Abstract
A coupled iterative approach based on quasilinearization and Krasnoselskii–Mann’s approximation for evaluating the solutions of nonlinear two point Dirichlet boundary value problems (BVPs) is illustrated. The nonlinear problems (including extremely nonlinear Troesch’s problem) are reduced to a sequence of linear equations by using the quasilinearization approach with Green’s function. Additionally, the Krasnoselskii–Mann’s approximation technique is applied to enhance the efficiency of the proposed approach. Some numerical problems, including Troesch’s problems, have been solved to exemplify the method’s ease of implementation. The comparison (in term of absolute errors) with existing techniques like Laplace, Sinc-Galerkin, homotopy perturbation method (HPM), Picard embedded Green’s function method (PGEM), Mann’s embedded Green’s function method (MGEM) and B-Spline method etc. demonstrate that the proposed method is extremely precise and converges quickly.
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References
Abbasbandy S, Magyari E, Shivanian E (2009) The homotopy analysis method for multiple solutions of nonlinear boundary value problems. Commun Nonlinear Sci Numer Simul 14:3530–3536
Adomian G (1988) A review of the decomposition method in applied mathematics. J Math Anal Appl 135(2):501–544
Al-Mazmumy M, Al-Mutairi A, Al-Zahrani K (2017) An efficient decomposition method for solving Bratu’s boundary value problem. Am J Comput Math 7:84–93
Ali F, Ali J, Uddin I (2021) A novel approach for the solution of BVPs via Green’s function and fixed point iterative method. J Appl Math Comput 66(1):167–181
Asaithambi A (2022) On solving the Troesch problem for large sensitivity parameter values using exact derivative evaluations. Int J Appl Comput Math 8:202
Bellman RE, Kalaba RE (1965) Quasilinearization and nonlinear boundary-value problems. Elsevier, New York
Chambre PL (1952) On the solution of the Poisson–Boltzmann equation with application to the theory of thermal explosions. J Chem Phys 20:1795–1797
Chandrasekhar S (1967) Introduction to the study of stellar structure. Dover Publications Inc., New York
Chang SH (2010) A variational iteration method for solving Troesch’s problem. J Comput Appl Math 234:3043–3047
Chawla MM, Katti CP (1985) A uniform mesh finite difference method for a class of singular two-point boundary value problems. J Numer Anal 22(3):561–565
Chun C, Sakthivel R (2010) Homotopy perturbation technique for solving two-point boundary value problems-comparison with other methods. Comput Phys Commun 181:1021–1024
Copple C, Hartree DR, Porter A, Tyson H (1939) The evaluation of transient temperature distributions in an alternating field. Inst Electr Eng 85:56–66
Coster CD, Habets P (2006) Two-point boundary value problems: lower and upper solutions. Math. Sci. Eng., vol 205. Elsevier, Amsterdam
Deeba E, Khuri SA, Xie S (2000) An algorithm for solving boundary value problems. J Comput Phys 159(2):125–138
EI-Ajou A, Ahmad, Arqub OA, Momani S (2012) Homotopy analysis method for second-order boundary value problems of integrodifferential equations. Discrete Dyn Nat Soc
EL-Gamel M, Sameeh M (2013) A Chebychev collocation method for solving Troesch’s problem. Int J Math Comput Appl Res 3(2):23–32
Falk TJ, Turcotte DL (1962) Current layer diffusion in one-dimensional pinch. Phys Fluids 5(10):1288–1292
Feng X, Mei L, He G (2007) An efficient algorithm for solving Troesch’s problem. Appl Math Comput 189(1):500–507
Frank-Kamenetski DA (1955) Diffusion and heat exchange in chemical kinetics. Princeton University Press, Princeton
Gidaspow D, Baker BS (1973) A model for discharge of storage batteries. J Electrochem Soc 120:1005–1010
Granas A (1976) Sur la methode de continuite de poincare. CR Acad Sci Paris 282(17):983–985
He JH (1999) Variational iteration method a kind of nonlinear analytical technique: some examples. Int J Non Linear Mech 34(4):699–708
Hosseini MM, Nasabzadeh H (2007) Modified Adomian decomposition method for specific second order ordinary differential equations. Appl Math Comput 186:117–123
Jacobsen J, Schmitt K (2002) The Liouville–Bratu–Gelfand problem for radial operators. J Differ Equ 184(1):283–298
Jalilian R (2010) Non-polynomial spline method for solving Bratu’s problem. Comput Phys Commun 181:1868–1872
Jang B (2008) Two-point boundary value problems by the extended Adomian decomposition method. J Comput Appl Math 219:253–262
Kafri HQ, Khuri SA, Sayfy A (2016) A new approach based on embedding Green’s functions into fixed-point iterations for highly accurate solution to Troesch’s problem. Int J Comput Methods Eng Sci Mech 17(2):93–105
Khuri SA (2003) A numerical algorithm for solving Troesch’s problem. Int J Comput Math 80(4):493–498
Khuri SA, Sayfy A (2011) Troesch’s problem: a B-spline collocation approach. Math Comput Model 54:1907–1918
Kreyszig E (1989) Introductory functional analysis with applications. Wiley, New York
Liao S (2003) Beyond perturbation: introduction to the homotopy analysis method. Chapman & Hall/CRC Press, Boca Raton
Lloyd NG (1978) Degree theory. Cambridge Univ. Press, Cambridge
Markin VS, Chernenko AA, Chizmadehev YA, Chirkov YG (1966) Aspects of the theory of gas porous electrodes in fuel cells: their electrochemical kinetics. Consultants Bureau, New York, pp 22–33
Masood Z, Majeed K, Samar R, Raja MAZ (2017) Design of Mexican Hat wavelet neural networks for solving Bratu type nonlinear systems. Neurocomputing 221:1–14
Mirmoradi SH, Hosseinpour I, Ghanbarpour S, Barari A (2009) Application of an approximate analytical method to nonlinear Troesch’s problem. Appl Math Sci 3(32):1579–1585
Mohsen A (2014) A simple solution of the Bratu problem. Comput Math Appl 67(1):26–33
Momani S, Abuasad S, Odibat Z (2006) Variational iteration method for solving nonlinear boundary value problems. Appl Math Comput 183(2):1351–1358
Raja MAZ, Samar R, Alaidarous ES, Shivanian E (2016) Bio-inspired computing platform for reliable solution of Bratu-type equations arising in the modeling of electrically conducting solids. Appl Math Model 40:5964–5977
Roberts SM, Shipman JS (1972) Solution of Troesch’s two-point boundary value problem by a combination of techniques. J Comput Phys 10:232–241
Roberts SM, Shipman JS (1976) On the closed form solution of Troesch’s problem. J Comput Phys 21:291–304
Shahni J, Singh Bernstein R (2021) Gegenbauer-wavelet collocation methods for Bratu-like equations arising in electrospinning process. J Math Chem 59:2327–2343
Shehu Y (2013) Modified Krasnoselskii–Mann iterative algorithm for nonexpansive mappings in Banach spaces. Arab J Math 2:209–219
Singh R, Singh M (2022) An optimal decomposition method for analytical and numerical solution of third-order Emden–Fowler type equations. J Comput Sci 63:101790
Singh M, Verma AK (2013) On a monotone iterative method for a class of three point nonlinear nonsingular BVPs with upper and lower solutions in reverse order. J Appl Math Comput 43(1):99–114
Singh R, Singh G, Singh M (2021) Numerical algorithm for solution of the system of Emden–Fowler type equations. Int J Appl Comput Math 7(4):136
Swati M, Singh K (2021) An advancement approach of Haar wavelet method and Bratu-type equations. Appl Numer Math 170:74–82
Taiwo OA (2002) Exponential fitting for the solution of two-point boundary value problems with cubic spline collocation tau-method. Int J Comput Math 79(3):299–306
Tirmizi IA, Twizell EH (2002) Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems. Appl Math Lett 15:897–902
Tomar S (2021) An effective approach for solving a class of nonlinear singular boundary value problems arising in different physical phenomena. Int J Comput Math 98(10):2060–2077
Tomar S, Pandey RK (2019) An efficient iterative method for solving Bratu-type equations. J Comput Appl Math 357:71–84
Tomar S, Singh M, Ramos H, Wazwaz AM (2022) Development of a new iterative method and its convergence analysis for nonlinear fourth-order boundary value problems arising in beam analysis. Math Meth Appl Sci, 1–9
Tomar S, Singh M, Vajravelu K, Ramos H (2023) Simplifying the variational iteration method: a new approach to obtain the Lagrange multiplier. Math Comput Simul 204:640–644
Troesch BA (1976) A simple approach to a sensitive two-point boundary value problem. J Comput Phys 21(3):279–290
Verma AK, Pandit B, Verma L, Agarwal RP (2020) A review on a class of second order nonlinear singular BVPs. Mathematics 8(7):1045
Wan YQ, Guo Q, Pan N (2004) Thermo-electro-hydrodynamic model for electrospinning process. Int J Nonlinear Sci Numer 5(1):5–8
Weibel ES (1959) On the confinement of a plasma by magnetostatic fields. Phys Fluids 2(1):52–56
Yusufoglu E (2007) Homotopy perturbation method for solving a nonlinear system of second order boundary value problems. Int J Nonlinear Sci Numer Simul 8(3):353–358
Zarebnia M, Sajjadian M (2012) The Sinc-Galerkin method for solving Troesch’s problem. Math Comput Model 56:218–228
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Jyoti, Singh, M. An iterative technique for a class of Dirichlet nonlinear BVPs: Troesch’s problem. Comp. Appl. Math. 42, 163 (2023). https://doi.org/10.1007/s40314-023-02303-z
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DOI: https://doi.org/10.1007/s40314-023-02303-z
Keywords
- Second order nonlinear BVPs
- Troesch’s problem
- Krasnoselskii–Mann’s approximation technique
- Quasilinearization method
- Green’s function
- Convergence analysis