Multiple-Scales Method and Numerical Simulation of Singularly Perturbed Boundary Layer Problems

Transcription

1 Appl. Math. Inf. Sci., No. 3, 9-7 (6) 9 Applied Mathematics & Infmation Sciences An International Journal Multiple-Scales Method and Numerical Simulation of Singularly Perturbed Boundary Layer Problems Parul Gupta and Manoj Kumar Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 4, India. Received: 7 Oct. 5, Revised: 8 Jan. 6, Accepted: 9 Jan. 6 Published online: May 6 Abstract: In this paper, Multiple-scales method is presented f solving second and third der singularly perturbed problems with the boundary layer at one end either left right. The iginal second and third der dinary differential equations are transfmed to partial differential equations. These problems have been solved efficiently by using multiple-scales method and Numerical simulations are perfmed on standard test examples to justify the robustness of the proposed method. Keywds: Singular perturbation problems, Multiple-scales method, Boundary layer problems. Introduction Differential equations with a small parameter multiplying the highest der derivative terms are said to be singularly perturbed which are often found in mathematical problems arising in sciences and engineering. Numerical solution of singularly perturbed boundary value problems in dinary differential equations is a well known research area. Many numerical methods have been developed f solving singularly perturbed problems. F detail analysis of this type of problems we refer []-[33]. The analysis of boundary layer problems and multiple scale phenomena which have been generalized under the notion of singular perturbation problems played a significant role in applied mathematics and theetical physics [, 5]. Regular perturbation they is often not applicable to various problems due to resonance effects the cancellation of degree of freedom. In der to obtain a unifmly valid asymptotic expansion of a solution f these singular perturbed problems a variety of methods has been developed such as boundary layer expansions, multiple scales methods, asymptotic matching, stretched codinates, averaging and WKB expansions. A physical system often involves multiple tempal spatial scales on which characteristics of the system change. In some cases the long time behavi of the system can depend on slowly changing time scales which have to be identified in der to apply multiple scale they. The choice of the slow fast changing scales is a nontrivial task. A naive expansion in a power series of the small parameter is often prevented by the appearance of resonant terms in higher ders. These terms have to be compensated by the introduction of counter terms. Boundary layers are also a common feature of singular perturbed systems. In these cases higher der derivatives disappear in the unperturbed equations which lead to the cancellation of degree of freedom of the system and finally in small regions where the system changes rapidly. Boundary layer they is a collection of perturbation methods f obtaining an asymptotic approximation to the solution of a differential equation whose highest derivative is multiplied by a small parameter. Solutions to such equations usually develop regions of rapid variation as. If the thickness of these regions approaches as, they are called boundary layer and boundary layer they may be used to approximate solution. These rapid changes cannot be handled by slow scales, but they can be handled by fast magnified stretched scales. In boundary layer they we treat the solution of the differential equation as a function of two independent variables x and i.e. y(x;). But the main target of this analysis is to obtain a global approximation to solution as a function of x, this is achieved by introducing the stretched scale ξ = x/, which in this case is the same as the inner variable x = x, which in this case is the outer variable. The unifm expansion of the solution of a singular perturbation problem cannot be expressed in Cresponding auth parul5@gmail.com

2 P. Gupta, M. Kumar: Multiple-Scales method and numerical... terms of a single scale i.e. a single combination of x and and, such as x x/ x/ / x/, making it an ideal problem f application of the method of multiple scales []. The idea behind introducing multiple scales is to keep expansions well dered, minimize the err of the approximation and avoid sometimes the appearance of secular terms. An expansion of a function that depends on an independent variable and a small parameter, such as y(x; ) depends strongly on the scale being used. In many situations, there exist multiple-boundary layers at one side, f which multiple calculations of inner and outer solutions and their asymptotic matching have to be made in different separated regions to obtain a unifmly valid solution. Again it turns out that the multiple scales method manages to produce the solution without any matching needed. Auth exples in [9] to solve the boundary layer second-der differential equations by using the interpolation perturbation method. An approximate boundary layer solution is presented in [7] f an axially moving beam with small flexural stiffness. The method of multiple scales is applied to the problem and the composite expansion including two inner solutions and one outer solution is found. In [9] the existence of the exponential series solution to the boundary value problem describing the boundary layer flows of Newtonian fluids has been described. The Crane s solution is generalized f stretching walls with a power law stretching velocity. In [] auths study a boundary value problem f a third der differential equation which arises in the study of self-similar solutions of the steady free convection problem f a vertical heated impermeable flat plate embedded in a pous medium. Jong-Shenq Guo et al. considered the structure of solutions of the initial value problem f this third der differential equation. R.A. Khan [] used the generalized approximation method (GAM) to investigate the temperature field associated with the Falkner-Skan boundary-layer problem. Auths applied in [] the modified variational iteration method (MVIM) f boundary layer equation in an unbounded domain and Pade approximants had been employed in der to make the wk me concise and f the better understanding of the solution behavi. Classical methods find the inner solution and outer solution separately and match the two solutions using physical constraints. The final solution is a composite expansion including the inner and outer solutions. On the other hand, using the method of multiple scales, the composite expansion can be retrieved at once using a single expansion [3, 7]. In this article, the method of multiple scales is applied to construct the solution of second and third der boundary layer problems. It is shown that the method of multiple scales provides the exact solution f second der singularly perturbed boundary layer problems and approximate solution f third der singularly perturbed boundary layer problems. In [8],[3]-[5] approximate solution has been obtained while our proposed method gives the exact solution f second der singularly perturbed boundary layer problems. The iginal second and third der dinary differential equations are transfmed to partial differential equations. These problems have been solved efficiently by using multiple scale method and numerical simulations are perfmed on standard test examples to justify the robustness of the proposed method. The paper is ganized as follows: Multiple scales method f the second and third der boundary layer problems is described in Section. Numerical example of second and third der boundary layer problems solved by Multiple scales method are presented in Section 3. In Section 4, concluding discussion is briefly mentioned. Multiple Scales Method f Second and Third der Boundary Layers Problems In this section, description of multiple scales method f general second and third der boundary layer problems has been presented.. Second Order Boundary Layer Problem To explain multiple scales method we consider the general second der singular perturbed equation y + a(x)y + b(x)y=, <x< () and boundary conditions y()=α, y()=β () Here, α, β are constants and is the perturbation parameter. It is well known that in equation () if a(x) >, then the boundary layer is at x = and if a(x)<, then the boundary layer is at x=. The point of interest is to solve problem () with boundary conditions () by using multiple scales method. Due to the presence of the boundary layer in the problem () we consider two scales; the outer scale at x = x and an inner boundary layer scale at ξ = x/. Then, using chain rule, the derivatives are defined as follows d dx = ξ + (3) d dx = ξ + + ξ d 3 dx 3 = 3 3 ξ x 3 x ξ ξ x F the existence of the two scales, we assume the following multi-scale expansion f the solution (4) (5) y=y (ξ,x )+y (ξ,x )+ y (ξ,x )+... (6)

3 Appl. Math. Inf. Sci., No. 3, 9-7 (6) / Substituting (3), (4) and (6) into iginal equation () we obtain ( ξ + + ) ξ x (y + y + y +...) ( +a(x) ξ + ) (y + y + y +...) +b(x)(y + y + y +...)= (7) Thus, the iginal dinary differential equation () is transfmed into the partial differential equation (7). Separating the coefficients of each der of, one obtain the set of equations (/) : y ξ + a y ξ = (8) ( ) : y ξ + a y ξ = y ξ a y by (9) ( ) : y ξ + a y ξ = y y ξ x a y by () The general solution of (8) is given by y = A(x )+B(x )e aξ () where A and B are undetermined at this level of approximation; they are determined at this next level of approximation by imposing the solvability conditions. Putting y in (9) gives; y ξ + a y ξ = ab e aξ aa ab e aξ ba bbe aξ () y ξ + a y ξ =(ab bb)e aξ (aa + ba) (3) A particular solution of (3) is y p = (ab bb) a ξ e aξ (aa + ba) ξ (4) a We seek a solution which is unifmly valid f x and ξ, but the latter implies ξ (as + ); thus we require, f unifmity the coefficients of ξ and ξ exp( aξ) in (4) must vanish independently. The result is ab bb=, aa + ba=. (5) Solving equation (5), find the value of A and B. Then substituting these values in () and imposing the boundary conditions from (), we find the values of a and b to obtain the approximate solution. This method has been illustrated me precisely by considering a specific example of second der singularly perturbed boundary layer problem in section 3.. Third Order Boundary Layer Problem. To discuss the method of multiple scales method we consider the general third der singular perturbed equation of the fm; y + a(x)y + b(x)y + c(x)y=, <x< (6) and initial conditions are y()=α, y ()=β and y ()=γ. (7) Here, α, β, γ are arbitrary constants and is the perturbation parameter. Substituting (3), (4), (5) and (6) into the iginal equation (6) then we have; ( 3 3 ξ x ( +a(x) ξ + 3 ) ξ + 3 ξ x (y + y + y +...) + ) ξ x (y + y + y +...) ( +b(x) ξ + ) (y + y + y +...) +c(x)(y + y + y +...)= (8) Equating the coefficients of each power of to zero of equation (8), we can write (/ ) : 3 y ξ 3 + a y ξ = (9) (/) : 3 y ξ 3 +a y ξ = 3 3 y ξ a y b y ξ ξ () ( ) : 3 y ξ 3 + a y ξ = 3 3 y ξ 3 3 y ξ x a y ξ x a y x The general solution of equation (9) is b y ξ b y cy () y = A(x )+B(x )ξ +C(x )e aξ, () where A,B and C are undetermined at this level of approximation; they are to be determined at the next level of approximation by imposing the solvability conditions. Substitute y from () in equation (), we can write () in view of 3 y ξ 3 + a y ξ = 3a C e aξ a(b C ae aξ ) b(b ace aξ ) (3)

4 P. Gupta, M. Kumar: Multiple-Scales method and numerical... 3 y ξ 3 +a y ξ = (a C abc)e aξ (ab +bb) (4) The particular solution of (4) is y p = (ac bc) a ξ e aξ (ab + bb) ξ (5) a which makes y, much larger than y as ξ. Hence, f a unifm expansion, the coefficients of ξ exp( aξ) and ξ in (5) should vanish independently. So, we have ac bc=, ab + bb=. (6) In the fthcoming section, we apply this procedure to find the approximate solution of a particular example of third der singular perturbed boundary layer problem. 3 Numerical Simulation To demonstrate the applicability and robustness of the Multiple scales method, we consider three linear singular perturbation problems; one with left-end boundary layer and two with right-end boundary layer. These examples have been chosen because they have been widely discussed in literature and exact solutions are available f comparison. Example : Consider the following homogenous singular perturbation problem y (x)+(+)y (x)+y(x)=; x [,] (7) with y()= and y()=. (8) As this problem has a boundary layer at x = i.e., at the left end of the underlying interval. Substituting (3), (4) and (6) into the iginal equation (7), we get ( ξ + + ) ξ x (y + y + y +...) ( +(+) ξ + ) (y + y + y +...) +(y + y + y +...)=. (9) Equating coefficients of each power of to zero of equation (9), we have (/) : y ξ + y ξ = (3) ( ) : y ξ + y ξ = y ξ y ξ y y (3) ( ) : y ξ + y ξ = y y ξ x y ξ y The general solution of (3) is y y (3) y = A(x )+B(x )e ξ. (33) Substitute the value of y from (33) in equation (3), we obtain y ξ + y ξ = ( B e ξ )+Be ξ (A + B e ξ ) (A+Be ξ ) (34) y ξ + y ξ = B e ξ (A + A). (35) A particular solution of (35) is y p = B ξ e ξ (A + A)ξ, (36) which makes y, much bigger than y as ξ. Hence, f a unifm expansion, the coefficients of ξ and ξ exp( ξ) in (36) must vanish independently. The result is B =, A + A=. (37) The solution of (37) is A=ae x, B=b (38) where a and b are arbitrary constants. Putting the values A and B from (38) in equation (33) gives y = ae x + be ξ (39), in terms of the iginal variable is y = ae x + be x/. (4) Substituting the value of y in equation (6) gives y=ae x + be x/ +... (4) Imposing the boundary conditions from (8) in (4) yields a+b= and ae + be / =. (4) Solving the equation (4) f a and b, we obtain a= (e / e ) and b= (e / e ) (43) Putting these values in (4), we obtain the final solution y= e x (e / e ) + e x/ (e / e ) (44)

5 Appl. Math. Inf. Sci., No. 3, 9-7 (6) / 3 y= (e x/ e x ) (e / e ) (45) Equation (45) is the exact solution of equation (7) which is given in [8] Example :Consider the following homogeneous singular perturbation problem y (x) y (x) (+)y(x)= ; x [,] (46) with y()=+exp( (+)/) and y()=+/e (47) which has a boundary layer at x = because the coefficient of y is negative i.e., the boundary layer will be situated at the right end of the underlying interval. F the solution near x =, we introduce the two scales, ξ =(x )/ the inner scale and the outer scale x = x. The derivatives can be defined in terms of these scales as d dx = ξ + (48) d dx = ξ + ξ x (49) Putting (48), (49) and (6) into the iginal equation (46) it becomes ( ξ + ) ξ x (y + y + y +...) ( ξ + ) (y + y + y +...) (+)(y + y + y +...)=.(5) Equating coefficients of each power of to zero of (5), we have (/) : y ξ + y ξ = (5) ( ) : y ξ + y ξ = y ξ + y + y (5) () : y ξ + y ξ = y y ξ x + y + y + y (53) The general solution of equation (5) is y = A(x )+B(x )e ξ (54) Substituting the value of y from (54) in equation (5), we obtain y ξ + y ξ = B e ξ + A + B e ξ + A+Be ξ (55) y ξ + y ξ =( B + B)e ξ +(A + A) (56) A particular solution of (56) is y p =(B B)ξ e ξ +(A + A)ξ (57) which makes y, much bigger than y as ξ. Hence, f a unifm expansion, the coefficients of ξ and ξ exp( ξ) in equation (57) must vanish independently. Then the result is The solution of (58) is B B=, A + A= (58) A=ae x, B=be x, (59) where a and b are arbitrary constants. Putting the values A and B from (59) in (54) and we get y=ae x +(be x )e ξ (6), in terms of the iginal variable is y = ae x + be x e (x )/ (6) Putting the value of y in equation (6), we get y=ae x + be x e (x )/ +... (6) Imposing the boundary conditions from (47) then equation (6) yields ( a+be / = +exp (+) ) and a+be = e+ (63) Solving these equations we obtain a= and b=/e (64) Putting these values in (6) and considering first two terms, we obtain y=e x + e ex(x )/ (65) y=e x + e (+)(x )/ (66) Equation (66) represents the exact solution of equation (46) which is given in [8], [3]-[5]. In these papers auths applied different numerical methods f solving second der singular perturbed two point boundary value problems with the boundary layers and obtained the numerical solution, but our proposed multiple scales method gives directly exact solution to these problems.

6 4 P. Gupta, M. Kumar: Multiple-Scales method and numerical... Example 3: Let us consider the following initial value problem [6] 3/ y +( / + + 3/ )y +(+ / + )y + y= (67) with initial conditions y() = 3, y ()= / and (68) y () = + + The coefficient of y is positive, the boundary layer will be situated at the left-hand edge of the domain i.e. near x =. Pretending we do not know how to solve it, we rest to conventional singular perturbation methods. It turns out that the conventional perturbation calculation is very tedious and rather challenging. But our proposed multiple scales method successfully finds the approximate solution without any matching by starting only with the thinnest innermost boundary layer by rescaling x = ξ and expanding y(x) in terms of. Substituting (3), (4), (5) and (6) into the equation (67) we have; ( 3/ 3 3 ξ x ξ ) ξ x (y + y +...) ( +( / + + 3/ ) ξ + + ) ξ x (y + y +...) ( +(+ / + ) ξ + ) (y + y +...) +(y + y +...)=. (69) Equating the coefficients of each power of to zero of (69), we get (/ 3/ ) : 3 y ξ 3 + y ξ = (7) (/ / ) : 3 y ξ 3 + y ξ = 3 3 y ξ y y ξ ξ y ξ ( / ) : 3 y ξ 3 + y ξ = 3 3 y ξ 3 3 y ξ x The general solution of equation (7) is y ξ y ξ y x (7) y ξ y. (7) y = A(x )+B(x )ξ +C(x )e ξ, (73) where A,B and C are undetermined at this level of approximation. They are determined at the next level of approximation by imposing the solvability conditions. Substituting the value of y from (73) in equation (7), we obtain 3 y ξ 3 + y ξ = 3C e ξ (B C e ξ ) Ce ξ B+Ce ξ (74) 3 y ξ 3 + y ξ = C e ξ (B + B) (75) A particular solution of (75) is y p = C ξ e ξ (B + B) ξ (76) which makes y, much bigger than y as ξ. Hence, f a unifm expansion, the coefficients of ξ exp( ξ) and ξ in (76) must vanish independently. The results are The solutions of (77) are B + B=, C = (77) B=b e x /, C= c, (78) where b and c are arbitrary constants. Putting the values of B and C from (78) in equation (73), which gives y = A+b e x / ξ + c e ξ (79), in terms of the iginal variable y = A+ x b e x/ + c e x/ (8) Substituting y in (6) we have y=a+ x b e x/ + c e x/ +... (8) Imposing the boundary conditions (68) in (8), which yields A+c = 3, b c = (+ / + ) and (8) b + c = (++ ) Solving equation (8) f A, b and c we obtain A= 3+, b = and c = ( ) ( ) ( ) (83) Then, putting these values in (8) and then equation (8) becomes ( 3+ y = ( ) ( + ( ) ) + x ( ) e x/ ( ) ) e x/, (84)

7 Appl. Math. Inf. Sci., No. 3, 9-7 (6) / MS Exact 3.5 MS Exact.5 y y x x Fig. : =.3 Fig. 3: = MS Exact 3.5 MS Exact y y x x Fig. : =.4 Fig. 4: =.8 y = then { ( 3+ )+ ( )x e x/ ( ) +( )e x/}. (85) The equation (85) represents the final approximate solution of equation (67). Numerical simulations are perfmed varying as it is shown in figure -4. The figures indicate that the solution of (85) is very close to the exact solution given in [6]. Figure -4 of the numerical solution (85) obtained by Multiple scales (MS) method with the exact solution given in [6]. 4 Conclusion We present a numerical method f solving second and third der singularly perturbed problems with the boundary layer at one end (left right) by Multiple scales method. The iginal second and third der dinary differential equations i.e. boundary value problems are transfmed to partial differential equations. This method is very easy f implementation. Numerical results of standard examples chosen from the references are presented in suppt of the proposed they. Using the method of Multiple scales, a single expansion is sufficient and requires no matching between the expansions. In references [8], [3]-[5] auths applied different numerical techniques to obtain the approximate solution of second der singularly perturbed boundary layer

8 6 P. Gupta, M. Kumar: Multiple-Scales method and numerical... problems while in this article exact solution has been obtained by our proposed method. Acknowledgement Auths express their sincere thanks to edit in chief, edit and reviewers f their valuable suggestions to revise this manuscript. References [] Y. Chen, N. Goldenfeld and Y. Oono, Renmalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation they, Physical Review E, 54() 996. [] A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New Yk, 98. [3] S. Johnson, Singular Perturbation They, Mathematical and Analytical Techniques with Applications to Engineering, Springer, 5. [4] M.K. Kadalbajoo, V. Gupta., A brief survey on numerical methods f solving singularly perturbed problems, Applied Mathematics and Computation, 7 () [5] F. 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9 Appl. Math. Inf. Sci., No. 3, 9-7 (6) / 7 [33] Basem S. Attili, Numerical treatment of singularly perturbed two point boundary value problems exhibiting boundary layers, Communications in Nonlinear Science and Numerical Simulation, 6, ,. Parul Gupta received the PhD degree on Mathematical Modeling and Numerical Simulation of Perturbation Problems Arising in Science and Engineering from Motilal Nehru National Institute of Technology, Allahabad-India. Her research interests are in the areas of Applied Mathematics and Mathematical Physics. She has published several research articles in reputed international journals. Manoj Kumar was bn in Aligarh, India, in 975. He received his Ph.D. degree from Aligarh Muslim University, Aligarh, India, in. He joined Motilal Nehru National Institute of Technology (MNNIT), Allahabad- India in 5. His current research interests are Numerical Analysis/Mathematical Modeling/Partial Differential Equations/Computational Fluid Dynamics. Currently he has published me than seventy research paper in reputed journals and guided me than ten Ph.D students in different areas of Applied Mathematics.