A Review on a Class of Second Order Nonlinear Singular BVPs
Abstract
:1. Introduction
1.1. The Boundary Conditions
- (i)
- is continuous on ,
- (ii)
- ,
- (a*)
- ,
- (b*)
where are constants.
- (a)
- If , , and
- (b)
- If , , and
- (c)
- If
- (d)
- If no singularity is present at the other end , then the mixed type of boundary conditions may be imposed.
1.2. Real Life Applications
1.2.1. Physiology
1.2.2. Electrohydrodynamics
1.2.3. Thermal Explosions
1.2.4. Polytropes as a Simple Model of a Star
1.2.5. Epitaxial Growth
1.2.6. Shallow Membrane Caps
1.2.7. Catalyst Diffusion Reaction
1.3. Test Examples
- BVP 1
- BVP 2
- BVP 3
- BVP 4
- BVP 5
- BVP 6
- BVP 7
- BVP 8
- BVP 9
- BVP 10
- BVP 11
- BVP 12
- BVP 13
- BVP 14
- BVP 15
- BVP 16
2. Survey
2.1. Survey on Numerical Results
2.1.1. Numerical Results when Nonlinear Term Is
- (i)
- formulation of Volterra integral equation or
- (ii)
- expansion of the dependent variable in the original ordinary differential equations.
2.1.2. Numerical Results when the Nonlinear Term Is
2.2. Survey on Analytical Results
2.2.1. Some Existing Analytical Techniques
Shooting Method
Nonlinear Alternative
Upper and Lower Solution Methods
2.2.2. Analytical Results when the Nonlinear Term Is
- (i)
- on ,
- (ii)
- , and for some ,
- (iii)
- is analytic in with Taylor series expansion
2.2.3. Analytical Results when the Nonlinear Term Is
- (a)
- in , .
- (b)
- , .
- (c)
- is continuous on , where are upper and lower solutions.
- (d)
- such that for all ,
- (e)
- such that for all ,
- (f)
- for and .
- (g)
- For all , , where is continuous and satisfies
- (i)
- in , and .
- (ii)
- in and .
- (iii)
- There exist functions and in , where are upper and lower solutions.
- (iv)
- is continuous on .
- (v)
- such that for all ,
- (vi)
- such that for all ,
- (vii)
- for , and .
- (viii)
- For all , , where is continuous and satisfiesFurthermore, they assume that and satisfy the following properties.
- (ix)
- , ∃, such that and .
- (x)
- in , and .
2.3. Survey on Monotone Iterative Techniques
3. Future Scope
- ADM is an iterative method that depends on Adomian’s polynomial. One may change the polynomial for better accuracy (see Remark 3).
- Coupled system of nonlinear SBVPs are not fully explored, so there are various possibilities related to the system of nonlinear SBVPs.
- Several theoretical as well as numerical works described in Section 2.1.1 may be explored for the case when the nonlinear term is derivative dependent.
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Dunninger, D.R.; Kurtz, J.C. Existence of Solutions for Some Nonlinear Singular Boundary Value Problems. J. Math. Anal. Appl. 1986, 115, 396–405. [Google Scholar] [CrossRef] [Green Version]
- Lin, H.S. Oxygen Diffusion in a Spherical Cell with Nonlinear Oxygen Uptake Kinetics. J. Theor. Biol. 1976, 60, 449–457. [Google Scholar] [CrossRef]
- McElwain, D.L.S. A Re-examination of Oxygen Diffusion in a Spherical Cell with Michaelis—Menten Oxygen Uptake Kinetics. J. Theor. Biol. 1978, 71, 255–263. [Google Scholar] [CrossRef]
- Anderson, N.; Arthurs, A.M. Analytical Bounding Functions for Diffusion Problems with Michaelis–Menten Kinetics. Bull. Math. Biol. 1985, 47, 145–153. [Google Scholar] [CrossRef]
- Duggan, R.C.; Goodman, A.M. Pointwise Bounds for Nonlinear Heat Conduction Model of the Human Head. Bull. Math. Biol. 1986, 48, 229–236. [Google Scholar] [CrossRef]
- Flesch, U. The Distribution of Heat Sources in the Human Head: A Theoretical Consideration. J. Theor. Biol. 1975, 54, 285–287. [Google Scholar] [CrossRef]
- Gray, B.F. The Distribution of Heat Sources in the Human Head—Theoretical Considerations. J. Theor. Biol. 1980, 82, 473–476. [Google Scholar] [CrossRef]
- Anderson, N.; Arthurs, A.M. Complementary Extremum Principles for a Nonlinear Model of Heat Conduction in the Human Head. Bull. Math. Biol. 1981, 43, 341–346. [Google Scholar] [CrossRef]
- Keller, J.B. Electrohydrodynamics I. The Equilibrium of a Charged Gas in a Containor. J. Rational Mech. Anal. 1956, 5, 715–724. [Google Scholar]
- Chamber, P.L. On the Solution of the Poisson-Boltzmann Equation with Application to the Theory of Thermal Explosions. J. Chem. Phys. 1952, 20, 1795–1797. [Google Scholar] [CrossRef]
- Kazutaka, N.; Toshiyuki, T.; Shin, S. A modified arrhenius equation. Chem. Phys. Lett. 1989, 160, 295–298. [Google Scholar]
- Verma, A.K.; Tiwari, D. On Some Computational Aspects of Hermite wavelets on a Class of SBVPs Arising in Exothermic Reactions. arXiv 2019, arXiv:1911.00495. [Google Scholar]
- Chandrashekhar, S. An Introduction to the Study of Stellar Structure; Dover: New York, NY, USA, 1939. [Google Scholar]
- Escudero, C.; Hakl, R.; Peral, I.; Torres, P.J. Existence and nonexistence Results for a Singular Boundary Value Problem Arising in the Theory of Epitaxial Growth. Math. Meth. Appl. Sci. 2014, 37, 793–807. [Google Scholar] [CrossRef] [Green Version]
- Rachünková, I.; Koch, O.; Pulverer, G.; Weinmxuxller, E. On a Singular Bundary Value Problem Arising in the Theory of Shallow Membrane Caps. J. Math. Anal. Appl. 2007, 332, 523–541. [Google Scholar] [CrossRef] [Green Version]
- Flockerzi, D.; Sundmacher, K. On Coupled Lane–Emden Equations Arising in Dusty Fluid Models. J. Phys. Conf. Ser. 2011, 268, 012006. [Google Scholar] [CrossRef]
- Agarwal, R.P.; Hodis, S.; O’Regan, D. 500 Examples and Problems of Applied Differential Equations; Springer: Berlin, Germany, 2019; p. 388. [Google Scholar]
- Ciarlet, P.G.; Natterer, F.; Varga, R.S. Numerical Methods of Higher-Order Accuracy for Singular Nonlinear Boundary Value Problems. Numer. Math. 1970, 15, 87–99. [Google Scholar] [CrossRef]
- Jamet, P. On the Convergence of Finite-Difference Apprxoimations to One-Dimensional Singular Boundary Value Problems. Numer. Math. 1970, 14, 355–378. [Google Scholar] [CrossRef]
- Russell, R.D.; Shampine, L.F. Numerical Methods for Singular Boundary Value Problems. SIAM J. Numer. Anal. 1975, 12, 13–36. [Google Scholar] [CrossRef]
- Shampine, L.F. Boundary Value Problems for Ordinary Differential Equations. II. Patch Bases and Monotone Methods. SIAM J. Numer. Anal. 1969, 6, 414–431. [Google Scholar] [CrossRef]
- Chawla, M.M.; Katti, C.P. Finite Difference Methods and Their Convergence for a Class of Singular Two Point Boundary Value Problems. Numer. Math. 1982, 39, 341–350. [Google Scholar] [CrossRef]
- Chawla, M.M. A Fourth Order Finite-Difference Method Based on Uniform Mesh for Singular Two-Point Boundary Value Problems. J. Comp. Appl. Math. 1987, 17, 359–364. [Google Scholar] [CrossRef] [Green Version]
- Chawla, M.M.; Katti, C.P. A Finite-Difference Method for a Class of Singular Two Point Boundary Value Problems. IMA J. Numer. Anal. 1984, 4, 457–466. [Google Scholar] [CrossRef]
- Chawla, M.M.; Katti, C.P. A Unifrom Mesh Finite Difference Method for a Class of Singular Two-Point Boundary Value Problems. SIAM J. Numer. Anal. 1985, 22, 561–565. [Google Scholar] [CrossRef]
- Chawla, M.M.; McKee, S.; Shaw, G. Order h2 Method for a Singular Two-Point Boundary Value Problem. BIT 1986, 26, 318–326. [Google Scholar] [CrossRef]
- Chawla, M.M.; Subramanian, R.; Sathi, H.L. A Fourth-Order Method for a Singular Two-Point Boundary Value Problem. BIT 1988, 28, 88–97. [Google Scholar] [CrossRef]
- Chawla, M.M.; Subramanian, R.; Sathi, H.L. A Fourth-Order Spline Method for Singular Two-Point Boundary Value Problems. J. Comp. Appl. Math. 1988, 21, 189–202. [Google Scholar] [CrossRef] [Green Version]
- Katti, C.P.; Pradhan, J. A New Finite Difference Method for a Class of Singular Two-Point Boundary Value Problems. J. Math. Phys. Sci. 1986, 20, 93–103. [Google Scholar] [CrossRef]
- Iyengar, S.R.K.; Jain, P. Spline Finite Difference Methods for Singular Two Point Boundary Value Problems. Numer. Math. 1987, 50, 363–376. [Google Scholar] [CrossRef]
- Sakai, M.; Usmani, R.A. Non Polynomial Splines and Weakly Singular Two-Point Boundary Value Problems. BIT 1988, 28, 867–876. [Google Scholar] [CrossRef]
- Sakai, M.; Usmani, R.A. An Application of Chawla’s Identity to a Different Scheme for Singular Problems. BIT 1989, 28, 566–568. [Google Scholar] [CrossRef]
- Jain, R.K.; Jain, P. Explicit Direct Method for a Class of Singular Two-Point Boundary Value Problems. J. Math. Phys. Sci. 1989, 23, 411–423. [Google Scholar]
- Pandey, R.K. On The Convergence of a Finite Difference Method for a Class of Singular Two Point Boundary Value Problems. Inter. J. Comput. Math. 1992, 42, 237–241. [Google Scholar] [CrossRef]
- Pandey, R.K. A Finite Difference Method for a Class of Singular Two Point Boundary Value Problems Arising in Physiology. Int. J. Comput. Math. 1997, 65, 131–140. [Google Scholar] [CrossRef]
- Pandey, R.K. A Note on a Finite Difference Method for a Class of Singular Boundary Value Problems in Physiology. Int. J. Comput. Math. 2000, 74, 127–132. [Google Scholar] [CrossRef]
- Pandey, R.K.; Singh, A.K. On the Convergence of a Finite Difference Method for a Class of Singular Boundary Value Problems Arising in Physiology. J. Comput. Appl. Math. 2004, 166, 553–564. [Google Scholar] [CrossRef] [Green Version]
- Abu-Zaid, I.T.; El-Gebeily, M.A. A Finite-Difference Method for the Spectral Approximation of a Class of Singular Two-Point Boundary Value Problems. IMA J. Numer. Anal. 1994, 14, 545–562. [Google Scholar] [CrossRef]
- Sen, R.N.; Hossain, M.B. Finite Difference Methods for Certain Singular Two Point Boundary Value Problems. J. Comp. Appl. Math. 1996, 70, 33–50. [Google Scholar] [CrossRef] [Green Version]
- Gustafsson, B. A numerical method for solving singular boundary value problems. Numer. Math. 1973, 21, 328–344. [Google Scholar] [CrossRef]
- El-Gebeily, M.A.; Abu-Zaid, I.T. On a Finite Difference Method for Singular Two-Point Boundary Value Problems. IMA J. Numer. Anal. 1998, 18, 179–190. [Google Scholar] [CrossRef]
- Guoqiang, H.; Jiong, W.; Hayami, K.; Yuesheng, X. Correction Method and Extrapolation Method for Singular Two-Point Boundary Value Problems. J. Comp. Appl. Math. 2000, 126, 145–157. [Google Scholar] [CrossRef] [Green Version]
- Pandey, R.K. On the Convergence of a Spline Method for Singular Two Point Boundary Value Problems Arising in Physiology. Int. J. Comput. Math. 2002, 79, 357–366. [Google Scholar] [CrossRef]
- Ha, S.N.; Lee, C.R. Numerical Study for Two-Point Boundary Value Problems Using Green’s Functions. Comput. Math. Appl. 2002, 44, 1599–1608. [Google Scholar] [CrossRef] [Green Version]
- El-Gebeily, M.A.; Attili, B.S. An Iterative Shooting Method for a Certain Class of Singular Two-Point Boundary Value Problems. Comput. Math. Appl. 2003, 45, 69–76. [Google Scholar] [CrossRef] [Green Version]
- Pandey, R.K.; Singh, A.K. On the Convergence of Finite Difference Method for General Singular Boundary Value Problems. Int. J. Comput. Math. 2003, 80, 1323–1331. [Google Scholar] [CrossRef]
- Pandey, R.K.; Singh, A.K. On the Convergence of Fourth-Order Finite Difference Method for Weakly Regular Singular Boundary Value Problems. Int. J. Comput. Math. 2004, 81, 227–238. [Google Scholar] [CrossRef]
- Pandey, R.K.; Singh, A.K. On the Convergence of Finite Difference Methods for Weakly Regular Singular Boundary Value Problems. J. Comput. Appl. Math. 2007, 205, 469–478. [Google Scholar] [CrossRef] [Green Version]
- Mittal, R.C.; Nigam, R. Solution of a Class of Singular Boundary Value Problems. Numer. Algorithms 2008, 47, 169–179. [Google Scholar] [CrossRef]
- Pandey, R.K.; Singh, A.K. On the Convergence of a Fourth-Order Method for a Class of Singular Boundary Value Problems. J. Comput. Appl. Math. 2009, 224, 734–742. [Google Scholar] [CrossRef] [Green Version]
- Kanth, A.S.V.R.; Aruna, K. Solution of singular two-point boundary value problems using differential transformation method. Phys. Lett. A 2008, 372, 4671–4673. [Google Scholar] [CrossRef]
- Bataineh, A.S.; Noorani, M.S.M.; Hashim, I. Approximate Solutions of Singular two-point BVPs by Modified Homotopy Analysis Method. Phys. Lett. A 2008, 372, 4062–4066. [Google Scholar] [CrossRef]
- Ramos, J.I. Series Approach to the Lane–Emden Equation and Comparison with the Homotopy Perturbation Method. Chaos Solitons Fractals 2008, 38, 400–408. [Google Scholar] [CrossRef]
- Adomian, G. Solving Frontier Problems of Physics: The Decomposition Method; Kluwer: Boston, MA, USA, 1994. [Google Scholar]
- He, J.H. Homotopy Perturbation Technique. Comput. Meth. Appl. Mech. Eng. 1999, 178, 257–262. [Google Scholar] [CrossRef]
- He, J.H. Homotopy Perturbation Method: A New Nonlinear Analytical Technique. Appl. Math. Comput. 2003, 135, 73–79. [Google Scholar] [CrossRef]
- He, J.H. Asymptotology by Homotopy Perturbation Method. Appl. Math. Comput. 2004, 156, 591–596. [Google Scholar] [CrossRef]
- Wazwaz, A.M. A New Method for Solving Singular Initial Value Problems in the Second-Order Ordinary Differential Equations. Appl. Math. Comput. 2002, 128, 45–57. [Google Scholar] [CrossRef]
- El-Gebeily, M.A.; Furati, K.M.; O’Regan, D.; Agarwal, R.P. On the Approximation of Nonlinear Singular Self-adjoint Second Order Boundary Value Problems. J. Comput. Appl. Math. 2009, 224, 360–372. [Google Scholar] [CrossRef] [Green Version]
- Caglar, H.; Caglar, N.; Ozer, M. B-spline Solution of Non-Linear Singular Boundary Value Problems Arising in Physiology. Chaos Solitons Fractals 2009, 39, 1232–1237. [Google Scholar] [CrossRef]
- Hasan, Y.Q.; Zhu, L.M. Solving Singular Boundary Value Problems of Higher-Order Ordinary Differential Equations by Modified Adomian Decomposition Method. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 2592–2596. [Google Scholar] [CrossRef]
- Bataineh, A.; Noorani, M.S.M.; Hashim, I. Homotopy Analysis Method for Singular IVPs of Emden-Fowler type. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 1121–1131. [Google Scholar] [CrossRef]
- Ebaid, A. A New Analytical and Numerical Treatment for Singular Two-Point Boundary Value Problems via the Adomian Decomposition Method. J. Comput. Appl. Math. 2011, 235, 1914–1924. [Google Scholar] [CrossRef] [Green Version]
- Mohammadi, F.; Hosseini, M.M. A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations. J. Frankl. Inst. 2011, 348, 1787–1796. [Google Scholar] [CrossRef]
- Xufeng, S.; Yubo, Y. Homotopy Perturbation Method Based on Green Function for Solving Non-Linear Singular Boundary Value Problems. In Proceedings of the 2011 International Conference on Machine Learning and Cybernetics, Guilin, China, 10–13 July 2011. [Google Scholar]
- Secer, A.; Kurulay, M. The Sinc–Galerkin Method and Its Applications on Singular Dirichlet-Type Boundary Value Problems. Bound. Value Probl. 2012, 2012, 126. [Google Scholar] [CrossRef] [Green Version]
- Bhrawy, A.H.; Alofi, A.S. A Jacobi–Gauss collocation Method for Solving Nonlinear Lane–Emden type Equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 62–70. [Google Scholar] [CrossRef]
- Iqbal, S.; Javed, A. Application of Optimal Homotopy Asymptotic Method for the Analytic Solution of Singular Lane–Emden Type Equation. Appl. Math. Comput. 2011, 217, 7753–7761. [Google Scholar] [CrossRef]
- Rismani, A.M.; Monfared, H. Numerical Solution of Singular IVPs of Lane–Emden Type Using a Modified Legendre-Spectral Method. Appl. Math. Model. 2012, 36, 4830–4836. [Google Scholar] [CrossRef]
- Randolph, R.; Duan, J.S.; Wazwaz, A.M. Solving Coupled Lane–Emden Boundary Value Problems in Catalytic Diffusion Reactions by the Adomian Decomposition Method. J. Math. Chem. 2014, 52, 255–267. [Google Scholar]
- Babolian, E.; Eftekhari, A.; Saadatmandi, A. A Sinc–Galerkin technique for the numerical solution of a class of singular boundary value problems. Comput. Appl. Math. 2015, 2015, 45–63. [Google Scholar] [CrossRef]
- Singh, R. Analytic Solution of Singular Emden-Fowler-type Equations by Green’s Function and Homotopy Analysis Method. Eur. Phys. J. Plus 2019, 134, 583. [Google Scholar] [CrossRef]
- Singh, R.; Das, N.; Kumar, J. The Optimal Modified Variational Iteration Method for the Lane–Emden Equations with Neumann and Robin Boundary Conditions. Eur. Phys. J. Plus 2017, 132, 251. [Google Scholar] [CrossRef]
- Singh, R.; Garg, H.; Guleria, V. Haar Wavelet Collocation Method for Lane–Emden Equations with Dirichlet. Neumann and Neumann-Robin boundary conditions. J. Comput. Appl. Math. 2019, 346, 150–161. [Google Scholar] [CrossRef]
- Singh, R.; Kumar, J.; Nelakanti, G. New Approach for Solving a Class of Doubly Singular Two-Point Boundary Value Problems Using Adomian Decomposition Method. Adv. Numer. Anal. 2012, 2012, 541083. [Google Scholar] [CrossRef] [Green Version]
- Singh, R.; Wazwaz, A.M.; Kumar, J. An Efficient Semi-Numerical Technique for Solving Nonlinear Singular Boundary Value Problems Arising in Various Physical Models. Int. J. Comput. Math. 2016, 93, 1330–1346. [Google Scholar] [CrossRef]
- Singh, R.; Wazwaz, A.M. Optimal Homotopy Analysis Method for Oxygen Diffusion in a Spherical Cell with Nonlinear Oxygen Uptake Kinetics. MATCH Commun. Math. Comput. Chem. 2018, 80, 369–382. [Google Scholar]
- Bobisud, L.E. Existence of Solutions for Nonlinear Singular Boundary Value Problems. Appl. Anal. 1990, 35, 43–57. [Google Scholar] [CrossRef]
- Singh, R. A Modified Homotopy Perturbation Method for Nonlinear Singular Lane–Emden Equations Arising in Various Physical Models. Int. J. Appl. Comput. Math. 2019, 5, 64. [Google Scholar] [CrossRef]
- Aydinlik, S.; Kiris, A. A High-Order Numerical Method for Solving Nonlinear Lane–Emden type Equations Arising in Astrophysics. Astrophys. Space Sci. 2018, 363, 264. [Google Scholar] [CrossRef]
- Madduri, H.; Roul, P. A Fast-Converging Iterative Scheme for Solving a System of Lane–Emden Equations Arising in Catalytic Diffusion Reactions. J. Math. Chem. 2019, 57, 570–582. [Google Scholar] [CrossRef]
- Roul, P. On the Numerical Solution of Singular Two Point Boundary Value Problem: Adomain Decomposition Homotopy Perturbation Approach. Math. Methods Appl. Sci. 2017, 40, 7396–7409. [Google Scholar] [CrossRef]
- Roul, P. A Fast and Accurate Computational Technique for Efficient Numerical Solution of Nonlinear Singular Boundary Value Problems. Int. J. Comput. Math. 2019, 96, 51–72. [Google Scholar] [CrossRef]
- Roul, P. A New Mixed MADM-Collocation Approach for Solving a Class of Lane–Emden Singular Boundary Value Problems. J. Math. Chem. 2019, 57, 945–969. [Google Scholar] [CrossRef]
- Roul, P.; Madduri, H. A new highly accurate domain decomposition optimal homotopy analysis method and its convergence for singular boundary value problems. Math. Meth. Appl. Sci. 2018, 41, 6625–6644. [Google Scholar] [CrossRef]
- Roul, P.; Madduri, H.; Agarwal, R.P. A fast-converging recursive approach for Lane–Emden type initial value problems arising in astrophysics. J. Comput. Appl. Math. 2019, 359, 182–195. [Google Scholar] [CrossRef]
- Roul, P.; Madduri, H.; Kassner, K. A sixth-order numerical method for a strongly nonlinear singular boundary value problem governing electrohydrodynamic flow in a circular cylindrical conduit. Appl. Math. Comput. 2019, 350, 416–433. [Google Scholar] [CrossRef]
- Roul, P.; Thula, K. A New High-Order Numerical Method for Solving Singular Two-Point Boundary Value Problems. J. Comput. Appl. Math. 2018, 343, 556–574. [Google Scholar] [CrossRef]
- Roul, P.; Thula, R.; Agarwal, R.P. Non-Optimal Fourth-Order and Optimal Sixth-Order B-Spline Collocation Methods for Lane–Emden Boundary Value Problems. Appl. Numer. Math. 2019, 145, 342–360. [Google Scholar] [CrossRef]
- Roul, P.; Warbhe, U. New Approach for Solving a Class of Singular Boundary Value Problem Arising in Various Physical Models. J. Math. Chem. 2016, 54, 1255–1285. [Google Scholar] [CrossRef]
- Thula, K.; Roul, P. A High-Order B-Spline Collocation Method for Solving Nonlinear Singular Boundary Value Problems Arising in Engineering and Applied Science. Mediterr. J. Math. 2018, 15, 176. [Google Scholar] [CrossRef]
- Roul, P. A New Efficient Recursive Technique for Solving Singular Boundary Value Problems arising in various physical models. Eur. Phys. J. Plus 2016, 131, 105. [Google Scholar] [CrossRef]
- Niu, J.; Xu, M.; Lin, Y.; Xue, Q. Numerical Solution of Nonlinear Singular Boundary Value Problems. J. Comput. Appl. Math. 2018, 331, 42–51. [Google Scholar] [CrossRef]
- Verma, A.K.; Kayenat, S. On the Convergence of Mickens’ Type Nonstandard Finite Difference Schemes on Lane—Emden Type Equations. J. Math. Chem. 2018, 56, 1667–1706. [Google Scholar] [CrossRef]
- Buckmire, R. Investigations of nonstandard. Mickens’-type, finite-difference schemes for Singular Boundary Value Problems in Cylindrical or Spherical Coordinates. Numer. Methods Partial Differ. Equ. 2003, 19, 380–398. [Google Scholar] [CrossRef]
- Singh, R. Analytical Approach for Computation of Exact and Analytic Approximate Solutions to the System of Lane–Emden-Fowler Type Equations Arising in Astrophysics. Eur. Phys. J. Plus 2018, 133, 320. [Google Scholar] [CrossRef]
- Singh, R.; Wazwaz, A.M. An Efficient Algorithm for Solving Coupled Lane–Emden Boundary Value Problems in Catalytic Diffusion Reactions: The Homotopy Analysis Method. MATCH Commun. Math. Comput. Chem. 2019, 81, 785–800. [Google Scholar]
- Lă pădat, M.; Paşca, M. Approximate Solutions of the Nonlinear Standard Lane–Emden Equation using the LSDQ Method. AIP Conf. Proc. 2019, 2116, 37000. [Google Scholar]
- Boubaker, K.; Van Gorder, R.A. Application of the BPES to Lane–Emden Equations Governing Polytropic and Isothermal Gas Spheres. New Astron. 2012, 17, 565–569. [Google Scholar] [CrossRef]
- Parand, K.; Dehghan, M.; Rezaeia, A.R.; Ghaderi, S.M. An Approximation Algorithm for the Solution of the Nonlinear Lane–Emden Type Equations Arising in Astrophysics Using Hermite Functions Collocation Method. Comput. Phys. Commun. 2010, 181, 1096–1108. [Google Scholar] [CrossRef] [Green Version]
- Parand, K.; Shahini, M.; Dehghan, M. Rational Legendre Pseudospectral Approach for Solving Nonlinear Differential Equations of Lane–Emden Type. J. Comput. Phys. 2009, 228, 8830–8840. [Google Scholar] [CrossRef]
- Parand, K.; Rezaei, A.R.; Taghavi, A. Lagrangian Method for Solving Lane–Emden Type Equation Arising in Astrophysics on Semi-Infinite Domains. Acta Astronaut. 2010, 67, 673–680. [Google Scholar] [CrossRef] [Green Version]
- Vasile, M.; Herisanu, N. Optimal Homotopy Asymptotic Method for Polytrophic Spheres of the Lane–Emden Type Equation. AIP Conf. Proc. 2019, 2116, 300003. [Google Scholar]
- Kaur, H.; Mittal, R.C.; Mishra, V. Haar Wavelet Approximate Solutions for the Generalized Lane–Emden Equations Arising in Astrophysics. Comput. Phys. Commun. 2013, 184, 2169–2177. [Google Scholar] [CrossRef]
- Singh, R.; Shahni, J.; Garg, H.; Garg, A. Haar Wavelet Collocation Approach for Lane–Emden Equations Arising in Mathematical Physics and Astrophysics. Eur. Phys. J. Plus 2019, 134, 548. [Google Scholar] [CrossRef]
- Verma, A.K.; Tiwari, D. Higher Resolution Methods Based on Quasilinearization and Haar Wavelets on Lane—Emden Equations. Int. J. Wavel. Multiresolut. Inf. Process. 2019, 17, 1950005. [Google Scholar] [CrossRef]
- Verma, A.K.; Tiwari, D. A Note on Legendre. Hermite, Chebyshev, Laguerre and Gegenbauer Wavelets with an Application on SBVPs Arising in Real Life. arXiv 2019, arXiv:1911.02004. [Google Scholar]
- Verma, A.K.; Kumar, N.; Tiwari, D. System of Lane–Emden Equations as IVPs BVPs and Four Point BVPs & Computation with Haar Wavelets. arXiv 2019, arXiv:1912.01395. [Google Scholar]
- Rasanan, A.H.H.; Rahmati, D.; Gorgin, S.; Parand, K. A Single Layer Fractional Orthogonal Neural Network for Solving Various Types of Lane–Emden Equation. New Astron. 2020, 75, 101307. [Google Scholar] [CrossRef]
- Verma, A.; Pandit, B.; Agarwal, R. On Approximate Stationary Radial Solutions for a Class of Boundary Value Problems Arising in Epitaxial Growth Theory. J. Appl. Comput. Mech. 2019. [Google Scholar] [CrossRef]
- Verma, A.; Pandit, B.; Escudero, C. Numerical Solutions for a Class of Singular Boundary Value Problems Arising in the Theory of Epitaxial Growth. Eng. Comput. 2020. [Google Scholar] [CrossRef]
- Singh, M.; Verma, A.K. An Effective Computational Technique for a Class of Lane–Emden Equations. J. Math. Chem. 2016, 54, 231–251. [Google Scholar] [CrossRef]
- Singh, M.; Verma, A.K.; Agarwal, R.P. On an Iterative Method for a Class of 2 Point & 3 Point Nonlinear SBVPS. J. Appl. Anal. Comput. 2019, 9, 1242–1260. [Google Scholar]
- Kanth, A.S.V.R.; Aruna, K. He’s Variational Iteration Method for Treating Nonlinear Singular Boundary Value Problems. Comput. Math. Appl. 2010, 60, 821–829. [Google Scholar] [CrossRef] [Green Version]
- Verma, A.K.; Kayenat, S.; Jha, G.J. A Note on the Convergence of Fuzzy Transformed Finite Difference Methods. J. Appl. Math. Comput. 2020. [Google Scholar] [CrossRef]
- Verma, L.; Pandit, B.; Verma, K.A. Taylor Series Solution of Some Real Life Problems: ODEs and PDEs. Preprint 2020. [Google Scholar] [CrossRef]
- Shahni, J.; Singh, R. An Efficient Numerical Technique for Lane–Emden-Fowler Boundary Value Problems: Bernstein Collocation Method. Eur. Phys. J. Plus 2020, 2020, 475. [Google Scholar] [CrossRef]
- Weinmueller, E. SBVP 1.0 Package. MATLAB Central File Exchange. Available online: https://www.mathworks.com/matlabcentral/fileexchange/1464-sbvp-1-0-package (accessed on 31 March 2020).
- Cabada, A.; Cid, J.A.; Maquez-Villamarin, B. Computation of Green’s functions for Boundary Value Problems with Mathematica. Appl. Math. Comput. 2012, 219, 1919–1936. [Google Scholar] [CrossRef]
- Cabada, Alberto Green’s Functions in the Theory of Ordinary Differential Equations; Springer: Berlin, Germany, 2014; Volume XIV.
- Jain, R.K. Single Step Methods for General Second Order Singular Initial Value Problems with Spherical Symmetry. BIT 1988, 28, 718–824. [Google Scholar] [CrossRef]
- Qu, R.; Agarwal, R.P. A Collocation Method for Solving a Class of Singular Nonlinear Two-Point Boundary Value Problems. J. Comp. Appl. Math. 1997, 83, 147–163. [Google Scholar] [CrossRef] [Green Version]
- Roul, P.; Goura, V.M.K.P. B-Spline Collocation Methods and Their Convergence for a Class of Nonlinear Derivative Dependent Singular Boundary Value Problems. Appl. Math. Comput. 2019, 341, 428–450. [Google Scholar] [CrossRef]
- Singh, R.; Kumar, J. The Adomian Decomposition Method with Green’s Function for Solving Nonlinear Singular Boundary Value Problems. J. Appl. Math. Comput. 2014, 44, 397–416. [Google Scholar] [CrossRef]
- Singh, R.; Kumar, J.; Nelakanti, G. Approximate Series Solution of Singular Boundary Value Problems with Derivative Dependence Using Green’s Function Technique. Comp. Appl. Math. 2014, 33, 451–467. [Google Scholar] [CrossRef]
- Roul, P. Doubly Singular Boundary Value Problems with Derivative Dependent Source Function: A Fast-Converging Iterative Approach. Math. Meth. Appl. Sci. 2019, 42, 354–374. [Google Scholar] [CrossRef] [Green Version]
- Roul, P.; Goura, V.M.K.P.; Agarwal, R.P. A Compact Finite Difference Method for a General Class of Nonlinear Singular Boundary Value Problems with Neumann and Robin Boundary Conditions. Appl. Math. Comput. 2019, 350, 283–304. [Google Scholar] [CrossRef]
- Roul, P.; Goura, V.M.K.P.; Agarwal, R.P. A New High Order Numerical Approach for a Class of Nonlinear Derivative Dependent Singular Boundary Value Problems. Appl. Numer. Math. 2019, 145, 315–341. [Google Scholar] [CrossRef]
- Zhang, Y. Positive Solutions of Singular Sublinear Dirichlet Boundary Value Problems. SIAM J. Math. Anal. 1995, 26, 329–339. [Google Scholar] [CrossRef]
- Taliaferro, S.D. A Nonlinear Singular Boundary Value Problem. Nonlinear Anal. 1979, 3, 897–904. [Google Scholar] [CrossRef]
- Leray, J.; Schauder, J. Topologie et équations functionnelles. Ann. Ecole Norm. Sup. 1934, 3, 45–78. [Google Scholar] [CrossRef]
- Lloyd, N.G. Degree Theory; Cambridge Tracts in Mathematics: London, UK, 1978; Volume 73. [Google Scholar]
- Granas, A. Sur la méthode de Continuité de Poincaré. C. R. Acad. Sci. Paris 1976, 282, 983–985. [Google Scholar]
- Duhoux, M. Nonlinear Singular Sturm–Liouville Problems. Nonlinear Anal. 1999, 38, 897–918. [Google Scholar] [CrossRef]
- Bobisud, L.E.; O’Regan, D. Positive Solutions for a Class of Nonlinear Singular Boundary Value Problems at Resonance. J. Math. Anal. Appl. 1994, 184, 263–284. [Google Scholar] [CrossRef]
- Agarwal, R.P.; O’Regan, D. Some New Results for Singular Problems with Sign Changing Nonlinearities. J. Comp. Appl. Math. 2000, 113, 1–15. [Google Scholar] [CrossRef] [Green Version]
- Cherpion, M.; Coster, C.D.; Habets, P. A Constructive Monotone Iterative Method for Second-Order BVP in the Presence of Lower and Upper Solutions. Appl. Math. Comp. 2001, 123, 75–91. [Google Scholar] [CrossRef]
- O’Regan, D.; El-Gebeily, M.A. Existence. Upper and Lower Solutions and Quasilinearization for Singular Differential Equations. IMA J. Appl. Math. 2008, 73, 323–344. [Google Scholar] [CrossRef]
- Marcelli, C.; Papalini, F. Boundary value problems for strongly nonlinear equations under a Wintner-Nagumo growth condition. Bound. Value Probl. 2017, 2017, 15. [Google Scholar] [CrossRef] [Green Version]
- Calamai, A.; Marcelli, C.; Papalini, F. Boundary value problems for singular second order equations. Fixed Point Theory Appl. 2018, 2018, 22. [Google Scholar] [CrossRef]
- Biagi, S.; Isernia, T. On the solvability of singular boundary value problems on the real line in the critical growth case. Disc. Cont. Dyn. Syst. (A) 2020, 40, 1131–1157. [Google Scholar] [CrossRef] [Green Version]
- Pandey, R.K.; Verma, A.K. Existence-uniqueness Results for a Class of Singular Boundary Value Problems Arising in Physiology. Nonlinear Anal. Real World Appl. 2008, 9, 40–52. [Google Scholar] [CrossRef]
- Pandey, R.K.; Verma, A.K. Existence-uniqueness Results for a Class of Singular Boundary Value Problems-II. J. Math. Anal. Appl. 2008, 338, 1387–1396. [Google Scholar] [CrossRef] [Green Version]
- Pandey, R.K.; Verma, A.K. A Note on Existence-Uniqueness Results for a Class of Doubly Singular Boundary Value Problems. Nonlinear Anal. Theory Methods Appl. 2009, 71, 3477–3487. [Google Scholar] [CrossRef]
- Pandey, R.K.; Verma, A.K. Monotone Method for Singular BVP in the Presence of Upper and Lower Solutions. Appl. Math. Comput. 2010, 215, 3860–3867. [Google Scholar] [CrossRef]
- Pandey, R.K.; Verma, A.K. On Solvability of Derivative Dependent Doubly Singular Boundary Value Problems. J. Appl. Math. Comput. 2010, 33, 489–511. [Google Scholar] [CrossRef]
- Pandey, R.K.; Verma, A.K. On a Constructive Approach for Derivative-Dependent Singular Boundary Value Problems. Int. J. Differ. Equ. 2011, 2011, 261963. [Google Scholar] [CrossRef] [Green Version]
- Cabada, A. An Overview of the Lower and Upper Solutions Method with Nonlinear Boundary Value Conditions. Bound. Value Probl. 2011, 2011, 893753. [Google Scholar] [CrossRef] [Green Version]
- Dunninger, D.R.; Kurtz, J.C. A Priori Bounds and Existence of Positive Solutions for Singular Nonlinear Boundary Value Problems. SIAM J. Math. Anal. 1986, 17, 595–609. [Google Scholar] [CrossRef]
- Chawla, M.M.; Shivkumar, P.N. On the Existence of Solutions of a Class of Singular Nonlinear Two-Point Boundary Value Problems. J. Comp. Appl. Math. 1987, 19, 379–388. [Google Scholar] [CrossRef] [Green Version]
- Pandey, R.K. On a Class of Weakly Regular Singular Two Point Boundary Value Problem I. Nonlinear Anal. 1996, 27, 1–12. [Google Scholar] [CrossRef]
- Pandey, R.K. On a Class of Weakly Regular Singular Two Point Boundary Value Problem, II. J. Differ. Equ. 1996, 127, 110–123. [Google Scholar] [CrossRef] [Green Version]
- Pandey, R.K. On a Class of Regular Singular Two Point Boundary Value Problems. J. Math. Anal. Appl. 1997, 208, 388–403. [Google Scholar] [CrossRef] [Green Version]
- Titchmarsh, E.C. Eigen Function Expansion, Part I; Oxford University Press: Oxford, UK, 1962. [Google Scholar]
- O’Regan, D. Theory of Singular Boundary Value Problems; World Scientific: Singapore, 1994. [Google Scholar]
- O’Regan, D. Existence Theorems for Certain Classes of Singular Boundary Value Problems. J. Math. Anal. Appl. 1992, 168, 523–539. [Google Scholar] [CrossRef] [Green Version]
- Granas, A.; Guenther, R.B.; Lee, J.W. Existence Principles for Classical and Carathéodory Solutions for Systems of Ordinary Differential Equations; Ohio University Press: Athens, OH, USA, 1988; pp. 353–364. [Google Scholar]
- El-Gebeily, M.A.; Boumenir, A.; Elgindi, M.B.M. Existence and Uniqueness of Solutions of a Class of Two-Point Singular Nonlinear Boundary Value Problems. J. Comp. Appl. Math. 1993, 46, 345–355. [Google Scholar] [CrossRef] [Green Version]
- O’Regan, D. Existence Theory for Nonresonant Singular Boundary Value Problems. Proc. Edindburg Math. Soc. 1995, 38, 431–447. [Google Scholar] [CrossRef] [Green Version]
- O’Regan, D. Nonresonant Nonlinear Singular Problems in the Limit Circle Case. J. Math. Anal. Appl. 1996, 197, 708–725. [Google Scholar] [CrossRef] [Green Version]
- Mawhin, J.; Ward, J.R. Nonuniform Nonresonance Conditions at the First, Two Eigenvalues for Periodic Solutions of Forced Liénard and Duffing Equations. Rocky Mt. J. Math. 1982, 112, 643–654. [Google Scholar] [CrossRef]
- Mawhin, J.; Ward, J.R. Periodic Solutions of Some Forced Liénard Differential Equations at Resonance. Arch. Math. 1983, 41, 337–351. [Google Scholar] [CrossRef]
- Agarwal, R.P.; O’Regan, D. Singular Boundary Value Problems for Superlinear Second Order Ordinary and Delay Differential Equations. J. Diff. Equ. 1996, 130, 333–355. [Google Scholar] [CrossRef] [Green Version]
- Bobisud, L.E.; O’Regan, D.; Royalty, W.D. Solvability of Some Nonlinear Boundary Value Problems. Nonlinear Anal. 1988, 12, 855–869. [Google