> 1) are relatively prime numbers. The proof of the theorem can be briefly outlined as follows: Taking z = y + p for some p ³ 1, z 2 = y 2 + x2 can be put into the form 2 2 1 1 + = + y x y p If p x a = , then the above equation can be put into the form ( )2 2 2 1+b = 1+a b ........................................................................ (1), where 2 1 = a 2 -1 b . Then the above equation can be reduced into 2 2 2 2 2 2 1 2 1 1 a a a + - = - + . In order to generate primitive triples, the above equation has to be multiplied by 4 if a is even and h 4 if h a = g . Now we are able to generate all the primitive Pythagorean triples if a satisfies the conditions of our theorem and 2 a 2 -1 is reduced to cancel 2 in the denominator whenever necessary. The condition a > 2 +1 and a is either integer or of the form = (h > 1) h a g with g and h are relatively prime odd be imposed after a careful study of the equation . In conclusion, an algorithm can be developed to determine p and y so that (( y + p), y, x) is a primitive Pythagorean triple in the order x < y < y + p for given x. A new theorem on primitive Pythagorean triples is found and it may be useful in understanding the Fermat’s Last Theorem.
On the validity of a practical three – body model
(University of Kelaniya, 2006) Piyadasa, R.A.D.; Karunathilake, N.G.A.
It is well known among Physicists that the classical three – body problem is not solvable, whereas, the Quantum Mechanical three–body problem is solvable due to the famous Faddeev’s work [1]. However, the problem in Faddeev’s method is not a practical method since it is not directly applicable to the simplest three–body problem. In particular, the important Coulomb potential cannot be included in a mathematically rigorous manner. Kushu group [2] developed a practical method based on [3] which has been remarkably well [4] in producing experimental results, and is now used all over the world, in case of elastic scattering of lights ions such as d, Li, etc. Which are easily breakable in scattering on composite nuclei. This method (CDCC) is simpler and it solves quantum mechanical Shrödinger equation corresponding to the three–body problem concerned. Very concise recap of this model is given in case of the three–body model n – p – A (d – A) in the following. The total wave function Ψ of the 3–body n–p–A system associated with the model, Hamiltonian H is expressed by =Σ − J M J M J M H a R 1ψ in the usual notation. Now J M ψ is expanded in the complete set of eigen functions of the deuteron sub Hamiltonians H K V (r) n p r n p = + in the usual notations. Here the ground–state wave function of (r) d φ and continuum set of wave functions { (k, r)} l φ play a vital role. Now ( ) [ ]J M l l L L l L J l l L J J M J M d J M (P , R) (r) Y (Rˆ) Y (rˆ) i (k, r) P(k),R dr Y (Rˆ) i Y (rˆ) i 0 0 0 00 = × +ΣΣ∫ × ∞ = ∞ ψ χ φ φ χ in the usual notation. t E , the total centre of mass energy, is given by d d N t m E P P k k 2 2 2 2 0 2 0 2 2 ( ) 2 h h h = + = + μ ε μ in the usual notation. Some assumptions, further, are needed. One of which in the cut off of the continuum and consider the Riemann sum over [0, k1 ], [k, k2 ],...... [ki , ki+1 ],...... [kN−1 − kN = km ]. Proceedings of the Annual Research Symposium 2006 - Faculty of Graduate Studies, University of Kelaniya 80 Still further, the following assumption is needed to do computer calculations. ( , ) ( ( ), ) ( ˆ ( ), ) ˆ ( ) 1 k r P k R dk P k R r i il J il L k k J l l L i i ∫φ χ = Δ χ φ + where ∫ + = Δ 1 ˆ ( ) ( , ) i i k k il l φ r φ k r dk This averaging procedure was drastically criticised by the experts [4] of Faddeev theory. The criticism was so drastic and that one had to answer at least on Physical grounds and which was done in [5]. The above criticism was fully answered, mainly on physical grounds, by the authors of [6] doing a the then gigantic numerical calculation. It has been now shown [7] also that CDCC method is the first order approximation to the Faddeev method. Then the question is why the first order method work so well. Answer to this question is mainly [8] and [9]. The main purpose of this paper is to justify, to a certain extent, CDCC, in a mathematical rigorous manner, by producing the correct form, which has been scrutinized by the authors, of the potential tails of CDCC and numerical support as in the following. Continuum – Continuum coupling potential ( ) , V R k k , in the usual notation, can be written as ∫ ∞ ′ = ′ 0 , 0 0 0 V (R) U (k, r) V (R, r)U (k , r) dr k k (1) in the usual notation for the simplest case of CDCC, where ( , ) ( , ) 0 0 U k r = r φ k r (2) Here ( , ) 0 φ k r defines the deuteron S – state breakup wave function of linear momentumk . Now ( , ) 2 sin( ( )) 0 U k r kr δ k π = + (3) Neglectingδ (k) , the phase shift, for the sake of simplicity, one writes V R [ (k k )r (k k )r ]V R r dr k k cos cos ( , ) 2 ( ) 1 , = ∫ − ′ − + ′ λ ′ (4) V (R, r) λ here has the usual meaning. In case of square well potential ( ) dr R r k R r k R r a r TR V V R a a k k ∫ − ′ + + ′ + − ( ) = sin 2 ( ) sin 2 ( ) ( ) 2 2 0,0 , (5) This can be readily simplified to ⎥⎦ ⎤ ⎢⎣ ⎡ ⎟⎠ ⎞ ⎜⎝ − ⎛ ⎟⎠ ⎞ ⎜⎝ = + ⎛ ′ c kR ka c kR ka R V a Vk k R cos 4 sin 4 64 cos 4 cos 4 3 16 1 3 3 2 ( ) 2 3 2 3 0,0 , π Proceedings of the Annual Research Symposium 2006 - Faculty of Graduate Studies, University of Kelaniya 81 (6) when k = k′ , under the assumption R >> a . Here c given by cka = 1. If ka >> 1, 2 3 0,0 , 3 2 ( ) R V a Vk k R π ′ = (A) When k = k′ ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ⎥⎦ ⎤ ⎢⎣ ⎡ − ′ ′ ′ − ⎥⎦ ⎤ ⎢⎣ ⎡ − ′ ′ ′ ′ = 2 2 2 3 3 3 0,0 , (2 ) cos 2 sin 2 (2 ) cos 2 sin 2 (2 ) cos 2 cos 2 (2 ) 2 cos 2 cos 2 ( ) ka kR ka k a k R k a ka kR ka k a k R k a R V a Vk k R π (B) where K′ = k′ − k and K = k′ + k (A) and (B) agrees with numerical calculations very nicely, which is depicted by the figures attached, in case of realistic potentials. In the figure 1, the diagonal potential (1 – 1) , (6 – 6) agree exactly the form, mathematically established, and figure 2 in case of non-diagonal potentials.
Periodical outbreak of Tuberculosis Epidemic: An epidemic model with dynamic host population
(Research Symposium 2009 - Faculty of Graduate Studies, University of Kelaniya, 2009) Madhuwanthi, K.H.P.; Karunathilake, N.G.A.
Simulation of blood flow through atherosclerotic intracranial arteries, veins, and aneurysms
(Faculty of Science, University of Kelaniya Sri Lanka, 2024) Rosheni, S. M. D.; Karunathilake, N.G.A.
Atherosclerosis is the formation of plaque in arterial walls, thickening the blood vessel wall, thereby hardening and obstructing blood flow through the arteries. An aneurysm is an atypical enlargement or bulge in the wall of a blood vessel inside an artery caused by a weakening of the arterial wall. Arise in fatalities due to atherosclerosis and aneurysms has been considered a hazard to health worldwide. This is mostly attributed to irregular blood flow in stenotic arteries. Thus, blood flow analysis through stenosed arteries and aneurysms is vital in the treatment process. Hence, various researchers have worked on finding the appropriate rheological model for blood flow simulations through arteries and aneurysms. This research study investigates various non-Newtonian rheological models to find the most appropriate model for blood flow simulations. A steady, incompressible, non-viscous, laminar blood flow through different geometrical domains on stenosis and an aneurysm is considered using five different non-Newtonian rheological models; Power law, Carreau law, Carreau-Yasuda law, HerschelBulkley law, and Casson law. COMSOL Multiphysics has been used to examine the velocity and pressure variations of blood flow for a range of simulated domains. The simulation through the ballonlike protrusion in a bulged arterial wall reveals that almost all models give very similar results for velocity and pressure distribution concluding that all five non-Newtonian models are suitable for the analysis. A striking change in pressure was observed in the area where the bulge is located. When considering an aneurysm type domain of a protrusion at the juncture between two arteries the velocity and pressure variations through the swelling area delineates a drastic change for the Herschel-Bulkley model with regard to other models. However, the remaining rheological models have approximately the same values outlining the minuscule deviations from one another. Hence, Herschel-Bulkley model is not appropriate in blood flow simulation of bulge near arterial junction aneurysms as for our work. The flow parameter variations in a stenosed arterial wall are also simulated and it was observed that all the rheological models are perfect for the modelling of velocity variation. As for the pressure variation determination it can be presumed that the values of the Power law, Herschel-Bulkley law, and Casson law models are exactly the same, while approximately similar values are observed in the remaining models. Further, a mathematical model describing plaque formulation has been analysed and simulated. The results of the simulations conclude that when the plaque growth obstructs the blood flow through an artery, the sporadic microscopic increase in pressure and a decrease in velocity near the lesion swelling area.