Spline Collocation Method for Solving Three Dimensional Parabolic Equations

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1 Spline Collocation Method for Solving Three Dimensional Parabolic Equations Dr A.K.Pathak H.O.D,H&S.S. Department S.T.B.S.College Of Diploma Engg. Surat, India an Dr. H.D.Doctor Prof. & H.O.D,Mathematics Department Veer Narmad South Gujarat Uni. Surat, India Abstract The present work describes spline collocation approximation for solving Heat Conduction In Rectangular problem. The mathematical problem gives rise to solve a linear partial differential equation which is of three dimensional parabolic type. The method involves the solution of algebraic linear equation which can be written in the matrix form is main advantage. The solution are obtained by this method compared with analytic solution to demonstrate the justification and simplicity of the spline approximation converge to analytic solution. Keywords: Spline Collocation Method, Partial Differential Equation.. INTRODUCTION: Two common questions are encountered while the numerical solution to the problem is obtained. The first is about its acceptance whether it is sufficiently close to the true solution or not. If one has an analytic solution then this can be answered very clearly but in either case it is not so easy. One has to be careful while concluding that a particular numerical solution is acceptable when an analytic solution is not available. Normally a method is selected which requires a minimum number of steps, consuming the shortest computational time and yet one that does not produce an excessive errors.. SPLINE COLLOCATION METHOD: For solving linear and nonlinear differential equation with the help of the numerical methods required much computational work and time. Bickley [968] Suggested the method of spline function containing truncated power polynomials to solve a linear boundary value problem Ahlberg et al [967] used cardinal splines for solving differential equations. Doctor et al [983,984] have shown that the method of spline collocation is quite useful for the solution of physical phenomena which give rise to linear parabolic one dimensional partial differential equation. The method demonstrates the use of spline function. Spline functions are piecewise polynomial and their successive derivatives are continuous. They were used for data interpolation initially. In 967, Blue [969] suggested the use of spline function for the solution of B.V.P. y = f(x,y,y ) (.) with boundary conditions G [Y(0),Y (0)] G [Y(),y ()] (.) The following recurrence relations were used. S ( x i- )+4s ( x i )+s ( x i+ ) = 6/ h (f( x i- )-f( x i )+f( x i+ ) (.3) 50

2 . Spline Formula To Solve Parabolic Partial Differential Equation With Three Space Variables The general form of linear parabolic partial differential equation with three space variables is of the form u = R (u + u u ) ; 0 < x < a,0 < y < b,0 < z < c (..) t xx yy + yy With Dirichlite condition prescribed on the boundaries x=0,y=0,z=0. We should divide the region 0 x a 0, y b,0 z c in to L,M and N sub-intervals of width Dx, Dy, Dz respectively such that L D x = a, MDy = b, and NDz = c. Let us denote the points of subdivisions by x i : i = 0,,,..., L, y j : j = 0,,,...,M, z k : k = 0,,,..., N. Let us u i, k, p denotes the values of u at the (i,k) th Mesh point at the time pd t.for simplicity, let us take cubic region with equal sides and D x = Dy = Dz = h (say) and L=M=N. Approximate the function u at time pdt by cubic spline S(x).Discretizing the left side of equation (..) by forward difference formula and replacing right hand side by three times of the second " derivative. i.e. 3s ( x i, k ) at p th level we get (u i, p+ + 4u i, k,p+ + u = (+ 8r) u ) i+, k,p+ i-, k,p + (4-36r) u i, k,p + (+ 8r) u i+, k,p i,k=,,n- (..) Where r = R Dt / h.these set of (N-) x (N-) x (N-) equations in (N-) x (N-) x (N-) unknowns can be solved by any well- known method. The above set of simultaneous equations gives a square matrix. The above equation (..) is known as cubic spline explicit formula to solve equation (..). For this explicit method maximum possible value of r, for stability and convergence of this method, maximum possible value of r is r /6. In Implicit method, he finite difference replacement of equation (...) is (u i, k,p+ - ui, k, p ) 3 = R [ ( s" i, k, p + s" i, k, p+ )] (..3) Dt Where s" i, k, p and s" i, k,p+ denote the second derivatives of s(x) at x = x" i, k, p at the time interval p and p+ respectively. we get (- 9r)ui,k, p+ + (4 + 8r)ui, k,p+ + ( - 9r) ui+, k,p+ (..4) = (+ 9r)u + (4-8r) u + (+ 9r)u i-, k,p i, k,p Where r = R Dt / h and i,k=,,3,..,n- Above relation known as cubic spline implicit formula to solve equation (...). Like explicit scheme we get (N-) x (N-) x (N-) equation in (N-) x (N-) x (N-) unknowns. These simultaneous equations with square matrix can be solved any standard method. In both methods once the value of u are known at (p+) th level, we can proceed to compute the next level (p+) by same techniques described as above. i+, k,p 3. HEAT CONDUCTION IN THIN RECTANGULAR VOLUME: Consider a flow of heat in a rectangular volume with sides of length axes x,y,z (figure(3.)) D x, Dy, Dz along the co-ordinate 5

3 R N z S M Dz y P(x,y,z) Q Dx L Dy O x Fig(3.) Rectangular element of solid The amount of heat entering the element through the face PQRS in time Dt is - k( D y D z )( u / x ) x D t in the element through the approximate face is - k( DyDz)( u / x) x+ Dx Dt Where is the thermal conductivity of the material and u(x,y,z,t) is the temperature function. The negative signs are taken because the heat flows in the direction of decreasing temperature. Hence the quantity of heat remaining in the solid as a result of entry through the PQRS and exit through the opposite face is - k{( u / x) x+ Dy - ( u / x) x} DyDzDt (3.) = { u / x ) Dx} DyDzDt Up to a first approximation. Similarly, the corresponding differences in the heat entering and heaving through the nearing two pairs of opposite face are k{( u / y ) DyDxDtDz and k{( u / y ) DzDxDyDt (3.) Hence the total heat retained by the solid in time Dt is the sum of (3. ) and (3.) which is equal to the heat required to the heat required to raise the temperature to raise the temperature of the element by D u.thus we have k( u / x + u / y + u / z ) DxDyDzDt = ( rdxdydz) sdu Where r is density and s be the specific heat of the solid. Dividing by Dt 0, weget DxDyDzs and limit as u / t = R ( u / x + u / x + u / x ); R = k / sr (3.3) Above equation (3.3) is the heat equation in three space variables. Also it is known as parabolic partial differential equation with three space variables x,y,z and time variable t. Solution of above equation (3.3) gives the temperature distribution in the rectangular volume. Consider the edges of rectangular volume of side are kept at temperature zero and faces are perfectly insulated. Hence the flow of heat in the volume governed by the equation (3.3) with boundary conditions U(x,0,z,t)=0, u(x,y,0,t)=0, u(0,y,z,t)=0 (3.4) 5

4 Where 0 x, y, z Let initial temperature distribution in the volume be u( x, y, z,0) = sin Pxsin Pysin Pz (3.5) Where 0 x, y, z The given Problem with boundary and initial conditions described as above is solved by explicit and implicit method. 4. SPLINE SOLUTION WITH EXPLICIT METHOD We shall determine the solution of equation (3.3) satisfying the initial and boundary conditions defined by the equation (3.4) and (3.5) respectively, using the explicit formula given by equation(3.3) as follows For that let h=/0 R = 0.00 and D t =0.00 which gives r= r=.08 and 4-36r = Substituting the value of +8r,4-36r with initial and boundary condition in equation (3.3) gives following result For p=0, k= and j= i= + 4u + u 0.74 u 0,,,,,,,,, = Since u0,,, = 0 4u,,, + u,,, = 0.74 u,,, + 4u,,, + u 3,,, = u,,, + 4u3,,, + u,,, = i= i= etc. Proceeding in this way for I,k =,3,,9 We get 9x9x9 equations in 9x9x9 unknowns, which can be solved by any standard method once the values of p=0 are obtained, the results for p=,.. are obtained by same process discussed as above. Due to symmetry of the solution the result obtained for 0 z 0.5,0 y 0.5,0 x 0. 5 at t=0.00 and t=0.003 Hear the result for z=0.5, y=0. at t=0.00 are given in table I and plotted in fig. 4. SPLINE SOLUTION WITH IMPLICIT METHOD: Using implicit formula given by equation(..4)the solution of equation (3.3) satisfying boundary and initial conditions For p=0, k=,j= i= (4.08)u,,, + (0.99)u,,, = i= (0.99)u,, + (4.08)u,,, + (0.99) u3,,, = i=3 (0.99)u,,, + (4.08)u3,,, + (0.99) u 4,,, = Similarly in this way we get 9x9x9 simultaneous equations we get the temperature distribution in rectangular volume at time t=0.00 Proceeding in the same way we get the temperature in the volume at t=0.00,0.003 etc. Result for z=0.5, y=0. at t=0.003 are given in table 6. and plotted in fig RESULT TABLE 5. Temperature Distribution In Rectangular Volume U At t=0.003, z=0.5 X,Y

5 TABLE 5. Temperature Distribution In Rectangular Volume U At t=0.003, z=0.5 X,Y TABLE 5.3 Error Analysis Of Temperature Distribution Rectangular Volume U at t=0.003,z=0.5 y=0. X UEXT UEXT -UE X0-6 UEXT - UI X Fig (5.) Temperature Distribution In Rectangular Volume through cubic spline explicit method (at z=0.5,y=0.) 54

6 Fig(5.) Temperature Distribution In Rectangular Volume through cubic spline implicit method (at z=0.5,y=0.) 6. CONCLUTION: Fig(5.3)Error Analysis (at t=0.003,z=0.5,y=0.) The solution of equation (3.3) obtained by spline explicit and implicit as well as explicit methods are compared with the exact solutions in table 5.3. Fig. 5.3 indicates the error analysis which compares the exact solutions with spline solutions obtained by both explicit as well as implicit method. From the fig. 5.3 and table 5.3 it is clear that spline solutions are quite reliable up to five digits of decimal places. 7. REFERENCES: [].Ahlberg J H, Nilson E N And Wals J N, The Theory of Spline And Their Application. Academic Press, N.Y,967. [].Berdyshev V I, Subbopim J N,Numerical Methods of Approximation of functions (Russian). Sverdlovsk. Ural.Publ.House, 979,8pp [3].Bhatt R V: An Extended Study Of The Spline Collocation Approach To Partial Differential Equations, Thesis, South Gujarat University, Surat, 005. [4].Bickley W G -Piecewise Cubic Interpolation and Two point Boundary Value Problem. Comp. J,, 968,06. 55

7 [5].Blue J L - Spline Function Methods for linear Boundary Value problem. Communications of the ACM., No.6, 969, 37. [6].Bruch J C And G Zyvoloski: Transient Two-Dimensional Heat Conduction Problems Solved by the Finite Element Method, International Journal for Numerical Methods in Engineering, 8, 974, PP [7].Clough R W: The Finite Element Method in Plane Stress Analysis J.Struct. Div., ASCE, Proc.d conf. Electronic Computation, 960,pp [8].Cook R D: Concepts and Applications of Finite Element Techniques for Fluid Flow, Butterworths, London, 976. [9].Courant R: Variational Methods for the Solution of Problems of Equilibrium and Vibration Bull. Am. Math. Soc., vol. 49, 943,pp-43. [0].Doctor H D And Kalathia,N L - Spline Collocation in the Flow of Non-Newtonian Fluids. Presented at the Annual conf. of National Academic of science [].Doctor H D And Kalthia N L - Spline Approximation of Boundary Value Problems. Presented at Annual Conf.of Indian math.soc, 49, 983. [].Doctor H D, And Kalathia N L -Spline Approximations of Boundary Value Problems Presented at Annual conf. of Indian Mathematical Society, 49, 983. [3].Doctor H D, Bulsari A B And Kalathia N L -Spline Collocation Approach to Boundary Value Problems. Int. J. Numer.Fluids, 4,6, 984,5. [4].Doctor H D, Bulsari A B And Kalthia N L - Spline Collocation Approach To Boundary Value Problems. Int. J. Number, Floids., 4, 6, 984,5. [5].Dr. Grewal B S:- Numerical Methods In Engineering And Science, Fifth ed, Khanna Publishers, 005. [6].Fyfe D J -The Use Of Cubic Splines In Solutions Of Two Point Boundary Value Problems, Com.P.J.,, 969,88-9. [7].Pathak A K and Doctor H D, Comparative Study of Collocation Method- Spline, Finite Difference and Finite Element Method for Solving Partial Differential Equation Presented at Conf. of Gujarat Science Congress, 00. [8].Reddy J N : An Introduction To The Finite Element Method. Mc Graw-Hill International Editions, New York, 984. [9].Sastry S S - Introductory Methods of Numerical Analysis. Prentice-Hall of India Private Limited. New Delhi,005. AUTHOR S BIOGRAPHY: Dr.Anant K Pathak: He is working as H.O.D. of Humanities and Social Science Department in S.T.B.S. College of Diploma Engineering. He has published 3 papers in international journal and in national journal.he has presented 3 papers in national conference and in International conference. He has 9 years of teaching experience. 56

Solve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method

Solve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method P.B. Choksi 1 and A.K. Pathak 2 1 Research Scholar, Rai University,Ahemdabad. Email:pinalchoksey@gmail.com 2 H.O.D.