Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Finite difference numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh

  • 248 Accesses

  • 2 Citations


Finding numerical solutions to the Troesch’s problem is known to be challenging, especially when the sensitivity parameter \(\lambda \) is large. In this manuscript, we propose a numerical method for solving the Troesch’s problem which combines efficiency and accuracy, even for large sensitivity parameter. Our method can be summarized as a finite difference method formulated on a Shishkin mesh, which is piecewise uniform with smaller step size in the boundary layer in the neighborhood of the point \(x=1\). In fact, we treat the Troesch’s problem as a singularly perturbed problem with a special transition point on the mesh. The transition point for the Shishkin mesh is first defined and computed. The application of the finite difference method on the piecewise uniform mesh leads to a nonlinear matrix system, which is solved numerically using the generalized Newton’s method. While, existing numerical solvers suffer significantly when sensitivity parameter \(\lambda \) becomes large and fail to provide accurate numerical solutions for \(\lambda >20\) (Chang, Appl Math Comput 216(11):3303–3306, 2010; Khuri and Sayfy, Math Comput Model 54(9):1907–1918, 2011; Raja, Inf Sci 279:860–873, 2014; Temimi, Appl Math Comput 219(2):521–529, 2012), our method provides accurate numerical solution for large values of this sensitivity parameter, up to \(\lambda =100\). Moreover, the implementation of the method is straight forward and the computation cost is low. Numerical experiments show that the method provides accurate solutions for different values of the sensitivity parameter \(\lambda \).

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2


  1. 1.

    Cash, J.R., Mazzia, F.: A new mesh selection algorithm, based on conditioning, for two-point boundary value codes. J. Comput. Appl. Math. 184(2), 362–381 (2005)

  2. 2.

    Cash, J.R., Hollevoet, D., Mazzia, F., Nagy, A. M.: Algorithm 927: The MATLAB code bvptwp.m for the numerical solution of two point boundary value problems. ACM Trans. Math. Softw. (TOMS), 39 (2): Article No. 15 (2013). http://calgo.acm.org/927.zip

  3. 3.

    Chang, S.-H.: Numerical solution of Troesch’s problem by simple shooting method. Appl. Math. Comput. 216(11), 3303–3306 (2010)

  4. 4.

    Chiou, J.P., Na, T.Y.: On the solution of Troesch’s nonlinear two-point boundary value problem using an initial value method. J. Comput. Phys. 19(3), 311–316 (1975)

  5. 5.

    Farrel, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust computational techniques for boundary layer. Chapman Hall / CRC, Stanford (1958)

  6. 6.

    Feng, X., Mei, L., He, G.: An efficient algorithm for solving Troesch’s problem. Appl. Math. Comput. 189(1), 500–507 (2007)

  7. 7.

    Jones, D.J.: Solution of Troesch’s and other two point boundary value problems by shooting techniques. J. Comput. Phys. 12(3), 429–434 (1973)

  8. 8.

    Khuri, S.A., Sayfy, A.: Troesch’s problem: A B-spline collocation approach. Math. Comput. Model. 54(9), 1907–1918 (2011)

  9. 9.

    Kopteva, N., O’Riordan, E.: Shishkin meshes in the numerical solution of singularly perturbed differential equations. Int. J. Num. Anal. Model. 7(3), 93–415 (2010)

  10. 10.

    Kubicek, M., Hlavacek, V.: Solution of Troesch’s two-point boundary value problem by shooting technique. J. Comput. Phys. 17(1), 95–101 (1975)

  11. 11.

    Mazzia, F., Trigiante, D.: A hybrid mesh selection strategy based on conditioning for boundary value ODE problems. Num. Algorithms 36(2), 169–187 (2004)

  12. 12.

    Miele, A., Aggarwal, A.K., Tietze, J.L.: Solution of two-point boundary-value problems with jacobian matrix characterized by large positive eigenvalues. J. Comput. Phys. 15(2), 117–133 (1974)

  13. 13.

    Raja, M.A.Z.: Stochastic numerical treatment for solving Troesch’s problem. Inf. Sci. 279, 860–873 (2014)

  14. 14.

    Roberts, S.M., Shipman, J.S.: On the closed form solution of Troesch’s problem. J. Comput. Phys. 21(3), 291–304 (1976)

  15. 15.

    Scott, M.: On the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms. In: Aziz A.K. (ed.) Numerical solutions of boundary-value problems for ordinary differential equations (1975)

  16. 16.

    Shishkin, G.I.: A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous boundary condition. Zh. Vychisl. Mat. Mat. Fiz. 28, 1649-1662 (1988a). (in Russian); translation in U.S.S.R. Comput. Math. and Math. Phys 28, 32-41, (1988)

  17. 17.

    Shishkin, G.I.: A difference scheme for a singularly perturbedequation of parabolic type with a discontinuous initial condition. Dokl. Akad. SSSR 300, 1066-1070 (1988b) (in Russian); translationin Soviet Math. Dokl. 37, 792-796, (1988)

  18. 18.

    Snyman, J.A.: Continuous and discontinuous numerical solutions to the Troesch problem. J. Comput. Appl. Math. 5(3), 171–175 (1979)

  19. 19.

    Temimi, H.: A discontinuous galerkin finite element method for solving the Troeschs problem. Appl. Math. Comput. 219(2), 521–529 (2012)

  20. 20.

    Temimi, H., Kurkcu, H.: An accurate asymptotic approximation and precise numerical solution of highly sensitive Troeschs problem. Appl. Math. Comput. 235, 253–260 (2014)

  21. 21.

    Troesch, B.A.: Intrinsec difficulties in the numerical solution of a boundary value problem. Internal report NN-142,TRW, Inc., Redondo beach (1960)

  22. 22.

    Troesch, B.A.: A simple approach to a sensitive two-point boundary value problem. J. Comput. Phys. 21(3), 279–290 (1976)

  23. 23.

    Vemuri, V., Raefsky, A.: On a method of solving sensitive boundary value problems. J. Franklin Inst. 307(4), 217–243 (1979)

  24. 24.

    Weibel, E.S., Landshoff, R.K.M.: The Plasma in Magnetic Field. Stanford University Press, Stanford (1958)

  25. 25.

    Zarebnia, M., Sajjadian, M.: The sinc-galerkin method for solving Troesch’s problem. Math. Comput. Model. 56(9), 218–228 (2012)

Download references


We acknowledge the support of the Mathematics and Natural Sciences Department at Gulf University for Science and Technology for funding and facilitating the visit of Professor G.I. Shishkin in November 2014.

Author information

Correspondence to Mohamed Ben-Romdhane.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Temimi, H., Ben-Romdhane, M., Ansari, A.R. et al. Finite difference numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh. Calcolo 54, 225–242 (2017). https://doi.org/10.1007/s10092-016-0184-1

Download citation


  • Troesch’s problem
  • Finite difference method
  • Piecewise uniform Shishkin mesh
  • Boundary layer

Mathematics Subject Classification

  • 34B15
  • 65L10
  • 65L12
  • 65G99