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Integral Transform Method CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Integral transform method R. M. Cotta and M. D. Mikhailov* Engenharia Mecbnica-EEICOPPEI UFRJ, Universidade Federal do Rio de Janeiro, Brazil The finite integral transform technique is interpreted as a powerful new general-purpose numerical method. The method transforms nonlinear partial differential equation models to a coupled nonlinear system of ordinary differential equations to be solved numerically. One class ofheat diffusion problems is considered, but the method is applicable to other models. A new approach for the derivation of the corresponding system of ordinary differential equations is used, and three examples are given. Keywords: integral transform, numerical method, nonlinear problem tally. One class of heat diffusion problems is consid- Introduction ered, but the method is applicable to many other Modelling of many engineering problems leads to par- models. A new approach for direct derivation of the tial differential equations subjected to corresponding corresponding system of ordinary differential equations initial and boundary conditions. The treatment of such is used, and three illustrative examples are given. The models is usually done by numerical methods. At pres- reader can find additional examples for other types of ent finite differences,’ finite elements,’ boundary ele- problems in Ref. 7. ments,3 and spectral methods dominate.4 The finite integral transform technique’ is mainly used for the exact solution of linear problems.6 Re- Classical integral transform technique cently, several linear and nonlinear models were suc- The application of the classical integral transform tech- cessfully solved by a generalized integral transform nique to heat transfer problems is described in detail in technique.’ Ref. 5. We consider here briefly the following boundary All methods transform the original partial differential value problem5 equations to a set of ordinary differential or algebraic equations that are to be solved by well-established NJ::+ YT= P inV*,t>O techniques. The finite differences and finite elements (la) methods consider from the very beginning the field variables in a limited number of points in the solution 53T=q5 onS (lb) region, whereas the boundary elements method uses a T= f att=O (lc) limited number of boundary points. The spectral method represents the solution in terms of truncated where series, which is analogous to considering a limited num- %= -V.(ZCV)+d (24 ber of points. The integral transform method uses trun- cated eigenfunctions expansions. %=a+/3K; WI This paper gives a new interpretation of the general- ized integral transform technique as a powerful general- purpose numerical procedure called integral transform Here V is the nabla operator and a/&z is the normal method. The method transforms nonlinear partial dif- derivative at the boundary surface S in the outward ferential equation models to a coupled nonlinear system direction. of ordinary differential equations to be solved numeri- The problem (1) is linear because the coefficients w, K, and dare known functions in V *, and (Y,/3 are known functions on S. The source terms P and 4 are known functions of time in V* and S, respectively. * Permanent address: Institute for Applied Mathematics and In- To solve problem (1) we need the solution of the formatics, P.O. Box 384, Sofia 1000, Bulgkia. following eigenvalue problem Address reprint requests to Dr. R. M. Cotta, Programade Engenharia S/~=hw* inV* @a) Mecsnica, COPPE-Universidade Federal do Rio de Janeiro, Cidade Universithria, C.P. 68503, Rio de Janeiro, RJ 21945, Brazil. ?i3*=0 0nS (3b) Received 2 May 1991; revised 3 July 1992; accepted 20 July 1992 We assume that the eigenvalues Ai (i = 1,2, . ,w) 156 Appl. Math. Modelling, 1993, Vol. 17, March 0 1993 Butterworth-Heinemann Integral transform method: R. M. Cotta and M. D. Mikhailov and the corresponding normalized eigenfunctions $i are To obtain homogeneous boundary conditions we known. separate the solution into two parts The classical integral transform technique trans- T=U+V (8) forms problem (1) to the following infinite system of ordinary differential equations’ The first part, U, is selected to satisfy the nonhomogeneous boundary conditions %l!J = 4 + 4, - c?&(U+ V) (9) (W The second part, V, is defined by s a+P v* subjected to the initial condition wf-J+w=P+P,,-wyy TAO) = j- w&f dv (4b) a(u + V) - wn - 2U - Jf$(U + V) inV* v* dt wherei= 1,2 ,..., 00. (104 The system (4) is solved analytically in Ref. 5 for the finite integral transform T,. To obtain the desired solu- 93V=O 0nS (lob) tion, the results for T; are used in the following inversion V=j- U att=O (LOc) formula We have some freedom in selecting the first part of T = -g l+fJ;i; (5) the solution. The computations are simplified if U is i= I selected in such a way that some ofthe terms in equation (10a) vanish. An additional term appears in this inversion formula We can transform problem (10) to the coupled system when d = 0 and a = 0, because then h, = 0 is also an of ordinary differential equations in the same way that eigenvalue and I,!I~= constant is the corresponding problem (1) is transformed to the system (4) in Ref. 5. eigenfunction. This is done for some special cases of problem (6) in Ref. The convergence of the series given by equation (5) is 7. But the final result will be the same as that obtained improved when the boundary conditions (1 b) are made directly from equation (4) if we consider the right-hand homogeneous. side of equation (10a) as a source term. Therefore we The above results can be readily written for the case can write directly the following infinite system of ordi- of one space variable as described in detail in Ref. 5. nary differential equations dvi (t) Integral transform method dt Let us consider the following problem - wn - %U - 2,,(U + V) do (lla) at inV*,t>O (6a) vi(O) = J W$i(f - U)dv (lib) (?i3 + 93,JT = 4 + c$, onS (6b) v* T=f att=O (6~) After using in this equation the truncated to the nth term inversion formula where -- .& = -V.(K,V) + d, + u;V (7a) v = i: (hi v; \ (12) i=l we obtain the coupled nonlinear system of ordinary differential equations that has to be solved numerically. Problem (6) is an extension of problem (1). Several In Ref. 7 numerical subroutines used for the solution of nonlinear terms having index n are added. The such systems are discussed, which incorporate auto- coefficients w,, K,, d,, and the vector u, are known matic relative error control procedures,’ allowing one functions of space, time, and temperature T in V *, to obtain a final converged solution within prescribed while LX,and & are known functions of space, time, and accuracy. temperature on S. The source functions P, and c#+,are Because nonlinear systems can have more than one known functions of time, space, and temperature in V * solution, for some problems analysis is needed analo- and S, respectively. gous to that explained in Ref. 9, allowing for quantita- Problem (6) has linear and nonlinear terms. We as- tive investigations of instability and bifurcation phe- sume that linear terms represent the problem in some nomena. average sense. Finally, the numerical solution of system (11) is used Appl. Math. Modelling, 1993, Vol. 17, March 157 Integral transform method: R. M. Cotta and M. D. Mikhailov in the inversion formula (12) to find the desired with boundary and initial conditions solution. By increasing the number of terms, we can WO,d obtain the results with prescribed accuracy, since the - = 0; 0(1,r) = 0; f3(X,O) = 1 (13b, c, d) error in the numerical solution of the O.D.E. system is ax fully controlled. In fact, an automatic adaptive The first two steps of the integral transform method procedure is used in the integration of system (I l), are not needed as the linear and nonlinear parts are which controls the number of differential equations, N, already separated, and the boundary conditions are retained at each step of integration. Therefore, the homogeneous. algorithm itself selects the number of terms required in The eigenvalues and the normalized eigenfunctions the expansions for a certain requested accuracy. This that correspond to the linear part of the problem are procedure is described in detail in Refs. 7 and 10. The integral transform method described has the /_&;= (2i - 1);) Ai = /_Ly;I/$ = lhcos (&X) following steps: (1% b) The problem has to be rewritten in the form having two parts. The first one is linear and represents the The system (11) for the case considered, after inte- problem in some average sense. The second part of gration by parts becomes the problem is nonlinear and considered as source d6(4 terms. It may also be convenient to consider as + &;a@ + s a$, = 0; source terms some of the linear terms that do not dr .j= I permit the direct application of the classical finite e,(O) = V?(-l)‘+‘lp; (15a,b) integral transform technique. The desired solution is also separated into two parts. where The first one satisfies the nonhomogeneous bound- ary conditions. The second part is defined by a uii = 2~~4 sin (p;X) sin (pjX)f(0) dX (15c) partial differential system with some additional / 0 source terms, but homogeneous boundary condi- tions. Once system (15) has been numerically solved the We have some freedom in selecting the first part desired solution is obtained from of the solution. In order to have a simpler problem for the second part of the solution some appropriate e(x,7) = 2 4;(x)ei(r) (16) additional conditions may be imposed on the first i=l part.
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