zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Solving nonlinear mixed Volterra-Fredholm integral equations with two dimensional block-pulse functions using direct method. (English) Zbl 1222.65149
Summary: Few numerical methods such as projection methods, time collocation method, trapezoidal Nystrom method, Adomian decomposition method and some else are used for mixed Volterra-Fredholm integral equations. The main purpose of this paper is to use the piecewise constant two-dimensional block-pulse functions (2D-BPFs) and their operational matrices for solving mixed nonlinear Volterra-Fredholm integral equations (VFIEs) of the first kind. This method leads to a system of linear equations by expanding unknown functions as 2D-BPFs with unknown coefficients. The properties of 2D-BPFs are then utilized to evaluate the unknown coefficients. The error analysis and rate of convergence are given. Finally, some numerical examples show the implementation and accuracy of this method.
MSC:
65R20Integral equations (numerical methods)
45B05Fredholm integral equations
45D05Volterra integral equations
45G10Nonsingular nonlinear integral equations
References:
[1]Brunner, H.: Collocation methods for Volterra integral and related functional equations, (2004)
[2]Diekman, O.: Thresholds and traveling waves for the geographical spread of infection, J math biol 6, 109-130 (1978) · Zbl 0415.92020 · doi:10.1007/BF02450783
[3]Pachpatte, B. G.: On mixed Volterra – Fredholm type integral equation, Indian J pure appl math 17, 488-496 (1986) · Zbl 0597.45012
[4]Hacia, L.: On approximate solution for integral equations of mixed type, ZAMM Z angew math mech 76, 415-416 (1996) · Zbl 0900.65389
[5]Kauthen, P. G.: Continuous time collocation methods for Volterra – Fredholm integral equations, Numer math 56, 409-424 (1989) · Zbl 0662.65116 · doi:10.1007/BF01396646
[6]Hacia, L.: On approximate solving of the Fourier problems, Demon math 12, 913-922 (1979) · Zbl 0434.65110
[7]Thieme, H. R.: A model for the spatio spread of an epidemic, J math biol 4, 337-351 (1977) · Zbl 0373.92031 · doi:10.1007/BF00275082
[8]Guoqiang, H.; Liqing, Z.: Asymptotic expantion for the trapezoidal Nyström method of linear Volterra – Fredholm equations, J comput appl math 51, No. 3, 339-348 (1994) · Zbl 0822.65123 · doi:10.1016/0377-0427(92)00013-Y
[9]Brunner, H.: On the numerical solution of nonlinear Volterra – Fredholm integral equation by collocation methods, SIAM J numer anal 27, No. 4, 987-1000 (1990) · Zbl 0702.65104 · doi:10.1137/0727057
[10]Guoqiang, H.: Asymptotic error expantion for the Nyström method for a nonlinear Volterra – Fredholm integral equations, J comput appl math 59, 49-59 (1995) · Zbl 0834.65137 · doi:10.1016/0377-0427(94)00021-R
[11]Maleknejad, K.; Hadizadeh, M.: A new computational method for Volterra – Fredholm integral equations, J comput math appl 37, 1-8 (1999) · Zbl 0940.65151 · doi:10.1016/S0898-1221(99)00107-8
[12]Adomian, G.: Solving frontire problems of physics-the decomposition method, (1994)
[13]Adomian, G.: A review of the decomposition method and some recent results for nonlinear equations, Math comput model 13, No. 7, 17-43 (1990) · Zbl 0713.65051 · doi:10.1016/0895-7177(90)90125-7
[14]Adomian, G.; Rach, R.: Noise terms in decomposition series solution, Comput math appl 24, No. 11, 61-64 (1992) · Zbl 0777.35018 · doi:10.1016/0898-1221(92)90031-C
[15]Cherruault, Y.; Saccomandi, G.; Some, B.: New results for covergence of Adomian’s method applied to integral equations, Math comput model 16, No. 2, 85 (1992) · Zbl 0756.65083 · doi:10.1016/0895-7177(92)90009-A
[16]Yee; Eugene: Application of the decomposition method to the solution of the reaction-convection-diffusion equation, Appl math comput 56, 1 (1993) · Zbl 0773.76055 · doi:10.1016/0096-3003(93)90075-P
[17]Wazwaz, A. M.: A reliable treatment for mixed Volterra – Fredholm integral equations, Appl math comput 127, 405-414 (2002) · Zbl 1023.65142 · doi:10.1016/S0096-3003(01)00020-0
[18]Cardone, A.; Messina, E.; Russo, E.: A fast iterative method for discretized Volterra – Fredholm integral equations, J comput appl math 189, 568-579 (2006) · Zbl 1092.65118 · doi:10.1016/j.cam.2005.05.018
[19]Maleknejad, K.; Yami, M. R. Fadaei: A computational method for system of Volterra – Fredholm integral equations, Appl math comput 183, 589-595 (2006) · Zbl 1107.65352 · doi:10.1016/j.amc.2006.05.105
[20]Jiang, Z. H.; Schaufelberger, W.: Block pulse functions and their applications in control systems, (1992)
[21]Beauchamp, K. G.: Applications of Walsh and related functions with an introduction to sequency theory, (1984)
[22]Deb, A.; Sarkar, G.; Sen, S. K.: Block pulse functions, the most fundamental of all piecewise constant basis functions, Int J syst sci 25, No. 2, 351-363 (1994) · Zbl 0834.42016 · doi:10.1080/00207729408928964
[23]Maleknejad, K.; Shahrezaee, M.; Khatami, H.: Numerical solution of integral equations system of the second kind by block-pulse functions, Appl math comput 166, 15-24 (2005) · Zbl 1073.65149 · doi:10.1016/j.amc.2004.04.118
[24]Maleknejad, K.; Shamloo, A. Salimi: Numerical solution of singular Volterra integral and integro-differential equations system of convolution type by using operational matrices, Appl math comput 195, 500-505 (2008) · Zbl 1132.65117 · doi:10.1016/j.amc.2007.05.001
[25]Babolian, E.; Shamloo, A. Salimi: Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions, J comput appl math 214, 495-508 (2008) · Zbl 1135.65043 · doi:10.1016/j.cam.2007.03.007
[26]Babolian, E.; Masouri, Z.: Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, J comput appl math 220, 51-57 (2008) · Zbl 1146.65082 · doi:10.1016/j.cam.2007.07.029
[27]Razzaghi, M.; Marzban, H. R.: Direct method for variational problems via hybrid of block-pulse and Chebyshev functions, Math problems eng 6, 85-97 (2000) · Zbl 0987.65055 · doi:10.1155/S1024123X00001265
[28]Maleknejad, K.; Mahmoudi, Y.: Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions, Appl math comput 149, 799-806 (2004) · Zbl 1038.65147 · doi:10.1016/S0096-3003(03)00180-2
[29]Marzban, H. R.; Razzaghi, M.: Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series, J sound vib 292, 954-963 (2006)
[30]Maleknejad, K.; Sohrabi, S.; Baranji, B.: Application of 2D-bpfs to nonlinear integral equations, Commun nonlinear sci numer simulat 15, No. 3, 528-535 (2010) · Zbl 1221.65339 · doi:10.1016/j.cnsns.2009.04.011
[31]Rao, G. P.: Piecewise constant orthogonal functions and their application to systems and control, (1983)