## A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations.(English)Zbl 1025.65044

Authors’ abstract: “Singularly perturbed two-point boundary value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative are considered. A numerical method is suggested to solve such problems. In this method, the given BVP is transformed into a system of two ODEs subject to suitable boundary conditions. Then the domain of definition of the differential equation (a closed interval) is divided into three non-overlapping sub-intervals, which we call them inner regions (boundary layers) and outer region. Then the DE is solved in these intervals separately. The solutions obtained in these regions are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval) we use mostly zero-order asymptotic expansion of the solution of the BVP. First, linear equations are considered and then nonlinear equations. To solve nonlinear equations, Newton’s method of quasi-linearization is applied. The present method is demonstrated by providing examples. The method is easy to implement and suitable for parallel computing”.
The method solves equations of the following form: $-\varepsilon y^{(iv)}(x) + b(x) y''(x) - c(x) y(x) = -f(x),\quad x\in D,$
$y(0)=p,\;y(1)=q,\;y''(0)=-r,\;y''(1)=-s,$ where $$\varepsilon$$ is a small positive parameter and $$b(x)$$, $$c(x)$$ and $$f(x)$$ are sufficiently smooth functions satisfying the following conditions: $b(x)\geq \beta > 0,$
$0\geq c(x)\geq -\gamma,\quad \gamma > 0,$
$\beta-2\gamma \geq \eta > 0\;\text{for some} \eta,$
$D=(0,1),\quad \bar{D}=[0,1],\quad y\in \text{ C}^4(D)\cap\text{ C}^2(\bar{D}).$
The sign of the function $$f$$ and the constants $$p$$, $$q$$, $$r$$ and $$s$$ does not seem to be arbitrary, because of the presence of the minus sign in the equations, but the authors do not clarify this point in the Introduction.

### MSC:

 65L10 Numerical solution of boundary value problems involving ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations 34E15 Singular perturbations for ordinary differential equations 65Y05 Parallel numerical computation 34B15 Nonlinear boundary value problems for ordinary differential equations 65L12 Finite difference and finite volume methods for ordinary differential equations
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### References:

 [1] Adams, E.; Spreuer, H., Über das vorligen der Eigenschaft von monotoner Art bei fortschreitend bzw. nur als ganzes losbarem systemen, Z. Angew. Math. Mech., 55, 191-193 (1975) · Zbl 0302.47045 [2] Doolan, E. P.; Miller, J. J.H.; Schilders, W. H.A., Uniform Numerical Methods for Problems with Initial and Boundary Layers (1980), Boole: Boole Dublin · Zbl 0459.65058 [3] Farrel, P. A., Sufficient conditions for uniform convergence of a class of difference schemes for a singular perturbation problems, IMAJ. Numer. Anal., 7, 459-472 (1987) · Zbl 0632.65084 [4] Gartlend, E. C., Graded-mesh difference schemes for singularly perturbed two-point boundary value problems, Math. Comput., 51, 631-657 (1988) · Zbl 0699.65063 [5] Howes, F. A., Differential inequalities of higher order and the asymptotic solution of nonlinear boundary value problems, SIAM J. Math. Anal., 13, 1, 61-80 (1982) · Zbl 0487.34066 [6] Howes, F. A., The asymptotic solution of a class of third-order boundary value problem arising in the theory of thin film flow, SIAM J. Appl. Math., 43, 5, 993-1004 (1983) · Zbl 0532.76042 [7] Israeli, M.; Ungarish, M., Improvement of numerical solution of boundary layer problems by incorporation of asymptotic approximation, Numer. Math., 3a, 309-324 (1984) · Zbl 0489.65054 [8] Jayakumar, J.; Ramanujam, N., A computational method for solving singular perturbation problems, Appl. Math. Comput., 55, 31-48 (1993) · Zbl 0773.65057 [9] Jayakumar, J.; Ramanujam, N., A numerical method for singular perturbation problems arising in chemical reactor theory, Comput. Math. Appl., 27, 5, 83-99 (1994) · Zbl 0792.65061 [10] Khadalbajoo, M. K.; Reddy, Y. N., Numerical treatment of singularly perturbed two-point boundary-value problems, Appl. Math. Comput., 21, 93-110 (1987) · Zbl 0626.65074 [11] Khadalbajoo, M. K.; Reddy, Y. N., Approximate methods for the numerical solution of singular perturbation problems, Appl. Math. Comput., 21, 185-199 (1987) · Zbl 0626.65075 [12] Khadalbajoo, M. K.; Reddy, Y. N., Perturbation problems via deviating arguments, Appl. Math. Comput., 21, 221-232 (1987) · Zbl 0626.65076 [13] Krishnamoorthy, E. V.; Sen, S. K., Numerical Algorithms-Computations in Science and Engineering (1986), Affiliated East-West Press: Affiliated East-West Press New Delhi [14] Feckan, Michel, Singularly perturbed higher order boundary value problems, J. Differential Equations, 3, 1, 79-102 (1994) · Zbl 0807.34068 [15] Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I., Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (1996), World Scientific: World Scientific Singapore · Zbl 0915.65097 [16] Nayfeh, A. H., Introduction to Perturbation Methods (1981), Wiley: Wiley New York · Zbl 0449.34001 [17] Niederdrenk, K.; Yserentant, H., The uniform stability of singularly perturbed discrete and continuous boundary value problems, Numer. Math., 41, 223-253 (1983) · Zbl 0526.65065 [18] Natesan, S.; Ramanujam, N., A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers, Appl. Math. Comput., 93, 259-275 (1998) · Zbl 0943.65086 [19] O’Malley, R. E., Introduction to Singular Perturbations (1974), Academic Press: Academic Press New York · Zbl 0287.34062 [20] O’Malley, R. E., Singular Perturbation Methods for Ordinary Differential Equations (1990), Springer: Springer New York [21] O’Malley, R. E., Singular Perturbation Methods for Ordinary Differential Equations (1991), Springer: Springer Berlin · Zbl 0743.34059 [22] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0153.13602 [23] Ramanujam, N.; Srivartsava, U. N., Singular perturbation problems for systems of PDEs of elliptic type, JMAA, 71, 1 (1979) · Zbl 0423.65061 [24] Ramanujam, N.; Srivartsava, U. N., Singular perturbation problems for systems of partial differential equations of parabolic type, Funkcialaj Ekvacioj, vol. 23, Numero 3, Decembro (1980) · Zbl 0455.35013 [25] Roberts, S. M., A boundary value technique for singular perturbation problems, J. Math. Anal. Appl., 87, 489-508 (1982) · Zbl 0481.65048 [26] Roberts, S. M., Further examples of the boundary value technique in singular perturbation problems, J. Math. Anal. Appl., 133, 411-436 (1988) · Zbl 0658.65076 [27] Roos, H. G.; Stynes, M., A uniformly convergent discretization method for a fourth order singular perturbation problem, Bonner Math. Schriften, 228, 30-40 (1991) · Zbl 0747.65066 [28] Roos, H. G.; Stynes, M.; Tobiska, L., Numerical methods for singularly perturbed differential equations- convection-diffusion and flow problems (1996), Springer · Zbl 0844.65075 [29] Semper, B., Locking in finite element approximation of long thin extensible beams, IMA J. Numer. Anal., 14, 97-109 (1994) · Zbl 0789.73074 [30] Sun, G.; Stynes, M., Finite element methods for singularly perturbed higher order elliptic two point boundary value problems I: reaction-diffusion type, IMA J. Numer. Anal., 15, 117-139 (1995) · Zbl 0814.65083 [31] Sun, G.; Stynes, M., Finite element methods for singularly perturbed higher order elliptic two point boundary value problems II: reaction-diffusion type, IMA J. Numer. Anal., 15, 117-139 (1995) · Zbl 0814.65083 [32] Weili, Zhao, Singular perturbations of boundary value problems for a class of third order nonlinear ordinary differential equations, J. Diff. Eqs., 88, 2, 265-278 (1990) · Zbl 0718.34010
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