ORIGINAL_ARTICLE
Derivations in semiprime rings and Banach algebras
Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,y\in R$ where $0\neq b\in R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. We also obtain some related result in case $R$ is a non-commutative Banach algebra and d continuous or spectrally bounded.
http://jlta.iauctb.ac.ir/article_510014_c0ecf3a2d537d24ca2b30bd50175a60b.pdf
2013-09-01T11:23:20
2018-04-23T11:23:20
129
135
prime ring
semiprime ring
Derivation
Utumi quotient ring
Banach algebra
Sh.
Sahebi
true
1
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
P. O. Box 14168-94351, Tehran, Iran
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
P. O. Box 14168-94351, Tehran, Iran
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
P. O. Box 14168-94351, Tehran, Iran
AUTHOR
V.
Rahmani
venosrahmani@yahoo.com
true
2
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
P. O. Box 14168-94351, Tehran, Iran
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
P. O. Box 14168-94351, Tehran, Iran
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
P. O. Box 14168-94351, Tehran, Iran
LEAD_AUTHOR
[1] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math., Vol. 196, New York, 1996.
1
[2] H. E. Bell, W. S. Martindale III, Centeralizing mappings of semiprime rings, Canadian Mathematical Bulletin, 30 (1) (1987), pp. 92-101.
2
[3] M. Bresar, M. Mathieu, Derivations mapping into the radical III, J. Funct. Anal., 133(1), (1995), pp. 21-29.
3
[4] C. L. Chung, GPIs having coecients in Utumi quotient rings, proc.Amer.Math.soc., 103 (1988), pp. 723-728.
4
[5] J. S. Ericson, W. S. Martindale III, J. M. Osborn, Prime nonassociative algebras, pascic J. math., 60 (1975), pp. 49-63.
5
[6] B. E. Jacobson, A. M. Sinclair, Continuity of derivations and problem of kaplansky, Amer. J. Math., 90 (1968), pp. 1067-1073.
6
[7] V. K. Kharchenko, Dierential identity of prime rings, Algebra and Logic., 17 (1978), pp. 155-168.
7
[8] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc., 118 (1993), pp. 731-734.
8
[9] T. K. Lee, Semiprime rings with dierential identities, Bull. Inst. Math. Acad. Sinica, 20 (1) (1992), pp. 27-38.
9
[10] W. S. Martindale III, Prime rings satistying a generalized polynomial identity, J. Algebra., 12 (1969), pp. 576-584.
10
[11] M. Mathieu, G. J. Murphy, Derivations mapping into the radical, Arch. Math., 57 (5) (1991), pp. 469-474.
11
[12] M. Mathieu, V. Runde, Derivations mapping into the radical II, Bull. london Math. soc., 24 (5)(1992), pp. 485-487.
12
[13] E. C. Posner, Derivation in prime rings, Proc. Amer. Math. Soc., 8 (1957), pp. 1093-1100.
13
[14] K. H. Park, On derivations in non-commutative semiprime rings and Banach algebras, Bull. Korean Math. Soc., 42 (4)(2005), pp. 671-678.
14
[15] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc., 20 (1969), pp. 166-170.
15
[16] I. M. Singer, J. Werner, Derivations on commutative normed algebras, Math. Ann., 129 (1955), pp. 260-264.
16
[17] M. P. Thomas, The image of a derivation is contained in the radical, math. Ann., 128 (2) (1988), pp. 435-460.
17
ORIGINAL_ARTICLE
Some results of semilocally simply connected property
If we consider some special conditions, we can assume fundamental group of a topological space as a new topological space. In this paper, we will present a number of theorems in topological fundamental group related to semilocally simply connected property for a topological space.
http://jlta.iauctb.ac.ir/article_510015_5062cb9c7c690e38b9dca1a0455c2dff.pdf
2013-09-01T11:23:20
2018-04-23T11:23:20
137
143
Semilocally simply connected
topological fundamental group
discrete space
A.
Etemad Dehkordya
ae110mat@cc.iut.ac.ir
true
1
Department of Mathematical sciences, Isfahan University of Technology, Isfahan, Iran
Department of Mathematical sciences, Isfahan University of Technology, Isfahan, Iran
Department of Mathematical sciences, Isfahan University of Technology, Isfahan, Iran
LEAD_AUTHOR
M.
Malek Mohamad
true
2
Department of Mathematical sciences, Isfahan University of Technology, Isfahan, Iran
Department of Mathematical sciences, Isfahan University of Technology, Isfahan, Iran
Department of Mathematical sciences, Isfahan University of Technology, Isfahan, Iran
AUTHOR
[1] D.K. Biss, The topological fundamental group and generalization covering space, Topology and its application 124 (2002), 355-371.
1
[2] W.A. Bogley and A.J. Sieradski, Wieghted combinatorial group theory and wiled metric complexces A.C. Kim(Ed) Groups-Korea'98 de Gruyter, Pusan(2005), 53-80.
2
[3] J.S. Calcut and J.D. McCarthy, Discreteness and homogeneity of the topological fundamental group, Topology Proc. 34(2009) 339-349.
3
[4] P. Fabel, Metric spaces with discrete topological fundamental group, Topology Appl., 154(2007), 635-638.
4
[5] R. Fritsch and R.A. Piccinini, Cellular Structures in Topology, Cambridge University press(1990).
5
[6] A. Hatcher, Algebraic Topology, Cambridge University press(2002).
6
[7] E.H. Spanier, Algebraic Topology, McGraw-Hill, New york (1966).
7
ORIGINAL_ARTICLE
A generalization of Bertrand's test
One of the most practical routine tests for convergence of a positive series makes use of the ratio test. If this test fails, we can use Rabbe's test. When Rabbe's test fails the next sharper criteria which may sometimes be used is the Bertrand's test. If this test fails, we can use a generalization of Bertrand's test and such tests can be continued innitely. For simplicity, we call ratio test, Rabbe's test, Bertrand's test as the Bertrand's test of order 0, 1 and 2, respectively. In this paper, we generalize Bertrand's test in order k for natural k > 2. It is also shown that for any k, there exists a series such that the Bertrand's test of order fails, but such test of order k + 1 is useful, furthermore we show that there exists a series such that for any k, Bertrand's test of order k fails. The only prerequisite for reading this article is a standard knowledge of advanced calculus.
http://jlta.iauctb.ac.ir/article_510016_819b520d1a7e6d00b75db94497f5b7f8.pdf
2013-09-01T11:23:20
2018-04-23T11:23:20
145
151
Bertrand's test
Convergence test
Series test
A. A.
Tabatabai Adnani
a.t.adnani@gmail.com
true
1
Islamic Azad University, Central Tehran Branch, Tehran, Iran
Islamic Azad University, Central Tehran Branch, Tehran, Iran
Islamic Azad University, Central Tehran Branch, Tehran, Iran
LEAD_AUTHOR
A.
Reza
true
2
Islamic Azad University, Central Tehran Branch, Tehran, Iran
Islamic Azad University, Central Tehran Branch, Tehran, Iran
Islamic Azad University, Central Tehran Branch, Tehran, Iran
AUTHOR
M.
Morovati
true
3
School of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran
School of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran
School of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran
AUTHOR
[1] J. M. H. Olmsted, Advanced Calculus, Prentice Hall. (1961).
1
[2] J. Wen, T. Han, C. Gao, Convergence tests on constant Dirichlet series, Computers and Mathematics with Applications. 62 (2011) 3472-3489.
2
[3] J. S. Chen, C. W. Liu, C. M. Liao, Two-dimensional Laplace-transformed power series solution for solute transport in a radially convergent flow field, Advances in Water Resources. 26 (2003) 1113-1124.
3
[4] P. Wonzy, Efficient algorithm for summation of some slowly convergent series, Applied Numerical Mathematics. 60 (2010) 1442-1453.
4
[5] E. Liflyand, S. Tikhonov, M. Zeltser, Extending tests for convergence of number series, Journal of Mathematical Analysis and Applications. 377 (2011) 194-206.
5
[6] A. Bartoszewicz, S. Glab, T. Poreda, On algebrability of nonabsolutely convergent series, Linear Algebra and its Applications. 435 (2011) 1025-1028.
6
[7] F. Moricz, A Quantitative Version of the Dirichlet-Jordan Test for Double Fourier Series, Journal of Approximation Theory. 71(1992) 344-358.
7
ORIGINAL_ARTICLE
On the Finite Groupoid G(n)
In this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid G(n). Also we show that G(n) contains commuting regular subsemigroup and give a necessary and sufficient condition for the groupoid G(n) to be commuting regular.
http://jlta.iauctb.ac.ir/article_510017_fb1252dd8598780a6b0d040a37bee02a.pdf
2013-09-01T11:23:20
2018-04-23T11:23:20
153
159
Commuting regular semigroup
semigroup
groupoid
M.
Azadi
meh.azadi@iauctb.ac.ir
true
1
Department of Mathematics, Islamic Azad University, Centeral Tehran Branch, Tehran, Iran
Department of Mathematics, Islamic Azad University, Centeral Tehran Branch, Tehran, Iran
Department of Mathematics, Islamic Azad University, Centeral Tehran Branch, Tehran, Iran
LEAD_AUTHOR
H.
Amadi
true
2
Department of Mathematics, Islamic Azad University, Centeral Tehran Branch, Tehran, Iran
Department of Mathematics, Islamic Azad University, Centeral Tehran Branch, Tehran, Iran
Department of Mathematics, Islamic Azad University, Centeral Tehran Branch, Tehran, Iran
AUTHOR
[1] H. Doostie, L. Pourfaraj, On the minimal ideals of commuting regularrings and semigroups, Internat. J. Appl. Math, 19, NO.2(2006), 201-216.
1
[2] H. Doostie, L. Pourfaraj, Finite rings and loop rings involving the commutingregular elements, International Mathematical Forum, Vol. 2, NO.52(2007), 2579-2586.
2
[3] J. M. Howie, Fundamental of Semigroup Theory, Clarendon Prees. Oxford,New York, 1995.
3
[4] L. Pourfaraj, On the nite groupoid, International Mathematical Forum Vol. 7,2012, no. 23, 1105-1114.
4
[5] W. B. Vasantha Kandasmy, Groupoids and Smarandachegroupoids, Publishedby the American Reserch Press, 2002, math. GM.
5
ORIGINAL_ARTICLE
OD-characterization of $S_4(4)$ and its group of automorphisms
Let $G$ be a finite group and $\pi(G)$ be the set of all prime divisors of $|G|$. The prime graph of $G$ is a simple graph $\Gamma(G)$ with vertex set $\pi(G)$ and two distinct vertices $p$ and $q$ in $\pi(G)$ are adjacent by an edge if an only if $G$ has an element of order $pq$. In this case, we write $p\sim q$. Let $|G= p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, where $p_1
http://jlta.iauctb.ac.ir/article_510018_04b780588dcc071ae9418ffe0226bd47.pdf
2013-09-01T11:23:20
2018-04-23T11:23:20
161
166
Finite simple group
OD-characterization
group of lie type
P.
Nosratpour
p.nosratpour@ilam-iau.ac.ir
true
1
Department of mathematics, Ilam Branch, Islamic Azad university, Ilam, Iran
Department of mathematics, Ilam Branch, Islamic Azad university, Ilam, Iran
Department of mathematics, Ilam Branch, Islamic Azad university, Ilam, Iran
LEAD_AUTHOR
[1] M. Akbari and A. R. Moghaddamfar, Simple groups which are 2-fold OD-characterizable, Bulletin of the Malaysian Mathematical Sciences Society, 35(1), 65-77(2012).
1
[2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford 1985.
2
160 P. Nosratpour / J. Linear. Topological. Algebra. 02(03) (2013) 155-160.
3
[3] G. Y. Chen, On structure of Frobenius and 2-Frobenius group, Jornal of Southwest China Normal University, 20(5), 485-487(1995).(in Chinese)
4
[4] M. R. Darafsheh, A. R. Moghaddamfar, and A. R. Zokayi, A characterization of nite simple groups by degrees of vertices of their prime graphs, Algebra Colloquium, 12(3), 431-442(2005).
5
[5] D. Gorenstein, Finite Groups, New York, Harpar and Row, (1980).
6
[6] B. Huppert, Endlichen Gruppen I, Springer-Verlag,(1988).
7
[7] D. S. Passman, Permutation Groups, New York, Benjamin Inc., (1968).
8
[8] J. S. Williams, Prime graph components of nite groups, J. Alg. 69, No.2,487-513(1981).
9
[9] A. V. Zavarnitsine, Finite simple groups with narrow prime spectrum, Siberian Electronic Math. Reports. 6, 1-12(2009).
10
ORIGINAL_ARTICLE
On the nonnegative inverse eigenvalue problem of traditional matrices
In this paper, at first for a given set of real or complex numbers $\sigma$ with nonnegative summation, we introduce some special conditions that with them there is no nonnegative tridiagonal matrix in which $\sigma$ is its spectrum. In continue we present some conditions for existence such nonnegative tridiagonal matrices.
http://jlta.iauctb.ac.ir/article_510019_85f53f76519a196781fdc2d15d2801fc.pdf
2013-09-01T11:23:20
2018-04-23T11:23:20
167
174
Inverse eigenvalue problem
Tridiagonal matrix
Nonnegative matrix
A. M.
Nazari
a-nazari@araku.ac.ir
true
1
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
LEAD_AUTHOR
S.
Kamali Maher
true
2
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
AUTHOR
[1] T. J. Laey, Helena. Smigoc, On a Classic Example in the Nonnegative Inverse Eigenvalue Problem, vol. 17, ELA, July 2008, pp. 333-342.
1
[2] R. Lowey, D. London, A note on an inverse problem for nonnegative matrices, Linear and Multilinear Algebra 6 (1978) 83-90.
2
[3] Helena Smigoc, The inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl. 393 (2004) 365-374.
3
[4] T. J. Laey, E. Meehan, A characterization of trace zero nonnegative 55matrices, Linear Algebra Appl. 302-303 (1999) 295-302.
4
[5] A. M. Nazari, F. Sherafat, On the inverse eigenvalue problem for nonnegative matrices of order two to five, Linear Algebra Appl. 436 (2012) 1771-1790.
5
[6] C. R. Johnson, Rowstochastic matrices similar to doubly stochasticmatrices, Linear and MultilinearAlgebra 10 (2) (1981) 113-130.
6
[7] M. T. Chu, G. H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms and Applications, Oxford University Press, New York, 2005.
7
[8] H. Hochstadt, On the construction of a Jacobi matrix from mixed given data, Linear Algebra Appl. 28 (1979) 113-115.
8
[9] H. Pickmann, R. L. Soto, J. Egana, M. Salas, An inverse eigenvalue problem for symmetrical tridiagonal matrices, Computers and Mathematics with Applications 54 (2007) 699-708.
9
ORIGINAL_ARTICLE
Some properties of band matrix and its application to the numerical solution one-dimensional Bratu's problem
A Class of new methods based on a septic non-polynomial spline function for the numerical solution one-dimensional Bratu's problem are presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Associated boundary formulas are developed. Convergence analysis of these methods is discussed. Numerical results are given to illustrate the efficiency of methods.
http://jlta.iauctb.ac.ir/article_510020_df91852bd2fbf4c39cb43b01bc47efbb.pdf
2013-09-01T11:23:20
2018-04-23T11:23:20
175
189
Two-point boundary value problem
Non-polynomial spline
Convergence analysis
Bratu's problem
R.
Jalilian
rezajalilian@iust.ac.ir
true
1
Department of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box 6714967346 Iran
Department of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box 6714967346 Iran
Department of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box 6714967346 Iran
LEAD_AUTHOR
Y.
Jalilian
true
2
Department of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box 6714967346 Iran
Department of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box 6714967346 Iran
Department of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box 6714967346 Iran
AUTHOR
H.
Jalilian
true
3
School of Mathematics, Iran University of Science and Technology Narmak, Tehran 16844, Iran
School of Mathematics, Iran University of Science and Technology Narmak, Tehran 16844, Iran
School of Mathematics, Iran University of Science and Technology Narmak, Tehran 16844, Iran
AUTHOR
[1] G. Akram, and S. S. Siddiqi, End conditions for interpolatory septic splines, International Journal of Computer Mathematics, Vol. 82, No. 12 (2005), pp. 1525-1540.
1
[2] G. Akram, and S. S. Siddiqi, Solution of sixth order boundary value problems using non-polynomial spline technique, Appl. Math. Comput. 181( 2006), pp. 708-720.
2
[3] Y.A.S. Aregbesola, Numerical solution of Bratu problem using the method of weighted residual, Electron. J. South. Afr. Math. Sci. 3(1)(2003), pp. 1-7.
3
[4] J. P. Boyd, Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Appl. Math. Comput. 142(2003), pp. 189-200.
4
[5] A. Boutayeb, and E. H. Twizell, Numerical methods for the solution of special sixth-order boundary-value problems, Intern. J. Computer Math. 45(1992), pp. 207-223.
5
[6] A. Boutayeb, and E. H. Twizell, Finite-dierence methods for the solution of special eighth-order boundary-value problems, International Journal of Computer Mathematics, Volume 48(1993 ), pp. 63-75.
6
[7] R., Buckmire, Application of a Mickens nite-dierence scheme to the cylindrical Bratu-Gelfand problem, Numer. Methods Partial Dieren. Eqns 20(3)(2004), pp. 327-337.
7
[8] H. Caglar, N. Caglar, and M. Ozer, Antonios Valaristos and Antonios N. Anagnostopoulos, B-spline method for solving Bratus problem, International Journal of Computer Mathematics, 87(2010), pp. 1885-1891.
8
[9] E. Deeba, S. A. Khuri, and S. Xie, An algorithm for solving boundary value problems, J. Comput. Phys. 159(2000), pp. 125-138.
9
[10] D. A. Frank-Kamenetski, Diusion and Heat Exchange in Chemical Kinetics, Princeton University Press, Princeton, NJ, 1955.
10
[11] P. Henrici, Discrete Variable Methods in Ordinary Dierential Equations, Wiley, New York, 1961.
11
[12] I.H.A.H. Hassan, and V. S. Erturk, Applying dierential transformation method to the one-dimensional planar Bratu problem, Int. J. Contemp. Math. Sci. 2(2007), pp. 1493-1504.
12
[13] J.H. He, Variational Approach to the Bratu's problem, Journal of Physics: Conference Series 96(2008), pp. 012-087.
13
[14] J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B 20(10)(2006), pp. 1141-1199.
14
[15] R. Jalilian, Non-polynomial spline method for solving Bratus problem, Computer Physics Communications, 181(2010),pp. 1868-1872.
15
[16] R. Jalilian, and J. Rashidinia, Convergence analysis of nonic-spline solutions for special nonlinear sixth-order boundary value problems, Commun Nonlinear Sci Numer Simulat, 15(2010), pp. 3805-3813.
16
[17] J. Jacobsen, and K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differen. Eqns. 184 (2002), pp. 283-298.
17
[18] S. A. Khuri, A new approach to Bratus problem, Appl. Math. Comput. 147(2004), pp. 131-136.
18
[19] S. Li, and S. J. Liao, An analytic approach to solve multiple solutions of a strongly nonlinear problem, Appl. Math. Comput. 169(2005), pp. 854-865.
19
[20] S. Liao, and Y. Tan, A general approach to obtain series solutions of nonlinear dierential equations, Stud. Appl. Math. 119(2007), pp. 297-354.
20
[21] J. S. McGough, Numerical continuation and the Gelfand problem, Appl. Math. Comput. 89(1998), pp. 225-239.
21
[22] A. S. Mounim, and B. M. de Dormale, From the tting techniques to accurate schemes for the Liouville-Bratu-Gelfand problem, Numer. Methods Partial Dieren.Volume 22, Issue 4(2006), pp. 761-775.
22
[23] A. Mohsen, L.F. Sedeek, and S.A. Mohamed, New smoother to enhance multigrid-based methods for Bratu problem, Applied Mathematics and Computation 204(2008), pp. 325-339.
23
[24] j. Rashidinia, and R. Jalilian, Non-polynomial spline for solution of boundary-value problems in plate defection theory, International Journal of Computer Mathematics, 84(2007), pp. 1483-1494.
24
[25] J. Rashidinia, R. Jalilian, and R. Mohammadi, Non-polynomial spline methods for the solution of a system of obstacle problems, Appl. Math. Comput. 188(2007), pp. 1984-1990.
25
[26] M. A. Ramadan, I. F. Lashien, and W. K. Zahra, A class of methods based on a septic non-polynomial spline function for the solution of sixth-order two-point boundary value problems, International Journal of Computer Mathematics Vol. 85, No. 5(2008) 759-770.
26
[27] M. Ramadan, I. Lashien, and W. Zahra, Quintic non-polynomial spline solutions for fourth order boundary value problem, Commun Nonlinear Sci Numer Simulat, 14(2009), pp. 1105-1114.
27
[28] S. S. Siddiqi, and G. Akram, Septic spline solutions of sixth-order boundary value problems, Journal of Computational and Applied Mathematics 215(2008), pp. 288-301.
28
[29] M. I. Syam, and A. Hamdan, An ecient method for solving Bratu equations, Appl. Math. Comput. 176(2006), pp. 704-713.
29
[30] E. H. Twizell, and A. Boutayeb, Numerical methods for the solution of special and general sixth-order boundary-value problems with applications to Bnard layer eigenvalue problems, Proc. R. Soc. Lond. A, 431(1990), pp. 433-450.
30
[31] I. A. Tirmizi, and E. H. Twizell, Higher-Order Finite-Dierence Methods for Nonlinear Second-Order Two-Point Boundary-Value Problems, Applied Mathematics Letters 15(2002), pp. 897-90.
31
[32] R. A. Usmani, and S. A. Wasrt, Quintic spline solutions of boundary value problems, Comput. Math. with Appl. 6(1980), pp. 197-203.
32
[33] R. A. Usmani, and M. Sakai, A connection between quartic spline and Numerov solution of a boundary value problem, Int. J. Comput. Math. 26(1989), pp. 263-273.
33
[34] S. Ul Islam, I. A. Tirmizi, F. Haq, and S. K. Taseer, Family of numerical methods based on non-polynomial splines for solution of contact problems, Commun Nonlinear Sci Numer Simulat, 13(2008), pp. 1448-1460.
34
[35] M. Van Daele, G. Vanden berghe, and H. A. De Meyer, Smooth approximation for the solution of a fourth-order boundary value problem based on non-polynomial splines, J. Comput. Appl. Math. Vol. 51(1994), pp. 383-394.
35
[36] A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput. 166(2005), pp. 652-663.
36