A new reproducing kernel approach for nonlinear fractional three-point boundary value problems

01/10/2020
by   Mehmet Giyas Sakar, et al.
0

In this article, a new reproducing kernel approach is developed for obtaining numerical solution of nonlinear three-point boundary value problems with fractional order. This approach is based on reproducing kernel which is constructed by shifted Legendre polynomials. In considered problem, fractional derivatives with respect to α and β are defined in Caputo sense. This method has been applied to some examples which have exact solutions. In order to shows the robustness of the proposed method, some numerical results are given in tabulated forms.

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1 Introduction

In this paper, a new iterative reproducing kernel approach will be constructed for obtaining the numerical solution of nonlinear fractional three-point boundary value problem,

(1)

with following boundary conditions,

(2)

Here, and are sufficiently smooth functions and fractional derivatives are taken in Caputo sense. Without loss of generality, we pay regard to , and . Because, , and boundary conditions can be easily reduced to , and .

Nonlinear fractional multi-point boundary value problems appear in a different area of applied mathematics and physics 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 and references therein. Many important studies have been concerned in engineering and applied science such as dynamical systems, fluid mechanics, control theory, oil industries, heat conduction can be well-turned by fractional differential equations 8 ; 9 ; 10 . Some applications, qualitative behaviors of solution and numerical methods to find approximate solution have been investigated for differential equation with fractional order 11 ; 12 ; 13 ; 14 .

More particularly, it is not easy to directly get exact solutions to most differential equations with fractional order. Hence, numerical techniques are utilised largely. Actually, in recent times many efficient and convenient methods have been developed such as the finite difference method 15 , finite element method 16 , homotopy perturbation method 17 , Haar wavelet methods 18 , Adomian decomposition method 19 , collocation methods 20 , homotopy analysis method 21 , differential transform method 22 , variational iteration method 23 , reproducing kernel space method 24 ; 25 and so on 26 ; 27 ; 28 .

In 1908, Zaremba firstly introduced reproducing kernel concept 29 . His resarches with regard to boundary value problems which includes Dirichlet condition. Reproducing kernel method (RKM) produces a solution in convergent series form for many differential, partial and integro-differential equations. For more information, we refer to 30 ; 31 . Recently, this RKM is applied for different type of problem. For example, fractional order nonlocal boundary value problems 32 , Riccati differential equations 33 , forced Duffing equations with nonlocal boundary conditions 34 , Bratu equations with fractional order Caputo derivative 35 , time-fractional Kawahara equation 36 , two-point boundary value problem 37 , nonlinear fractional Volterra integro-differential equations 38 .

Recently, Legendre reproducing kernel method is proposed for fractional two-point boundary value problem of Bratu Type Equations 39 . The main motivation of this paper is to extend the Legendre reproducing kernel approach for solving nonlinear three-point boundary value problem with Caputo derivative.

The remainder part of the paper is prepared as follows: some fundamental definitions of fractional calculus and the theory of reproducing kernel with Legendre basis functions are given in Section 2. The structure of solution with Legendre reproducing kernel is demonstrated in Section 3. In order to show the effectiveness of the proposed method, some numerical findings are reported in Section 4. Finally, the last section contains some conclusions.

2 Preliminaries

In this section, several significant concepts, definitions, theorems, and properties are provided which will be used in this research.

Definition 2.1 Let and . Then, the order left Riemann-Liouville fractional integral operator is given as 8 ; 12 ; 13 :

here is Gamma function, and .

Definition 2.2 Let and . Then, the order left Caputo differential operator is given as 8 ; 12 ; 13 :


Definition 2.3 In order to construct polynomial type reproducing kernel, the first kind shifted Legendre polynomials are defined over the interval . For obtaining these polynomials the following iterative formula can be given:

The orthogonality requirement is

(3)

here, weighted function is taken as,

(4)

Legendre basis functions can be established so that this basis function system satisfy the homogeneous boundary conditions as:

(5)

Eq. (5) has a advantageous feature for solving boundary value problems. Therefore, these basis functions for can be defined as;

(6)

such that this system satisfy the conditions

(7)

It is worth noting that the basis functions given in Eq. (6) are complete system. For more information about orthogonal polynomials, please see 41 ; 42 ; 43 .

Definition 2.4 Let , and with its inner product be a Hilbert space of real-valued functions on . Then, the reproducing kernel of is iff

  1. .

The last condition is known as reproducing property. Especially, for any , ,

If a Hilbert space satisfies the above two conditions then is called reproducing kernel Hilbert space. Uniqueness of the reproducing kernel can be shown by use of Riesz representation theorem 40 .

Theorem 2.1 Let be an orthonormal basis of -dimensional Hilbert space , then

(8)

is reproducing kernel of 30 ; 31 .

Definition 2.5 Let polynomials space be pre-Hilbert space over with real coefficients and its degree and inner product as:

(9)

with described by Eq. (4), and the norm

(10)

With the aid of definiton of Hilbert space, for any fixed , is a subspace of and ,

Theorem 2.2 Hilbert space is a reproducing kernel space.

Proof. From Definition 2.5, it is quite apparent that functions space is a finite-dimensional. It is well known that all finite-dimensional pre-Hilbert space is a Hilbert space. Herewith, using this consequence and Theorem 2.1, is a reproducing kernel space.

For solving problem (1)-(2), it is required to describe a closed subspace of so that satisfy homogeneous boundary conditions.

Definition 2.6 Let

One can easily demonstrate that is a reproducing kernel space using Eq. (6). From Theorem 2.1, the kernel function of can be written as

(11)

Here, is complete system which is easily obtained from basis functions in Eq. (6) with the help of Gram-Schmidt orthonormalization process. Eq. (11) is very useful for implementation. In other words, and can readily re-calculated by increasing .

3 Main Results

In this section, some important results related to reproducing kernel method with shifted Legendre polynomials are presented. In the first subsection, generation of reproducing kernel which is satify three-point boundary value problems is presented. In the second subsection, representation of solution is given . Then, we will construct an iterative process for nonlinear problem in third subsection.

3.1 Generation of reproducing kernel for three-point boundary value problems

In this subsection, we shall generate a reproducing kernel Hilbert space in which every functions satisfies , and .

is defined as .

Obviously, reproducing kernel space is a closed subspace of . The reproducing kernel of can be given with the following theorem.

Theorem 3.1 The reproducing kernel of ,

(12)

Proof. Frankly, not all elements of vanish at . This shows that 0. Hence, it can be easily seen that and therefore . For , clearly, , it follows that

Namely, is of reproducing kernel of . This completes the proof.

3.2 Representation of solution in Hilbert space

In this subsection, reproducing kernel method with Legendre polyomials is established for obtaining numerical solution of three-point boundary value problem. For Eqs. (1)-(2), the approximate solution shall be constructed in . Firstly, we will define linear operator as follow,

such that

The Eqs.(1)-(2) can be stated as follows

(13)

Easily can be shown that linear operator is bounded. We will obtain the representation solution of Eq. (13) in the space. Let be the polynomial form of reproducing kernel in space.

Theorem 3.2 Let be any distinct points in open interval for Eqs. (1)-(2), then
Proof. For any fixed , put

(14)

It is quite obvious that . Therefore . Here, shows the adjoint operator of . For any fixed and , .

Theorem 3.3 Let be any distinct points in open interval for , then is complete in .

Proof. For every fixed , let

this result shows, for ,

(15)

In Eq. (15), by use of inverse operator, it is decided that . Thus, is complete in . This completes the proof.

Theorem 3.3 indicates that in Legendre reproducing kernel approach, using a finite distinct points are enough. But, in traditional reproducing kernel method need to dense sqeuence on the interval. Namely, this new approach is vary from traditional method in 28 ; 32 ; 33 ; 34 ; 35 ; 38 .

The orthonormal system of can be derived with the help of the Gram-Schmidt orthogonalization process using ,

(16)

here show the coefficients of orthogonalization.

Theorem 3.4 Suppose that is the exact solution of Eqs. (1)-(2) and shows any distinct points in open interval , in that case

(17)

Proof. Since from Theorem 3.3 can be written

On the other part, using Eq. (14) and Eq. (16), we obtain which is the precise solution of Eq. (10) in as,

The proof is completed.

Theorem 3.5 If , then for , where is a constant.

Proof. We have for any , From the expression of , it pursue that
So,

Therefore, .

Theorem 3.6 and its derivatives are respectively uniformly converge to and ().

Proof By using Theorem 3.5 for any we get

where are positive constants. Therefore, if in the meaning of the norm of as , and its derivatives are respectively uniformly converge to and its derivatives . This completes the proof.

If considered problem is linear, numerical solution can be directly get from (17). But, for nonlinear problem the following iterative procedure can be construct.

3.3 Construction of iterative procedure

In this subsection, we will use the following iterative sequence to overcome the nonlinearity of the problem, , inserting,

(18)

here, orthogonal projection operator is defined as and shows the -th iterative numerical solution of (18). Then, the following important theorem will be given for iterative procedure.

Theorem 3.7 If is distinct points in open interval , then

(19)

Proof. Since , is the complete orthonormal system in ,

This completes the proof.
Taking and define the iterative sequence

(20)

4 Numerical applications

In this section, some nonlinear three-point boundary value problems are considered to exemplify the accuracy and efficiency of proposed approach. Numerical results which is achieved by L-RKM are shown with tables.

Example 4.1 We consider the following fractional order nonlinear three-point boundary value problem with Caputo derivative:

(21)
(22)

Here, a known function such that the exact solution of this problem is .

By using proposed approach for Eqs. (21)-(22), and choosing nodal points as , the approximate solution is computed by Eq. (20). For (21)-(22), comparison of absolute errors for different , values are demonstrated in Table 1 and Table 2 and comparison of exact solution and numerical solution for and is given in Table 3.

Example 4.2 We take care of the following nonlinear three-point boundary value problem with Caputo derivative

(23)
(24)

Here, a known function such that the exact solution of this problem is .

By using proposed approach for Eqs. (23)-(24), and choosing nodal points as , the approximate solution is computed by Eq. (20). For (23)-(24), comparison of absolute errors for different , values are demonstrated in Table 4 and Table 5 and comparison of exact solution and numerical solution for and is given in Table 6.

5 Conclusion

In this research, a novel numerical approach which is called L-RKM has been proposed and successfully implemented to find the approximate solution of nonlinear three-point boundary value problems with Caputo derivative. For nonlinear problem, a new iterative process is proposed. Numerical findings show that the present approach is efficient and convenient for solving three-point boundary value problems with fractional order.

References

  • (1) Y. Lin, J. Niu, M. Cui. A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space. Applied Mathematics and Computation, 218(14) (2012) 7362-7368.
  • (2) M. Rehman, R. A. Khan, N. A. Asif. Three point boundary value problems for nonlinear fractional differential equations. Acta Mathematica Scientia, 31 (4) (2011) 1337-1346.
  • (3) F. Geng, Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method. Applied Mathematics and Computation (2009) 215(6) 2095-2102.
  • (4) C. P. Zhang, J. Niu, Y. Z. Lin. Numerical solutions for the three-point boundary value problem of nonlinear fractional differential equations. Abstract and Applied Analysis (2012) Volume 2012, Article ID 360631, 16 pages.
  • (5) S. Etemad, S. K. Ntouyas, J. Tariboon. Existence results for three-point boundary value problems for nonlinear fractional differential equations. J. Nonlinear Sci. Appl. 9 (2016) 2105-2116.
  • (6) B. Wu, X. Li. Application of reproducing kernel method to third order three-point boundary value problems. Applied Mathematics and Computation (2010) 217 (7) 3425-3428.
  • (7) B. Ahmad, M. Alghanmi, S. K. Ntouyas, A. Alsaedi. A study of fractional differential equations and inclusions involving generalized Caputo-type derivative equipped with generalized fractional integral boundary conditions. AIMS Mathematics, 4 (1) (2018) 26-42.
  • (8) I. Podlubny. Fractional differential equations. Academic Press, New York, 1999.
  • (9) V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of fractional dynamic systems. Cambridge Scientific Publishers, 2009.
  • (10) R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000
  • (11) V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, HEP, 2011.
  • (12) K. Diethelm, The analysis of fractional differential equations. Lecture notes in mathematics. Berlin Heidelberg: Springer-Verlag, 2010.
  • (13) A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, B.V: Elsevier Science, 2006.
  • (14) A. Khalouta, A. Kadem. A new numerical technique for solving Caputo time-fractional biological population equation. AIMS Mathematics, 4 (5) (2019) 1307-1319.
  • (15) C. Tadjeran, M. M. Meerschaert. A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007) 813-823.
  • (16) A. Esen, O. Tasbozan. Numerical solution of time fractional Burgers equation by cubic B-spline finite elements. Mediterr. J. Math. 13 (2016) 1325-1337.
  • (17) M. G. Sakar, F. Uludag, F. Erdogan, Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method. Appl. Math. Model. 40 (2016), 6639-6649.
  • (18) U. Saeed, M. Rehman, Haar wavelet-quasilinearization technique for fractional nonlinear differential equations. Applied Mathematics and Computation 220 (2013) 630-648.
  • (19) E. Babolian, A. R. Vahidi, A. Shoja, An efficient method for nonlinear fractional differential equations: combination of the Adomian decomposition method and spectral method, Indian J. Pure Appl. Math. (2014) 1017-1028.
  • (20) L. Pezza, F. Pitolli, A multiscale collocation method for fractional differential problems. Mathematics and Computers in Simulation 147 (2018) 210-219.
  • (21) M. G. Sakar, F. Erdogan, The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian’s decomposition method, Appl. Math. Model. 37 (20-21) (2013) 1634-1641.
  • (22)

    H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M. Ariyan, F. Tchier, Reduced differential transform method for partial differential equations within local fractional derivative operators, Advances in Mechanical Engineering, 8 (4) (2016) 1–6.

  • (23) M. G. Sakar, O. Saldır, Improving variational iteration method with auxiliary parameter for nonlinear time-fractional partial differential equations. Journal of Optimization Theory and Applications 174 (2) (2017) 530-549.
  • (24) M. Q. Xu, Y. Z. Lin, Simplified reproducing kernel method for fractional differential equations with delay, Applied Mathematics Letters 52 (2016) 156-161.
  • (25) Y. L. Wang, M. J. Du, C. L. Temuer, D. Tian, Using reproducing kernel for solving a class of time-fractional telegraph equation with initial value conditions, International Journal of Computer Mathematics 95 (8) (2018) 1609-1621.
  • (26) A. Kadem, D. Baleanu. Fractional radiative transfer equation within Chebyshev spectral approach. Computers and Mathematics with Applications 59 (2010) 1865-1873.
  • (27) S. S. E. Eldien, R. M. Hafez, A. H. Bhrawy, D. Baleanu, A. A. E.Kalaawy. New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials. Journal of Optimization Theory and Applications 174 (2017) 295-320.
  • (28) M. G. Sakar, A. Akgül, D. Baleanu, On solutions of fractional Riccati differential equations, Advances in Difference Equations (2017) 2017:39.
  • (29) S. Zaremba, Sur le calcul numérique des fonctions demandées dans le probléme de Dirichlet et le problème hydrodynamique. Bulletin International de l’Académie des Sciences de Cracovie (1908) pp. 125-195.
  • (30) M. Cui, Y. Lin. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science, New York, 2009.
  • (31) S. Saitoh, Y. Sawano. Theory of Reproducing Kernels and Applications, Springer, Singapore, 2016.
  • (32) F. Geng, M. Cui, A reproducing kernel method for solving nonlocal fractional boundary value problems. Applied Mathematics Letters 25 (5) (2012) 818-823.
  • (33) M. G. Sakar, Iterative reproducing kernel Hilbert spaces method for Riccati differential equation. Journal of Computational and Applied Mathematics. 309 (2017) 163-174.
  • (34) F. Geng, M. Cui, New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions. Journal of Computational and Applied Mathematics. 233 (2009) 165-172.
  • (35) E. Babolian, S. Javadi, E. Moradi, RKM for solving Bratu-type differential equations of fractional order. Mathematical Methods in the Applied Sciences. 39 (6) (2016) 1548-1557.
  • (36)

    O. Saldır, M. G. Sakar, F. Erdogan. Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate. Computational and Applied Mathematics, 38 (4) (2019) 98.

  • (37) M. Khaleghi, E. Babolian, S. Abbasbandy. Chebyshev reproducing kernel method: application to two-point boundary value problems. Advances in Difference Equations (2017) 26. DOI: 10.1186/s13662-017-1089-2
  • (38) W. Jiang, T. Tian. Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method. Applied Mathematical Modelling 39 (16) (2015) 4871-4876.
  • (39) M. G. Sakar, O. Saldır, A. Akgül. Numerical solution of fractional Bratu type equations with Legendre reproducing kernel method. International Journal of Applied and Computational Mathematics (2018) 4:126.
  • (40) N. Aronszajn. Theory of reproducing kernels. Trans. Am. Math. Soc. (1950), 68:337-404.
  • (41) W. Kaplan, Advanced Calculus (5E). Pearson Education 2002.
  • (42) E. D. Rainville. Special Functions. Chelsea Publishing Co., New York (1960).
  • (43) G. Szegö. Orthogonal Polynomials. American Mathematical Society Colloquium Publications, Providence, Rhode Island, 1939.

Tables

0.1 3.25E-11 6.08E-11 3.70E-12 4.45E-10 3.14E-9
0.2 5.45E-11 8.75E-11 4.93E-12 8.70E-10 5.51E-9
0.3 6.97E-11 9.00E-11 4.34E-12 1.32E-9 7.36E-9
0.4 8.17E-11 7.81E-11 2.55E-12 1.85E-9 8.96E-9
0.5 0 0 0 0 0
0.6 1.10E-10 5.14E-11 2.03E-12 3.36E-9 1.24E-8
0.7 1.34E-10 5.65E-11 3.56E-12 4.44E-9 1.49E-8
0.8 1.70E-10 8.70E-11 3.73E-12 5.80E-9 1.81E-8
0.9 2.21E-10 1.53E-11 1.89E-12 7.49E-9 2.25E-8
Table 1: Comparison absolute error of Example 4.1 for various (, )
0.1 3.78E-17 1.33E-16 2.49E-19 4.86E-15 7.48E-14
0.2 5.27E-17 1.94E-16 3.36E-19 1.13E-14 1.36E-13
0.3 5.10E-17 2.03E-16 3.80E-19 1.99E-14 1.91E-13
0.4 3.91E-17 1.81E-16 5.20E-19 3.09E-14 2.43E-13
0.5 0 0 0 0 0
0.6 1.05E-17 1.36E-16 1.56E-18 6.18E-14 3.60E-13
0.7 6.65E-18 1.57E-16 2.74E-18 8.25E-14 4.36E-13
0.8 1.81E-17 2.34E-16 4.50E-18 1.07E-14 5.30E-13
0.9 5.14E-17 3.91E-16 7.02E-18 1.36E-14 6.48E-13
Table 2: Comparison absolute error of Example 4.1 for various (, )
Exact Sol. Approximate Sol. Absolute Error
0.0 0.000000000000000000000 0.000000000000000000000 0
0.1 0.036000000000000000000 0.036000000000000000018 1.80E-20
0.2 0.048000000000000000000 0.048000000000000000044 4.40E-20
0.3 0.042000000000000000000 0.042000000000000000061 6.10E-20
0.4 0.024000000000000000000 0.024000000000000000071 7.10E-20
0.5 0.000000000000000000000 0.000000000000000000000 0
0.6 -0.024000000000000000000 -0.023999999999999999899 1.01E-19
0.7 -0.042000000000000000000 -0.041999999999999999819 1.81E-19
0.8 -0.048000000000000000000 -0.047999999999999999694 3.06E-19
0.9 -0.036000000000000000000 -0.035999999999999999491 5.09E-19
1.0 0.000000000000000000000 0.000000000000000000000 0
Table 3: Numerical results of Example 4.1 for , values (, )
0.1 5.00E-11 4.11E-15 2.78E-13 6.10E-12 1.40E-12
0.2 8.39E-11 3.45E-15 1.87E-13 1.17E-11 2.42E-11
0.3 1.06E-10 1.24E-15 1.87E-13 1.45E-11 7.08E-11
0.4 1.23E-10 9.21E-15 7.60E-13 1.20E-11 1.32E-10
0.5 1.39E-10 1.97E-14 1.44E-12 1.75E-12 2.02E-10
0.6 0 0 0 0 0
0.7 1.88E-10 4.52E-14 2.81E-12 5.17E-11 3.45E-10
0.8 2.31E-10 5.88E-14 3.32E-12 9.98E-11 4.05E-10
0.9 2.93E-10 7.18E-14 3.60E-12 1.65E-10 4.49E-10
Table 4: Comparison absolute error of Example 4.2 for various (, )
0.1 3.49E-13 1.93E-18 4.48E-16 2.68E-14 4.75E-15
0.2 5.87E-13 8.32E-19 2.99E-16 5.58E-14 7.68E-14
0.3 7.47E-13 2.90E-18 3.06E-16 7.18E-14 2.25E-13
0.4 8.65E-13 8.81E-18 1.23E-15 6.00E-14 4.21E-13
0.5 9.76E-13 1.65E-17 2.34E-15 5.54E-15 6.45E-13
0.6 0 0 0 0 0
0.7 1.31E-12 3.55E-17 4.55E-15 2.91E-13 1.09E-12
0.8 1.61E-12 4.60E-17 5.37E-15 5.64E-13 1.29E-12
0.9 2.05E-12 5.65E-17 5.83E-15 9.39E-13 1.43E-12
Table 5: Comparison absolute error of Example 4.2 for various (, )
Exact Sol. Approximate Sol. Absolute Error
0.0 0.000000000000000000000 0.000000000000000000000 0
0.1 0.045000000000000000000 0.045000000000480782793 4.80E-13
0.2 0.064000000000000000000 0.064000000000580045412 5.80E-13
0.3 0.063000000000000000000 0.063000000000488602930 4.88E-13
0.4 0.048000000000000000000 0.048000000000398645450 3.98E-13
0.5 0.025000000000000000000 0.025000000000468872800 4.68E-13
0.6 0.000000000000000000000 0.000000000000000000000 0
0.7 -0.021000000000000000000 -0.020999999998651962040 1.34E-12
0.8 -0.032000000000000000000 -0.031999999998006864020 1.99E-12
0.9 -0.027000000000000000000 -0.026999999997598991320 2.40E-12
1.0 0.000000000000000000000 0.000000000000000000000 0
Table 6: Numerical results of Example 4.2 for , values (