Third Order Cnoidal Wave Solutions in Shallow Water and Its Horizontal and Vertical Fluid Velocity Components

International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051

Third Order Cnoidal Wave Solutions in Shallow Water and its Horizontal and Vertical Fluid Velocity Components

Parvin, M. S. Sultana, M. S. Sarker, M. S. A. Assistant Professor, Dept. of Applied Associate Professor, Dept. of Applied Assistant Professor, Dept. of Applied Mathematics, University of Rajshahi, Mathematics, University of Rajshahi, Mathematics, University of Rajshahi, Rajshahi-6205, Rajshahi-6205, Rajshahi-6205, , E-mail: E-mail: [email protected] E-mail: [email protected] [email protected]

Abstract – Third order cnoidal wave solutions in shallow real wave in shallow water. Higher order solutions of water are developed where waves progress steadily without cnoidal waves have been derived by Laitone [15], who any change of form. Shallow water wave problems are solved provided a number of results, re-casting the series in terms at the bottom and at the free surface. The boundary of the wave height/depth. Tsuchiya and Yasuda [21] conditions are also taken from Navier-Stokes equation of obtained a third order solution with the introduction of motion. Using these boundary conditions, three nonlinear ordinary differential equations are derived which can be another definition of wave celerity based on assumptions solved using series expansion method. Taking Jacobi elliptic concerning the Bernoulli constant. Nishimura et al. [16] function, third order cnoidal wave solutions have been devised procedures for generating higher order theories for established from which wave elevation, horizontal wave both Stokes and cnoidal theories, making extensive use of velocity and acceleration due to gravity for cnoidal wave are recurrence relations. Nishimura et al. [17] continued the expressed. Assuming Taylor expansion for the stream work of Nishimura et al. [16] and presented a unified view function about the bed, the fluid velocity components are of Stokes and cnoidal theories. Karabut [12] solved an derived in terms of Jacobi elliptic function. ordinary quadratic nonlinear differential difference

equation of the first order containing an unknown function Keywords – Navier-Stokes Equation, Jacobi Elliptic Function, Cnoidal Wave, Elliptic Parameter, Velocity under certain conditions. Halasz [10] discussed on higher Components. order corrections for shallow water solitary waves. Chen et al. [4] studied the cnoidal wave solutions of the I. INTRODUCTION Boussinesq systems in two different techniques using the Jacobi elliptic function series. Carter et al [3] discussed the In recent years, the capabilities for calculating shallow kinematics and stability of solitary and cnoidal wave water wave problems have advanced a great deal. The solutions of the Serre equations which are a pair of computation of higher order cnoidal wave solutions for strongly nonlinear, weakly, dispersive, Boussinesq type steady, incompressible flow is an important aspect in partial differential equations. They [3] also described the coastal engineering. Several higher order approximations model of the surface elevation and the depth averaged to irrotational water waves of constant form have appeared horizontal velocity of an inviscid, irrotational, in recent years, often based on Fourier series, but where incompressible shallow water. Jain et al. [11] derived convergence is slow, if at all, for shallow water. These coupled evolution equations for first and second order solutions are often numerical and of an inverse potentials using reductive perturbation method with formulation, and are generally of such higher order, that it appropriate boundary conditions. Oh and Watanabe [18] is difficult to obtain expressions for physical quantities as obtained the second order solution by taking into account functions of position for practical use. The presentation of the unstable and dissipative effects based on a cnoidal results has been limited to tables of integral quantities for wave solution to the KdV equation. Xu et al. [22] a range of wave lengths and heights. However, these calculated the cnoidal function in cnoidal wave theory methods have achieved real success in obtaining based on the precise integration method and also provided numerically exact solutions for the first time Schwartz a trigonometric function approximation for the cnoidal [20] and Cokelet [5]. A survey and comparison of the function. Parvin et al. [19] derived first and second order methods is given Cokelet [5]. cnoidal wave solutions using Jacobi elliptic function. Fenton [6] presented a fifth order cnoidal wave theory Khater et al. [13] used a suitable ansatz and Jacobi elliptic where boundary condition comes from Bernoulli’s function expansion method to construct new exact cnoidal equation which was both apparently complicated, wave solutions of the modified fifth order KdV equation requiring the presentation of many coefficients as and generalized fifth order KdV equation which included unattractive floating point numbers and also gave poor as special cases, some well known equations. Fu et al. [8] results for fluid velocities under high waves. However, in applied Jacobi elliptic function in Jacobi elliptic function a later work, Fenton [7] showed that instead of fluid expansion method to construct the exact periodic solutions velocities being expressed as expansions in wave height, of nonlinear wave equations. In this paper, shallow water the original spirit of cnoidal theory were retained. Cnoidal wave problems have been solved using boundary theory obtained its name in 1895 when Korteweg and_de conditions, at the bottom Y = 0 and at the free Vries [14] obtained their eponymous equation for the surface Y = η(X ). Also the boundary conditions from propagation of waves over a flat bed. The cnoidal solution Navier-Stokes equation of motion generate third order shows the familiar long flat troughs and narrow crests of Copyright © 2016 IJISM, All right reserved 164 International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051

cnoidal wave solutions. Then horizontal and vertical fluid ∂U ∂U  ∂ 2U ∂ 2U  velocity components have been established in terms of U +V = υ +  on free ∂ ∂  ∂ 2 ∂ 2  Jacobi elliptic function. X Y  X Y  surface Y = η (X ) ,pressure p = 0 , (3) II. MATHEMATICAL FORMULATION and ∂ ∂  ∂ 2 ∂ 2  V + V = +υ V + V  (4) Consider the wave as shown in figure-1, with a U V g  2 2  ∂X ∂Y  ∂X ∂Y  stationary frame of reference (x, y) , x in the direction of We assume a Taylor expansion for ψ about the bed of propagation of the waves and y vertically upwards with the following form the origin on the flat bed. The waves travel in the x  d   d Y 3 d 3 Y 5 d 5  ψ ()X ,Y = −sin Y  f (X ) = − Y + − + ...... f ()X direction at speed c relative to this frame. Consider also a  3 5   dX   dX !3 dX !5 dX  frame of reference (X ,Y ) moving with the waves at (5) velocity c , such that x = X + ct , where t is time df = as in Fenton [7], where is the horizontal velocity on and y Y . The fluid velocity in the (x, y) frame is dX (u,v) and that in the (X ,Y ) frame is (U,V ) . The the bed. Now, the velocity components anywhere in the fluid are velocities are related by u = U + c and v = V . ∂ψ  d  In the (X ,Y ) frame, all fluid motion is steady and U = = −cos Y  f ′()X ∂  dX  consists of a flow in the negative X direction, roughly of Y (6) the magnitude of the wave speed, underneath the ∂ψ =  d  ′() stationary wave profile. The mean horizontal fluid velocity V = − sin Y  f X ∂X  dX  in this frame, for a constant value of Y over one (7) wavelength λ is denoted by -U . It is negative because From Eqs. (3) and (4), we get the apparent flow is in the - X direction. For the convenience of our calculation, the velocities in this frame  d   d     ′   ′′  are used to obtain the solutions.  cos Y f (X) cos Y f (X)  dX   dX   λ Y  d  ′  d  ′′  +sin Y f (X)sin Y f (X) = 0  dX   dX   H c (8) and X  d  ′  d  ′′  −cos Y f (X)sin Y f (X)  dX   dX   y h  d  ′  d  ′′  +sin Y f (X)cos Y f (X) = g x d  dX   dX   (9) At the free surface = η Eqs. (2), (8) and (9) Fig. 1. Wave train, showing important dimensions and Y (X ) , coordinates. become

 d  For irrotational flow, stream function satisfies Laplaces sin η  f (X ) = Q (10) ∂ 2ψ ∂ 2ψ  dX  equation + = 0 (1) ∂X 2 ∂Y 2  d  ′  d  ′′  cos η f (X)cos η f (X) The boundary conditions at the bottom Y = 0 is a  dX   dX   stream line on which ψ (X ,Y ) is constant and at the free  d  ′  d  ′′  surface Y = η (X ) is also a stream line. +sin η f (X)sin η f (X) =0  dX   dX   ∴ψ (X )0, = 0 (Taking zero constant) and ψ (X ,η(X )) = −Q (2) (11) and where Q is the volume flux underneath the wave train per  d  ′  d  ′′  unit span. The negative sign is for the flow which is in the −cos η f (X)sin η f (X)  dX   dX  negative X-direction, such that the wave will also    propagate in the positive X- direction.  d  ′  d  ′′  (12) For two components, Navier-Stokes equation of motion +sin η f (X)cos η f (X) = g for steady incompressible flow is  dX   dX   Copyright © 2016 IJISM, All right reserved 165 International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051

Differentiating Eq. (10) and then substituting this value IV. THIRD ORDER CNOIDAL WAVE SOLUTION in Eqs. (11) and (12), we have  d  ′  d  ′′  To derive third order cnoidal wave solutions, we need cos η f (X)cos η f (X)  dX   dX   some values from Parvin et al. [19] + = η F1 Y1 0  d  d  ′  d  ′′  − cos η f (X)sin η  f (X) = 0 ′ =  (21) dX  dX   dX   F1 0  ′′+ =  F1 g1 0 (13) and + − 2 − 1 ′′=  F2 Y2 F1 F1 0  d    d   6 cosη fX′ ()   sin η  fX ′′ ()   dX    dX   F ′′′  (22) F F′+ F′ − 1 = 0  dη  d    d   1 1 2 + cosη fX′ ()   cos η  fX ′′ ()  = − g 2  dX dX    dX    ′′ − ′2 − 1 iv + = F2 F1 F1 g 2 0  (14) 6  Eqs. (10), (13) and (14) are three nonlinear ordinary and + + + − 1 ′′ − 1 ′′ + 1 iv =  differential equations in the unknowns Y3 Y2 F1 Y1 F2 F3 F2 Y1 F1 F1 0 η( X ), f ′( X ) and g . These ordinary differential 6 2 120  1 1 1 equations can be solved using power series method. F ′ − Y F ′′′ − F ′′′ + F v + F F ′ − F F ′′′ + F ′F  3 1 1 2 2 24 1 1 2 2 1 1 1 2   − 1 ′ ′′ − ′ ′′ =  III. POWER SERIES SOLUTION F1 F1 Y1 F1 0 2   1 Assuming constant depth η()X= handf′ (), X = U the F ′Y ′ + F ′Y ′ − F Y ′′′′ + F F ′Y ′ + F ′′ + Y F ′′  1 2 2 1 2 1 1 1 1 1 3 1 2  derived equations can be solved using series expansion  method about the state of a uniform critical flow. Let the + ′′ − 1 iv − 1 iv + 1 vi + ′′  Y2 F1 F2 Y1 F1 F1 F1 F2 scaled horizontal variable be 6 2 120   θh − 1 iv + ′′ + ′′ − 1 ′′ 2 + = X = ,η =η h and f = f Q Eqs. (10), (13) and (14) can F1 F1 F2 F1 Y1 F1 F1 F1 g 3 0  α * * 6 2  be rewritten in terms of these dimensionless quantities (23) 1  d  1 Using the series expansions (18), (19) and (20) and η α θ − = ≈ (15) 8 sin * f*( ) 1 0as 1 taking (α ) Eqs. (15), (16) and (17) can be written as α  dθ  h o ,

 d   d    ηα ′θ  ηα ′′θ  cos * f*( )cos * f*( )  1   dθ   dθ   α 2 ()Y + F + α 4 Y + F Y + F − F ′′ 1 1  2 1 1 2 6 1  dη  d   d   −α *  ηα  ′θ  ηα  ′′θ  =  cos * f*( ) sin * f*( ) 0   dθ  dθ   dθ   + + + − 1 ′′ Y3 Y2 F1 Y1 F2 F3 F2  + α 6  6  (16)  1 1  and  − Y F ′′+ F iv   2 1 1 120 1  dη  d   d   *  η α ′ θ  η α ′′ θ  cos *  f* ( )cos * . f* ( )  1  dθ  dθ   dθ    F + F Y + F Y + Y F + Y − Y F ′′ 4 3 1 2 2 3 1 4 2 2 1 1  d   d     + cos η α  f ′(θ )sin η α  f ′′(θ) = −g   α  * θ *  * θ *  * + α 8 − 1 2 ′′− 1 ′′ − 1 ′′ + 1 iv =  d   d    Y1 F1 Y1 F2 F3 F2  0 2 2 6 120 1 gh   as ≈ 1and = g 2 2 *  1 1  α Q  + Y F iv − F vi   24 1 1 5040 1  (17) (24) Using α 2 as the expansion parameter, the series   α2′+ α 4 ′ + ′ − 1 ′′′ expansions are in the following F1 FFF 112 F 1  2  N η = + α 2 j θ 1 1  * 1 ∑ Y j ( ) (18) ′−− ′′′ ′′′ +v + ′ FYF311 F 2 F 1 FF 12  j=1 +α6 2 24  N −1′′′ +− ′ 1 ′ ′′ − ′ ′′  2 j FF11 FF 12 FF 11 YF 11  ′ = + α θ 2 2  f* 1 ∑ F j ( ) (19) j=1 N = + α 2 j g * 1 ∑ g j (20) j=1 where N is the order of solution required. Copyright © 2016 IJISM, All right reserved 166 International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051

′−−++ ′′′ ′′′ ′ ′ +− ′ ′′′ − ′ ′′ ′ − ′′′− ′′′+ ′ + ′ + ′− ′′′  F YF YF FF FF FF FYF FYF  F4 Y2 F1 Y1F2 F1F3 F2 F2 F3 F1 F1Y1F1  4 21 12 13 22 31 111 111     ′′′+ ′′′ + ′′ ′ − 1 ′′ ′′′   ′′′ + ′′′+ ′′ ′ − 1 ′′ ′′′+ ′ ′′ 1 FF12 FF 21 FF 12 FF 11  1  F F F F F F F F F F   − 2  − F′Y F ′′− 1 2 2 1 1 2 2 1 1 1 2  2    1 1 1   +′ ′′ +2 ′′′ + ′′′  2  2  8  FF12 YF 11 F 3   + ′′′+ ′′′ +α = 0  Y1 F1 F3   1 1   +()v +++ v v′ iv − vii − ′ ′′  1 1 4YF11 F 2 FF 11 FF 11 F 1 YYF 111 + ()v + v + v + ′ iv − vii  24 720  4Y1F1 F2 F1F1 F1F1 F1 24 720  1   −YF′ ′′ + YF ′iv − Y ′ F′′− FFY ′′ ′  12 11 21 111 − ′ ′′− ′ ′′ + 1 ′ iv − ′ ′′− ′′′ =  6  Y1Y1F1 Y1F2 Y1F1 Y2 F1 F1F1Y1 0 6 (25) (28) and and F′Y ′− Y Y ′F′′′+ F F′Y ′+ F F′Y ′+ F F′Y ′ + F′Y ′ + F′Y ′   3 1 1 1 1 1 2 1 2 1 1 1 1 2 1 3 2 2 α2′′+ α 4 ′ ′ ++− ′′ ′′1 iv + ′′ F1 FYYFF 11112 F 1 FF 11  1 1 6  − ()F Y ′′′′ + F F Y ′′′′ + F′F Y ′′′ + F Y ′′′′ + F vY ′+ F ′′ + Y F ′′ 2 2 1 1 1 1 1 1 1 2 24 1 1 4 3 1  1  ′ ′− ′′′ ′ + ′ ′ ++ ′ ′ ′′ 1  FY21 FY 11 FFY 111 FY 12 YF 21  + ′′ + ′′ − ()iv + iv + iv 2 Y2 F2 Y1F3 3Y2 F1 3Y1F2 F3   6 1 +++−α 6  ′′ ′′ ()iv ++ iv iv  1 1  YF12 F 33 YF 11 F 2 FF 11  + ()vi + vi − viii + ′′+ ′′ + ′′ 6 5Y1F1 F2 F1 Y2 F1F1 Y1F1F2 F1F3   120 5040 1vi 1 2  + ++′′ ′′ +− ′′ ′′  1 iv iv 1 vi  F1 FF 12 YFF 111 FF 12 F 1  − ()3Y F F + F F + F F + Y F ′′F + F F ′′  120 2  6 1 1 1 1 2 120 1 1 1 1 2 2 2  FY′′− YYF ′′′′ + FFY ′′ + FFY ′′ + FFY ′′  31 111 121 211 11 1 1  1    − F iv F + F ′′F − 3Y F ′′ 2 + 2F F ′′′′ − F ′′F iv  = −g  1 FY′′′ ′+ FFY ′′′ ′ + FFY ′ ′′ ′   1 2 1 3 1 1 1 2 1 1 4 +FY′′ + FY ′′ − 21 111 11  6 2  4   2 13 22 2 +F Y′′′ ′   1 2   (29)   +1 v ′ +++ ′′ ′′ ′′ + ′′ From Eqs. (27), (28) and (29), we have  FY11 F 4 YF 31 YF 22 YF 13   24  1 1 1  1 1  F iv − F F ′′ − F iv F − F F ′′′′ − 2F ′ 2 − F F ′′ −()3YFiv +++ 3 YF iv F iv() 5 YF vi + F vi 3 1 3 1 2 1 2 2 2 2  621 12 3 120 11 2  3 6 3 +α 8   = 1 0 7 1 iv 1 vi 2 7 iv  −Fviii + Y F F′′+ YFF ′′ + FF ′′  − F ′′F ′′ − F F − F + F F ′′ + F ′′F  5040 1 2 11 112 13  6 1 2 6 1 2 30 2 1 1 60 1 1   1 1  −()3YFFiv ++ FF iv FF vi + YFF′′  1 2 iv 11 vi 1 viii  111 12 11 112  + F F + F F + F + g = 0 6 120 2 1 1 120 1 1 840 1 4  1   +FF′′ − FFiv + FF ′′   226 12 13   1 1    −′′2 + ′′ ′′ − ′′ iv  (30)  3YF11 2 FF 12 FF 11    2 4   According to the Parvin et al. [19], the solutions for =−+α2 + α 4 + α 6 + α 8 2 ()1 g1 g 2 g 3 g 4 (θ ) F1 and F2 in terms of cn m are

(26) = − 2 (θ ) F1 5mcn m (31) 8 Equating the coefficients of α from Eqs. (24), (25) and and (26), we have = 1 ()− + 2 F2 10192 55927 m 55927 m 1 1 2 675 F + F Y + F Y + Y F + Y − Y F ′′− Y F ′′ 4 3 1 2 2 3 1 4 2 2 1 2 1 1 1838 209 + m()1 − 2m cn 2 ()θ m + m 2 cn 4 ()θ m 1 1 1 1 45 15 − Y F ′′ − F ′′ + F iv + Y F iv 2 1 2 6 3 120 2 24 1 1 (32) 1 − vi = Now, assuming F3 in terms of Jacobi elliptic function as F1 0 5040 Fenton [6], we have (27) F = b cn 2 ( θ m ) + b cn 4 ( θ m ) + b cn 6 ( θ m) 3 1 2 3 (33)

2 2 [3b (1− 2m + m )− b (1− 3m + 2m )]  919 2  2 1  ()()1−−mm 1 2() 10192 − 55927 mm + 55927    675   1− 3m     45 b ()1− 2m + m 2 − 30 b    1 18732896− 222233392m + 839954703 m 2   + 3 2  + 2  2 ()θ  −   cn2 ()θ m    2m cn m  3 4     675  −1235442622m + 617721311 m    + ()− + 2      b1 2 17 m 17 m  4 1   −−−mmm ()1 2() 6955636 − 37176844 m + 37176844 mcnm2 4 ()θ   −195 b ()1− 3m + 2m 2   45 45  3       1 2 2 6 ∴ F iv = 8+ + b ()32 − 212 m + 212 m 2 cn 4 ()θ m  −m()12959624 − 56186114 m + 56186114 m cn()θ m  3  2   45      +15 b m()1− 2m  60421 9    1   − m3()1− 2 mcn 8()θ m − 43681 mcn 4 10 () θ m     3   ()− + 2     b3 162 1017 m 1017 m  +  cn 6 ()θ m  2   3676− 16585 m   +130 b m()1− 2m +15 b m  9192 1 2   2 1    ()()1−mm 12 −−() 1 − m  cnm(θ ) + 2 + []()− + 8 ()θ   45 45 16585 m   105 m 5b3 1 2m b2 m cn m     1 + 2 10 ()θ +−()1 2m() 3676 − 26488 m + 26488 mcnm2 4 ()θ   378 b m cn m  140 2 3 + m  45  (34) 3  1  +m()5348 − 24318 m + 24318 mcnm2 6 ()θ  Substituting these values in Eq. (30), we have  15    [ ( − + 2 )− ( − + 2 )]  1471 2 8 3 10 3b2 1 2m m b1 1 3m 2m +m()1 − 2 mcn()θ m + 209 m cn()θ m     3   ()()− + 2 − − + 2   45 b3 1 2m m 30 b2 1 3m 2m    + 2 ()θ 1 2   cn m   ()1−m() 1838 − 9233 m + 9233 m  + ()− + 2 45   b1 2 17 m 17 m      2 2 2   2  −−()1 2m() 1838 − 25028 m + 25028 m cn()θ m − ()− +  20 2 2   195 b3 1 3m 2m − m cn()θ m 45     3   + + ()− + 2 4 ()θ  −2 − + 2 4 ()θ  8  b2 32 212 m 212 m cn m m()23817 121602 m 121602 m cn m    45      +15 b m()1− 2m 2 6 38 3   1   −2424m() 1 − 2 m cn()θ m − 1463 m cn() θ m      () − + 2   ()()1−−mm 1 2( 1838 − 25028 mm + 25028 2 )  b3 162 1017 m 1017 m 6   +  cn ()θ m  + ()− + 2  1838− 103831m + 568749 m 2     130 b2m 1 2m 15 b1m   −2   cn2 ()θ m   −929836m3 + 464918 m 4   + []()− + 8 ()θ    105 m 5b3 1 2m b2m cn m 16     −−−mmm ()1 2() 196050 − 1866000 m + 1866000 mcnm2 4 ()θ  2 10 + 378 b m cn ()θ m  675   3 −10m2 () 121989 − 606459 m + 606459 m2 cn 6 ()θ m  2   21b()()()−+ m 121 b −− mb 412 − mcn  ()θ m  3 8 410 1 2 1  −2198700m() 1 − 2 m cn()θ m − 1185030 m cn() θ m      +5mcn2 ()θ m + 30 b()() 1 −−−− m 16 b 1 2 m 6 bmcn  4 ()θ m     3 2 1     2 +−−−3612b() m 20 bmcn  6()θ m − 42 bmcn 8 () θ m  ()1−m − 413( − m + 2 m2) cn 2 ()θ m   3 2  3     10192− 86503m + 244092 m   −500m3 cn 2 ()θ m +−+ 2() 2 11 m 11 m2 cn 4 () θ m  − 1    3 4    675 −279635m + 111854 m   +12m() 1 − 2 m cn6()θ m + 9 m 2 cn 8 () θ m      − + 2   1 20384 312688m 1373727 m  2 +   cn2 ()θ m ()()()− −−−() − + 2 2 ()θ   3 4   1m 1 2 m 1 m 4 25 mmcnm 25 675 −2122078m + 1061039 m       2 4    140 +−4() 1 2m() 1 − 13 m + 13 m cn()θ m 20 1 2 4 − 2   + m +−mm()1 2() 13868 − 87800 m + 87800 mcnm()θ  m 3 45 3 + − + 2 6 ()θ     6m() 6 31 m 31 m cn m   2  +2m − + 2 6 θ  2 8 310   ()19508 88433m 88433 m cn() m  +75m() 1 − 2 m cn()θ m + 45 m cn() θ m   45 

+2465 3 − 8θ + 4 10 θ   m()1 2 m cn() m 209 m cn() m  3 −−+(13mm 22) +−( 217 m + 17 mcn 2) 2 ()θ m    3 4 −500 m cn()θ m      4 2 6  +1512m() − m cn()θ m + 15 m cn() θ m  18381( − 5m + 8 m2 − 4 m 3) cn 2 ()θ m    22()− 19m + 34 m2 − 17 m 3    2 −− − + 2 4 θ  16 m 21() 2m() 919 7060 m 7060 m cn() m  2 2  −   55 −−81() 2m() 131 − m + 31 m cn()θ m + m2 cn 2 ()θ m   27  2 6  −2m() 4303 − 19093 m + 19093 mcnm()θ 3 −() − + 2 4 ()θ     252m 1 6 m 6 m cn m  −2 − 8θ − 3 10 θ  −2 − 6θ − 38 θ   10530m() 12 mcn() m 3762 mcn() m   840m() 12 m cn() m 630 m cn() m  8445611()()−m 1 − 2 m2 cn2 ()θ m  ()1−m() 133 − mm + 932 − 62 m 3      −−()1 2m() 844561 − 4530670 m + 4530670 m2 cn 4 ()θ m  − ()2−+ 259m 1641 m2 − 2764 m 3 + 1382 mcnm 42 ()θ      32 2  2 6  16  2 3 4  −mm −()1996987 − 8381077 m + 8381077 mcnm()θ + m −−+−5m() 51 738 m 1908 m 1272 mcnm()θ += g 0 2025   21   4  2 8  −1545555m() 1 − 2 m cn()θ m −2 − + 2 6 θ     315m() 7 37 m 37 m cn() m  −393129 m3 cn 10 ()θ m      −4725m3() 12 − m cn 8()θ m − 2835 m 4 cn 10 () θ m     (35)

International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051

Now equating the coefficients of like power of cn 2 (θ m) f ′ = 1 − 5mα 2cn 2 (θ m) from Eq. (35), we obtain the values of b ,b and b , i. e., * 1 2 3  1  ()10192 − 55927 m + 55927 m 2 1 2   b = − (2274828166 −1282861191 1m +1282861191 1m )m 675 1 4039875    1838  (36) + α 4 + m()1 − 2m cn 2 ()θ m   45   86755418  2 = − ()−  209 2 4  b2  m 1 2m + m cn ()θ m   269325   15   1  2274828166 − 1282861191 1m   (37) − m cn 2 ()θ m   2   and  4039875  + 1282861191 1m       6  86755418  2 4 = − 3769211 3 + α −  m ()1 − 2m cn ()θ m  b3  m   269325    89775    (38)  3769211  −  m 3cn 6 ()θ m  Substituting the values in Eq. (33), we get   b ,1 b2 and b3   89775   1 2274828166 −1282861191 1m (42) F = − m cn 2 ()θ m 3  2  Also, from Parvin et al. [19], Eq. (22) can be written 4039875 +1282861191 1m  16 40  86755418   3769211  g = 1− α 2 ()m 2 − m +1 − α 4 m(1− 2m )(1− m ) −  m 2 ()1− 2m cn 4 ()θ m −  m3cn 6 ()θ m *  269325   89775  3 3 64 − α 6 m()1− m ()20 −173 m +173 m 2 27 (39) (43) From Eq. (23), the value of Y3 is Eqs. (41), (42) and (43) are the third order cnoidal wave solutions i.e. which express wave elevation, horizontal = 1793 (− )(− ) Y3 m 1 m 1 2m wave velocity and acceleration due to gravity for cnoidal 135 wave. 1 1557525916 − 9086341036 m  + m cn 2 ()θ m  + 2  4039875  9086341036 m  V. RESULTS AND DISCUSSION 24954607 7739261 − m 2 ()1 − 2m cn 4 ()θ m + m 3cn 6 ()θ m 269325 89775 In Eqs. (41), (42) and (43) we have obtained the third (40) order cnoidal wave solutions. These solutions are obtained Substituting these value in Eqs. (18) and (19), we obtain for elliptic parameter which is less than one (assuming five numbers say 0.1, 0.2, 0.3, 0.4 and 0.5) and Jacobi elliptic  1 ()− + 2   10192 54802 m 54802 m  function which tends to one. In Eqs. (41), (42) and (43)  675    η = + α 2 2 ()θ − α 4 + 1688 ()− 2 ()θ taking expansion parameter as positive and negative and * 1 5m cn m  m 1 2m cn m   45  so on, it is noted that y1, y2, y3, y4 and y5 are solution  241 2 4  − m cn ()θ m  curves of Eqs. (41), (42) and (43) shown in Fig. 2-7. In  15  Fig. 2-7, we observe that the variations of elliptic 1793 (− )(− )   m 1 m 1 2m  parameter (m=0.1, 0.2, 0.3, 0.4 and 0.5) cause significant  135  changes in the third order cnoidal wave solutions. In Figs.  1 1557525916 − 9086341036 m   + α 6 + m cn 2 ()θ m 2 and 3, it is mentioned that the first two curves of third   + 2    4039875  9086341036 m   order cnoidal wave increase with the increases of elliptic  24954607 7739261  − m 2 ()1 − 2m cn 4 ()θ m + m 3 cn 6 ()θ m  parameter and the last three curves of third order cnoidal   269325 89775 wave decrease with the increases of elliptic parameter. In (41) Figs. 4 and 5, the first two curves of third order cnoidal wave decrease with the increases of elliptic parameter and the last three curves of third order cnoidal wave increase with the increases of elliptic parameter. In Figs. 6 and 7, the first curve of third order cnoidal wave decreases with the increases of elliptic parameter and the last four curves firstly decrease and finally increase rapidly with the increases of elliptic parameter.

15 15 m=0.4 m=0.5 m=0.1 10 10 m=0.2 y5 y1 y4 y3 5 5 y2 m=0.3

0 0

-5 -5 y3 Third order cnoidal wave solution m=0.3 Third order cnoidal wave solution -10 y4 y2 y1 y5 -10 m=0.4 m=0.5 m=0.2 m=0.1 -15 0 0.5 1 1.5 -15 Expansion parameter -1.5 -1 -0.5 0 Expansion parameter Fig. 2: Third order cnoidal wave solution due to Jacobi elliptic function cn 2 (θ m) = 1 Fig. 5: Third order cnoidal wave solution due to Jacobi 2 15 elliptic function cn (θ m) = 1.

6 10 m=0.1 m=0.2 4 5 y1 m=0.5 y2 2 y5 m=0.4 0 y4 m=0.3 0 y3

-5 -2 y2 m=0.2 y3 Third order cnoidal wave solution y1 y4 Third order cnoidal wave solution -10 y5 -4 m=0.3 m=0.1 m=0.4 m=0.5 -15 -6 -1.5 -1 -0.5 0 0 0.5 1 1.5 Expansion parameter Expansion parameter Fig. 6: Third order cnoidal wave solution due to Jacobi Fig. 3: Third order cnoidal wave solution due to Jacobi elliptic function cn 2 (θ m) = 1. elliptic function cn 2 (θ m) = 1. 6

15 4 m=0.5 m=0.5 m=0.4 10 m=0.4 2 y5 y4 y5 m=0.3 y4 m=0.3 5 y3 0 y3

0 m=0.2 -2 y2 y1 Third order cnoidal wave solution -5 -4 m=0.1

Third order cnoidal wave solution y2 -10 y1 -6 -1.5 -1 -0.5 0 m=0.1 m=0.2 Expansion parameter -15 Fig. 7: Third order cnoidal wave solution due to Jacobi 0 0.5 1 1.5 Expansion parameter 2 (θ ) = elliptic function cn m 1. Fig. 4: Third order cnoidal wave solution due to Jacobi elliptic function cn 2 (θ m) = 1.

  − + 2   VI. VELOCITY COMPONENTS 22192 82619 m 82146 m   8   −11054 m 3 + 8527 m 4 − 9000 m5   253125   To derive velocity components in fluid, Taylor  + 6     12000 m   expansion for the stream function is assumed. Then from   2050270966  Eq. (6), we obtain for dimensionless form,      −1180481781 1m   consider X = hX , Y = hY , f ′ = fQ ′,     * * * − m + 2 2 ()α   1175095281 1m cn X * m     504984375  U = −  d  ′ + 718439400 m 3  Q *  cos Y*  f * ( X * )      4  g  dX *   + 215460000 m   h    2m 2 38374249 − 81751958 m   2  −  cn 4 ()αX m 1  Y    2  *  ′ − ′′′ 33665625 − 5003460 m   f * ( X * )   f * (X * )    1 −  2  h    3769211  = −()g 2 − 3 6 ()α  *  4  3 m cn X * m   − δ  11221875  + 1 Y v −    f * ( X * ) ......      4 ()− − 2 ()−   24  h     619 1538 m 300 m m 1 m  (44)  1125    1238 − 7433 m +   − 4   2 ()α Hence from Eq. (43) and neglecting higher order term, we  2  mcn X * m  1  Y  1125  + 3  −    7433 m 1200 m   get  10  h    4 2 2 4 1  − ()97 − 214 m − 20 m m cn ()αX m  − 8  25 *  2 = + δ ()2 − +   g * 1 m m 1    − 836 3 6 ()α 15   m cn X * m    75  + 4 δ 2 ()− + 2 − 3 + 4  m(1 − m )(1 − 2m )   8 11 m 9m 6m 8m      4 − ()− + 2 2 ()α  75 1  Y   m 2 17 m 17 m cn X * m  +    2 3 − 2 ()− 4 ()α   40 − 55 m − 133 m + 336 m   75  h  15 m 1 2m cn X * m  32 3     + δ  − 3 6 ()α    4 5 6    15 m cn X * m   3375  − 158 m − 30 m + 40 m  (45) θh Since X = hX = , then Eq. (42), by differentiating * α in four times and substituting these necessary values in Eq.(44) , we have   − + δ  8 ()2 − + − 2 ()α   1  m m 1 mcn X * m    15    2   4 17392 − 65827 m + 64027 m         3 4   16875 − 5400 m + 7200 m      2 U  + m()619 −1538 m − 300 m2 cn 2 ()αX m  = −  1125 *  g    + δ 2  209  h  + m2cn 4 ()αX m    375 *     − 2m()1− m     2   1  Y   2    −   + 4m()1− 2m cn ()αX m     10  h  *   + 2 4 ()α      6m cn X * m  

 Y  EFERENCES = 2α cn ()αX m sn ()αX m dn ()αX m R  h  * * * [1] M. Abramowitz, and I. A. Stegun, “Handbook of Mathematical      2 ()− − 2  Functions”, Dover, New York, 1965.  m 619 1538 m 300 m   1125   [2] M. Abramowitz, and I. A. Stegun, “Handbook of Mathematical     Functions with Formulas, Graphs and Mathematical Tables” , δ −δ 2 + 418 2 2 ()α NBS Applied Mathematics Series, Vol. 55, National Bureau of  m  m cn X * m     375   Standard, Washington DC, 1964.   2   [3] J. D. Carter, and R. Cienfuegos, “The Kinematics and Stability − 1  Y  {}()− + 2 2 ()α of Solitary and Cnoidal Wave Solutions of the Serre     4m 1 2m 12 m cn X * m     30  h    equations”, European journal of Mechanics b: Fluids, Vol. 30,   2011, pp. 259-268.    [4] H. Chen, C. Min, and N. Nghiem, “Cnoidal Wave Solutions to    Boussinesq Systems” , IOP Publishing, Nonlinearity, Vol. 20,    2006, pp. 1443-1461.    [5] E. D. Cokelet, “Steep Gravity Waves in Water of Arbitrary    Uniform Depth” , Phil. Trans. R. Soc. A, Vol. 286, 1977, pp.    2050270966 −1180481781 1m   183-230.      m 2 3  [6] J. D. Fenton, “A Higher Order Cnoidal Wave Theory” , J. Fluid   +1175095281 1m + 718439400 m   Mech. Vol. 94, 1979, pp 129-161. 504984375     + 4  [7] J. D. Fenton, ”Nonlinear Wave Theories” , The Ocean    215460000 m     Engineering Science, Part A,B. Le, 1990.  2  −   [8] Z. Fu, S. Liu, S. Liu, and Q. Zhao, “New Jacobi Elliptic  4m 38374249 81751958 m 2   +  cn ()αX m  Function Expansion and New periodic Solutions of Nonlinear   2  *   33665625 − 5003460 m   Wave Equations”, Physics Letter A, Vol . 290, ,Issue 1-2, 2001,      pp. 72-76. + δ 3 + 3769211 3 4 ()α   m cn X * m  [9] I. S. Gradshteyn, and I. M. Ryzhik, “Table of Integrals, Series   3740625  and Products”, Fourth Edn. Academic, 1965.     [10] G. B. Halasz, “Higher Order Corrections for Shallow Water 4m ()− + + 3    1238 7433 m 7433 m 1200 m  Solitary Waves: Elementary Derivation and Experiments”,   1125  Eur. J. Phys. Vol. 30, 2009, pp. 1311-1323.  2   1  Y   8 2 2 2  [11] S. L. Jain, R. S. Tiwary, and M. K. Mishra, “Ion-Acoustic  −   + ()97 − 214 m − 20 m m cn ()αX m   30  h   25 *  Cnoidal Wave and Associated Nonlinear Ion Flux in Dusty     Plasma”, Physics of Plasmas, Vol. 19, No. 10, 2012, DOI.  836   + m3cn 4 ()αX m  1063/1.4757222.    *    25   [12] E. A. Karabut, “Higher Order Approximations of Cnoidal Wave   Theory”, Journal of Applied Mechanics and Technical Physics,   m()2 −17 m +17 m2    4    Vol. 41 , Issue 1, 2000, pp. 84-94. + 1  Y  + 2 ()− 2 ()α  [13] A. H. Khater, M. M. Hasan, and R. S. Temsah, “Cnoidal Wave      30 m 1 2m cn X * m    375  h     Solution for a Class of Fifth Order KdV Equations,  + 45 m3cn 4 ()αX m  Mathematics and Computers in Simulations”, Vol. 70, Issue 4,    *   2005, pp. 221-226. To obtain the vertical component of fluid velocity, we [14] D. J. Korteweg, and G. de_Vries, “On The Change of Form of can use the equation that the fluid motion is Long Waves Advancing in a Rectangular Canal and on a New incompressible, Type of Long Stationary Waves” , Fhil. Mag.(5), Vol. 39, 1895, pp 422-443. ∂U ∂V + = 0 [15] E. V Laitone, “The Second Approximation to Cnoidal and ∂ ∂ Solitary Waves”, J. Fluid Mech. Vol. 9, 1960, pp 430-444. X Y [16] H. Nishimura, M. Isobe, and K. Horikawa, “Higher Order   Solutions of the Stokes and the Cnoidal Waves” , J. Faculty of V Y* ∂  U  (46) Eng., The University of Tokyo, Vol. 34, 1977, pp 267-293. ∴ = −  dY [17] H. Nishimura, M. Isobe, and K. Horikawa, “Theoratical ∫ ∂X * g 0 *  g  Considerations on Perturbation Solutions for Waves of h  h  Permanent Type” , Bull. Faculty of Engng., Yokohama National University, Vol. 31, 1982, pp29-57. [18] H. G. Oh, .and S. Watanabe, “Higher Order Solution of VII. CONCLUSION Nonlinear Waves. I.Cnoidal wave in Unstable and Dissipative media”, Journal of Phys. Soc. JPN, Vol. 66, No. 4, 1997, pp. Cnoidal wave theory is useful for studying wave motion 979-983. [19] M. S. Parvin, M. S. A. Sarker, and M. S. Sultana, “Cnoidal in shallow water. Here, third order cnoidal wave solutions Wave Solutions in Shallow Water and Solitary Wave Limit”, are formulated using boundary conditions at the bottom Journal of Pure and Applied Mathematics, Advances and and at the free surface. Also the boundary conditions are Applications, Vol. 11, No. 1, 2013, pp. 63-93. used from Navier-Stokes equation of motion. Then taking [20] L. W. Schwartz, “Computer Extension and Analytic Continuation of Stokes Expansion for Gravity Waves” , J. Fluid Jacobi elliptic function, third order cnoidal wave solutions Mech. Vol. 62, 1974, pp. 553-578. have been derived. Finally, the horizontal and vertical [21] Y. Tsuchiya, and T. Yasuda, “Cnoidal Waves in Shallow Water fluid velocity components are established using Taylor and their Mass Transport, Advances in Nonlinear Waves” , L. expansion for the stream function about the bed. Devnath (ed), Pitman, 1985, pp. 57-76. [22] Y. Xu, X. Xia, and J. Wang, “Calculation and Approximation of The Cnoidal Function in Cnoidal Wave Theory, Computers and Fluids”, Vol. 68, 2012, pp. 244-247.

AUTHORS ' PROFILES Year : 1996 Graduation : Mst. Shahana Parvin B. Sc. Honors in Mathematics Institution : University of Rajshahi, Rajshahi-6205, Mst. Shahana Parvin M.Sc. (Raj) Assistant Professor Bangladesh Department of Applied Mathematics FIELD OF SPECIALIZATION University of Rajshahi, Rajshahi-6205, • Teaching at Graduate Level: Calculus, Mathematical Methods, Tel:+880-721-711155 (Off) Classical Mechanics Mobile: +88-01710437663, Fax: +880-721-711155 • Teaching at Post Graduate Level: Magneto-hydrodynamics and Email: [email protected] Turbulence, Water Waves. ACADEMIC QUALIFICATION • ACADEMIC QUALIFICATION Current Main Research Field: Water Waves. Year :2002 Post Graduation: Master of Science (M. Sc.) in Awards/Scholarships (Held in 2004) Applied Mathematics(Thesis) 1. Fellowships for M. Phil degree from Rajshahi University.. Institution : University of Rajshahi, Rajshahi-6205, 2. Gold Medal for the Faculty first awarded by Rajshahi University. Bangladesh 3. Gold Medal for the 1st class 1st position in B.Sc (Hons.) Degree Year : 2001 Graduation : B. Sc. Honors in Mathematics awarded by Rajshahi University (Held in 2002) Institution : University of Rajshahi, Rajshahi-6205, 4. Awarded scholarship for the 1st class 1st position in B.Sc (Hons.) Bangladesh Degree. FIELD OF SPECIALIZATION 5. Awarded the ‘Rajshahi University Prize’ along with a Certificate for • M.Sc. Degree. Current Main Research Field : Water Waves, Fluid Dynamics, 6. Awarded the ‘Rajshahi University Prize’ along with a Certificate for Meteorology. B. Sc. Degree. Awards/Scholarships 7. Awarded “Engineer Akbar Hossain Scholarship” for B.Sc. (Hons.) 1. Scholarship in B.Sc. Program given by the University of Rajshahi, Degree. Bangladesh. THESIS SUPERVISION 2. Scholarship in M.Sc. Program given by the University of Rajshahi, • Bangladesh. 6 M. Sc. Thesis students 3. Scholarship for highest marks in Science Faculty given by Saida PUBLICATIONS Khatun Merit Scholarship, University of Rajshahi, Bangladesh. • Published 12(Twelve) research papers in National and International 4. Agrani Bank Gold Medal award for M.Sc. Program, University of Journals Rajshahi, Bangladesh. Languages Published Articles • Writing and speaking (fluently): English and Bengali. • Published 4(Four) research papers in National and International • Arabic: Proficiency-Good Journals PERSONAL DETAILS Computer Literacy Name: Dr. Mst. Shamima Sultana • MATLAB Date of Birth : 11th June, 1976 • FORTRAN 77, 90 Present Address: Associate Professor • Department of Applied Mathematics, PROGRAMMING WITH C Univrsity of Rajshahi, Rajshahi-6205, • C++. Bangladesh. Languages Permanent Address: House Name: Matrogriho, House No. 593/A • Writing and speaking (fluently): English and Bengali. ,Dharampur, Binodpur-6206; Rajshahi; Bangladesh • Arabic: Proficiency-Good Nationality: Bangladeshi

Medium of Education: English PERSONAL DETAILS Name: Mst. Shahana Parvin Date of Birth : 25th June, 1979 Dr. M. Shamsul Alam Sarker Present Address: Assistant Professor, DR. M. SHAMSUL ALAM SARKER M.Sc. (Raj), PhD, (Banaras) Department of Applied Mathematics, Univrsity of Professor Rajshahi, Rajshahi-6205, DEPARTMENT OF APPLIED MATHEMATICS Bangladesh. University of Rajshahi, Rajshahi-6205, Nationality: Bangladeshi Tel: 0721-711155(Off.), 0721-750745 (Res.) Medium of Education: English Mobile: 01715844017, Fax: 0088-0721-750064 Email: [email protected] ACADEMIC QUALIFICATION Dr. Mst. Shamima Sultana Year :1992 Doctor of Philosophy (Ph. D.) Dr. Mst. Shamima Sultana M.Sc., MPhil (Raj), PhD (Raj) Associate Professor Institution : Department of Mathematics, Banaras Hindu University, India. Department of Applied Mathematics Thesis Title : Some Theoretical Investigations in University of Rajshahi, Rajshahi-6205, Tel:+880-721-751217 (Res.) Magneto-hydrodynamic Turbulence. Year :1981 Mobile: +88-01719542052, Fax: 0088-0721-750064 Post Graduation: Master of Science (M. Sc.) in Applied Mathematics Email:[email protected] Institution : University of Rajshahi ACADEMIC QUALIFICATION Result : First Class First Year :2013 Doctor of Philosophy (Ph. D.) Year : 1979 Institution : University of Rajshahi, Rajshahi-6205, Graduation : Bangladesh Bachelor of Science (B. Sc.) Institution : Gaibandha Govt. College, Gaibandha Year :2004 Master of Philosophy (M. Phil) University : Rajshahi Institution : University of Rajshahi, Rajshahi-6205, Bangladesh Result : First Division (5th Place) ACADEMIC QUALIFICATION FIELD OF SPECIALIZATION Year :1997 Post Graduation: Master of Science (M. Sc.) in Applied Mathematics • Fluid dynamics (Turbulence, MHD Turbulence and Laminar Flow ) Institution : University of Rajshahi, Rajshahi-6205, • Teaching at Graduate Level: Business Mathematics and Bangladesh Hydrodynamics. Copyright © 2016 IJISM, All right reserved 173 International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051

• Teaching at Post Graduate Level: Fluid dynamics. 7. Reviewer, Ganit, Journal of Bangladesh Math. Soc., Dhaka Univ., • Khulna Univ. Journal, Khulna and Islamic univ. Journal, Khustia Also conducting research on Turbulence, MHD Turbulence and etc. Laminar Flow . 8. Reviewer, Journal of Scientific Research, Faculty of Science, ACADEMIC EXPERIENCES Rajsahhi University, Bangladesh

1. Professor, Dept. of Applied Mathematics, R ajshahi University, 9. Member, Faculty of Applied Sciences and Technology, Islamic Rajshahi, from July 19, 2004 to date. University, Kustia, from Sept. 9, 2004 to Sept. 8, 2006. 2. Professor, Dept. of Mathematics, Rajshahi University, from June 10. Member, Project Evaluation Committee, University Grants 11,1999 to July 18, 2004 Commission, from July 2006 to June 2007 and July 2008-June 2009. 3. Associate Professor, Dept. of Mathematics, Rajshahi University, 11. Member , Governing Body, Dental College, Rangpur (Nominatated from Nov.6,1993 to June 10,1999. by Rajshahi University), from 27/08/ 2002 to 29/07/.2010. 4. Assistant Professor, Dept. of Mathematics, Rajshahi University, 12. Expert Member, Selection Board for Assistant Professor, from July 16, 1989 to Nov. 1993. Associated Professor & Professor in Applied Mathematics, 5. Lecturer, Dept. of Mathematics, Rajshahi University, from July 16, South Asian University, New Delhi, India, July 02-03, 2012 1986 to July 15, 1989. CONFERENCE/ SEMINAR ATTENDED 6. Lecturer in Mathematics, Sundargonj D.W. College, Sundargonj, 1. Summer Science Institute in Mathematics, Dept. of Mathematics, Gaibandha, from 25 Sept., 1983 to July, 11, 1984. Dhaka Univ. (1987). (Secured Second Position in Performance).

7. Adjunct Faculty Member : 2. Summer Science Institute in Mathematics, Dept. of Mathematics, (i) Northern University, Bangladesh (Rajshahi Campus). Rajshahi Univ. (1988). (Secured First Position in Performance) From February 2005 to Dec. 2010. 3. Bangladesh Association for Science (BAAS) International (ii) Ahsanullah University of Science and Technology,Bangladesh Conference, Jahangirnagar University, Dhaka (1987) (Rajshahi Campus). 4. BHU Mathematical Conference, Dept. of Mathematics, Banaras From April, 2006 to September , 2007 and April 2008 to September, Hindu Univ. India (1989) 2008. 5. Bangladesh A ssociation for Science (BAAS) International (iii) Part time course teacher: Institute of Business Administration ( IBA), Conference, Bangobandhu Sheikh Mujib Agriculture University, Rajshahi University. Gazipur, 1992 From July,2002 to December 2004 and July, 2007 to Nov.2008 6. Bangladesh International Mathematical Conference, Dept. of ADMINISTRATIVE EXPERIENCES Mathematics, Rajshahi Univ. (1993)

1. Senior Officer, Janata Bank, Lalmonirhat Br. from July 12, 1984 to 7. Bangladesh International Mathematical Confe rence, Dept. of July12, 1986. Mathematics, Dhaka University Univ., Dhaka (1995) 2. House Tutor, Shaheed Suhrawardi Hall, Rajshahi University, from 8. Bangladesh Association for Advancement of Sciences (BAAS) March 15, 1992 to Dec. 04, 1994. International Conference, Jahangirnagar University, Dhaka (1996)

3. Provost , Sher-E-Bangla Fazlul Haque Hall, Rajshahi University, 9. Bangladesh International Mathematical Conference, RCMPS, from Oct. 17, 1996 to Oct. 16, 1999. Chittagong, Univ. (1999)

4. Convener, Provost Council, Rajshahi University, from January 10. 25th International Nathiagali Summer College on Physics and 1999 to Oct. 1999. Contemporary Needs, Islamabad, Pakistan (2000) 5. Proctor , Rajshahi University, from March 09, 2003 to March 08, 11. International Conference on Geometry, Analysis and Applicatoions, 2006. Banaras Hindu Univ. India (2000).

6. Chairman , Department of Applied Mathematics, Rajshahi 12. Bangladesh International Mathemat ical Conference, Dept. of University, Bangladesh. From October 1, 2008 to Sept.30, 2011. Mathematics, Dhaka University Univ., Dhaka (2007).

7. Senate Member, Rajshahi University Senate, from Feb. 26, 2005 to 13. Bangladesh International Mathematical Conference, Dept. of date. Mathematics, Bangladesh University of Engineering and 8. Member, Academic Council, Rajshahi Univ., Rajshahi, from June Technology (BUET) Dhaka (2009) 11, 1999 to date MEMBERSHIP 9. Member, Faculty of science, Rajshahi University, Rajshahi, from 1. Vice-President, Bangladesh Mathematical society, from Jan-2008 - June 11, 1999 to date Dec-2009.

10. Member, Planning and Academic Committee, Dept. of Applied 2. Life-Member, Bangladesh Mathematical Society. Math. Rajshahi Univ .from July 19, 2004 to date. 3. Member, Bangladesh Association for Advancement of 11. Member, Appeal and Arbitration Council, Board of Intermediate Science(BAAS). and Secondary Education, Rajshahi, from Dec. 11, 2004 to Dec., 10, THESIS SUPERVISION 2007. • OTHER EXPERIENCES Supervised 7(Seven) Ph. D. 1. Member, Editi onal Board, Ganit, Journal of Bangladesh • 3(Three) M. Phil. and 10(Ten) M. Sc. Students. Mathematics Society, January, 2008 to December, 2009. • At present supervising 1(one) Ph. D. and 1(M. Sc.) M. Sc. student. 2. Member, Editorial Board, Journal of Science, Rajshahi university PUBLICATIONS studies, from April 8, 2006 to date. • 3. Expert Member, Selection Board for Lecturer/ Assistant Pro fessor, Published 65(Si xty five) research papers in National and International Journals in Mathematics, Shahjalal University of Science & Technology, Sylhet,(01/10/2005 to 29/03/2009) Haji Danesh Univ. of Science & PERSONAL DETAILS Technology, Dinajpur,(01/03/2004 to 21/11/2010) Rajshahi Univ. of Father’s Name: Ahmed Ali Sarker Mother’s Name : Sarvan Nesa Science & technology( RUET), Rajshahi,( 01/07/2005 to Date of Birth : 01- 01- 1957 30/06/2009), Jahangirnagar University ( 06/07/2002 to 05/07/2004) and Bangladesh Open University, Dhaka( 2006). Present Address: Professor Department of Applied Mathematics, 4. Expert Member, Selection Board for Professor/ Associate Professor Univrsity of Rajshahi, Rajshahi-6205, Bangladesh. in Mathematics, Dhaka University, Dhaka,(18/03/2003 to 6/02/2011) Shahjalal Uni v. of Science & Tech.(SUST), Permanent Vill: Kanchibari, P.O.: Dhubni Bazar Address: P.S. : Sundargonj, Dist.: Gaibandha, Bangladesh. Sylhet,(26/12/2006 to 29/03/2009) and Khulna Univ. of Science & Religion : Islam Tech.(KUET) Khulna (07/03/ 2006 to 11/03/ 2008 and 11/03/ 2010 to 10/03/2012) Nationality: Bangladeshi ID No.: 8194030127428 5. Member, Selection Committee in Applied Mathematics, Accounting Passport No.: AC 9205839 & Information System a nd Population Science & Human Resource Dev. Departments, Rajshahi University ( 01/11/2006 to 30/10/2007) Place of Birth : Gaibandha,Bangladesh. Marital Status : Married. 6. External Examiner of M. Phil. and Ph. D. in Mathematics, Medium of English Bangladesh Univ. of Science & Tech.(BUET) Dhaka, Khulna Univ. Education: of Science & Tech.(KUET) Khulna and RCMPS, Chittagong Univ. Chittagong. Copyright © 2016 IJISM, All right reserved 174