Numerical solution of integro-differential equations of high-order Fredholm by the simplified reproducing kernel method. (English) Zbl 07474797

Summary: The key of the reproducing kernel method (RKM) to solve the initial boundary value problem is to construct the reproducing kernel meeting the homogenous initial boundary conditions of the considered problems. The usual method is that the initial boundary conditions must be homogeneous and put them into space. Another common method is to put homogeneous or non-homogeneous conditions directly into the operator. In addition, we give a new numerical method of RKM for dealing with initial boundary value problems, homogeneous conditions are put into space, and for nonhomogeneous conditions, we put them into operators. The focus of this paper is to further verify the reliability and accuracy of the latter two methods. Through solving three numerical examples of integral-differential equations and comparing with other methods, we find that the two methods are useful.


34K05 General theory of functional-differential equations
34K30 Functional-differential equations in abstract spaces
47G20 Integro-differential operators
45Axx Linear integral equations
45Bxx Fredholm integral equations


Full Text: DOI


[1] Abu Arqub, * O., The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations, Math. Methods. Appl. Sci., 39, 4549-4562 (2016) · Zbl 1355.65106
[2] Abu Arqub, * O., Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions, Comput. Math. Appl., 73, 1243-1261 (2017) · Zbl 1412.65174
[3] Akram, G.; Ur Rehman, H., Numerical solution of eighth order boundary value problems in reproducing kernel space, Numer. Algorithms, 62, 527-540 (2013) · Zbl 1281.65101
[4] Babaaghaie, A.; Maleknejad, K., Numerical solution of integro-differential equations of high order by wavelet basis, its algorithm and convergence analysis, Appl. Math. Comput., 325, 125-133 (2017) · Zbl 1367.65195
[5] Beyrami, H.; Lotfi, T., Stability and error analysis of the reproducing kernel Hilbert space method for the solution of weakly singular Volterra integral equation on graded mesh, Appl. Numer. Math., 120, 197-214 (2017) · Zbl 1370.65075
[6] Chen, Z.; Jiang, W., An approximate solution for a mixed linear Volterra-Fredholm integral equation[J], Appl. Math. Lett., 25, 8, 1131-1134 (2012) · Zbl 1246.65249
[7] Du, M.; Wang, Y.; Chaolu, T., Reproducing kernel method for numerical simulation of downhole temperature distribution, Appl. Math. Comput., 297, 19-30 (2017) · Zbl 1411.80006
[8] Gelmi, C. A.; Jorquera, H., A general purpose solver for nth-order integro-differential equations, Appl. Comput. Phys. Commun., 185, 392-397 (2014) · Zbl 1344.45001
[9] Geng, F., A numerical algorithm for nonlinear multi-point boundary value problems, J. Comput. Appl. Math., 236, 1789-1794 (2012) · Zbl 1241.65067
[10] Jiang, W.; Chen, Z., A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation, Numer. Methods. Partial. Differ. Equ., 30, 1, 289-300 (2014) · Zbl 1285.65065
[11] Jiang, W.; Liu, N., A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model, Appl. Numer. Math., 119, 18-32 (2017) · Zbl 1432.65155
[12] Jiang, W.; Tian, T., Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method, Appl. Math. Model., 39, 16, 4871-4876 (2015) · Zbl 1443.65113
[13] Li, Z.; Wang, Y.; Tan, F., Solving a class of linear nonlocal boundary value problems using the reproducing kernel, Appl. Math. Comput., 265, 1098-1105 (2015) · Zbl 1410.65270
[14] Lin, Y.; Lin, J., A numerical algorithm for solving a class of linear nonlocal boundary value problems, Appl. Math. Lett., 23, 997-1002 (2010) · Zbl 1201.65130
[15] Momani, S.; Noor, M. A., Numerical comparison of methods for solving a special fourth-order boundary value problem, Appl. Math. Comput., 191, 218-224 (2007) · Zbl 1193.65135
[16] Wang, W.; Han, B., Inverse heat problem of determining time-dependent source parameter in reproducing kernel space, Nonlinear Anal. Real World Appl., 14, 875-887 (2013) · Zbl 1256.35206
[17] Wang, Y.; Chaolu, T.; Pang, J., New algorithm for second-order boundary value problems of integro-differential equation, J. Comput. Appl. Math., 229, 1-6 (2009) · Zbl 1361.34079
[18] Wang, Y.; Chaolu, T.; Chen, Z., Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems, Int. J. Comput. Math., 87, 367-380 (2010) · Zbl 1185.65134
[19] Wu, B.; Li, X., A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernei space, Appl. Math. Lett., 24, 156-159 (2011) · Zbl 1215.34014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.