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Research ArticleThe Applications of Cardinal Trigonometric Splines inSolving Nonlinear Integral Equations
Jin Xie1 Xiaoyan Liu2 and Lixiang Xu1
1 Department of Mathematics and Physics Hefei University Hefei 230601 China2Department of Mathematics and Physics University of La Verne La Verne CA 91750 USA
Correspondence should be addressed to Xiaoyan Liu xliulaverneedu
Received 3 December 2013 Accepted 15 January 2014 Published 4 March 2014
Academic Editors Y M Cheng and L You
Copyright copy 2014 Jin Xie et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The cardinal trigonometric splines on small compact supports are employed to solve integral equations The unknown function isexpressed as a linear combination of cardinal trigonometric splines functionsThen a simple system of equations on the coefficientsis deducted When solving the Volterra integral equations the system is triangular so it is relatively straight forward to solve thenonlinear system of the coefficients and a good approximation of the original solution is obtained The sufficient condition for theexistence of the solution is discussed and the convergence rate is investigated
1 Introduction
Trigonometric splines were introduced by Schoenberg in [1]Univariate trigonometric splines are piecewise trigonometricpolynomials of the form
119899
sum119896=0
119886119896cos (120572
119896119909) + 119887
119896sin (120572
119896119909) (where 119887
0= 0) (1)
(where 1205720= 0 lt 120572
1lt 1205722lt sdot sdot sdot lt 120572
119899are real numbers)
in each interval and they are nature extensions of polynomialsplines Needless to say trigonometric splines have their ownadvantages A number of papers have appeared to study theproperties of the trigonometric splines and trigonometric B-splines (cf [2ndash4]) since then
In my previous papers (cf [5ndash7]) low degree orthonor-mal spline and cardinal spline functions with small compactsupports were constructed The method can be extendedto construct higher degree orthonormal or cardinal splinesUnlike in the book (cf [1]) by the cardinal splines wemean the specific splines satisfying cardinal interpolationconditions which means that the cardinal function has thevalue one at one interpolation point and value zero at all otherinterpolation points Cardinal splines are not only useful ininterpolation problems but they are also useful in deduction
of numerical integration formulas [6] and in solving integralequations
Integral equations appear in many fields includingdynamic systems mathematical applications in economicscommunication theory optimization and optimal controlsystems biology and population growth continuum andquantum mechanics kinetic theory of gases electricity andmagnetism potential theory and geophysics Many differen-tial equations with boundary value can be reformulated asintegral equations There are also some problems that can beexpressed only in terms of integral equations
In this paper we focus on the Volterra integral equationsof the second kind
where 120582 is a complex number the kernel 119870(119909 119905) 119891(119910) and119892(119909) are known functions and 119910(119909) is an unknown functionto be determined
This paper has six sections In Section 2 a univariatetrigonometric cardinal spline on a small compact supportis constructed and properties are studied In Section 3 theapplications of trigonometric cardinal splines on solving theVolterra integral equations are explored The unknown func-tion is expressed as a linear combination of trigonometric
cardinal spline functions Then a simple system of nonlinearequations on the coefficients is deducted It is relatively simpleto solve the linear system since the system is triangular anda good approximation of the original solution is obtainedThe sufficient condition for the existence is discussed and theconvergence rate is investigated In Section 4 the applicationsof trigonometric cardinal splines on solving the systems ofVolterra integral equations are explored In Section 5 numer-ical examples are given on solving the nonlinear Volterraintegral equations and a system of nonlinear Volterra integralequations Section 6 contains the conclusion remarks
2 A Cardinal Trigonometric Spline witha Small Support
To construct cardinal trigonometric splines let
120583ℎ(119909) =
1 minusℎ2lt 119909 le ℎ
2
0 elsewhere(3)
This is the zero degree polynomial or trigonometric B-spline
Let 1198790ℎ(119909) = 120583
ℎ(119909) A continuous univariate cardinal
trigonometric spline with a small support is
1198791ℎ(119909) =
11986811198790(119909)
11986811198790(0)
= 12 sin (ℎ2)
intℎ2
minusℎ2
120583ℎ(119909 + 119905) cos 119905 119889119905
(4)
Explicitly
1198791ℎ(119909) =
12 sin (ℎ2)
(sin(ℎ2) minus sin(119909 minus ℎ
2))
for 0 le 119909 le ℎ1
2 sin (ℎ2)(sin(ℎ
2) + sin(119909 + ℎ
2))
for minus ℎ le 119909 le 00 for 119909 gt ℎ or 119909 lt minusℎ
(5)
The graph of 1198791ℎ(119909) is Figure 1
Proposition 1 If 119910(119909) isin 1198621[119886 119887]11991010158401015840(119909) exists and is boundedon the finite interval [119886 119887] (where 119886 lt 119887) for any 119909 isin [119886 119887] andany integer 119899 such that ℎ = (119887 minus 119886)119899 lt 1 let
119879119871119910 (119909) =119899
sum119895=0
119910 (119886 + 119895ℎ) 1198791ℎ(119909 minus 119886 minus 119895ℎ) (6)
then |119910(119909) minus 119879119871119910((119909))| le 6ℎ2Max119909isin[119886119887]
|11991010158401015840(119909)|If 119910(119909) isin 1198621(minusinfininfin) 11991010158401015840(119909) exists on (minusinfininfin) and both
1199101015840(119909) and 11991010158401015840(119909) are bounded for any 119909 isin (minusinfininfin) and anychosen ℎ lt 1 let
119879119871 (119910 (119909)) =infin
sum119895=minusinfin
119910 (119895ℎ) 1198791ℎ(119909 minus 119895ℎ) (7)
then |119910(119909) minus 119879119871119910((119909))| le 6ℎ2Max119909isin(minusinfininfin)
|11991010158401015840(119909)|
x
y
minush h
1
Figure 1 The graph of 1198791ℎ(119909)
3 Numerical Method SolvingIntegral Equations
To solve the Volterra integral equations (2) in an interval(119886 119887) we let ℎ = (119887 minus 119886)119899 119909
119894= 119886 + 119894ℎ 119894 = 0 1 119899
Furthermore let
119910 (119909) =119899
sum119896=0
1198881198961198791ℎ(119909 minus 119909
119896)
119891 (119910 (119909)) =119899
sum119896=0
119891 (119888119896) 1198791ℎ(119909 minus 119909
119896)
119870 (119909 119905) =119899
sum119894=0
119899
sum119895=0
119870(119909119894 119909119895) 1198791ℎ(119909 minus 119909
119894) 1198791ℎ(119905 minus 119909
119895)
119892 (119909) =119899
sum119896=0
119892 (119909119896) 1198791ℎ(119909 minus 119909
119896)
(8)
plugging in (2) we get119899
sum119896=0
1198881198961198791ℎ(119909 minus 119909
119896)
minus 120582119899
sum119894=0
119899
sum119895=0
119899
sum119896=0
1198791ℎ(119909 minus 119909
119894) int119909
119886
119870(119909119894 119909119895) 1198791ℎ(119905 minus 119909
119895)
119891 (119888119896) 1198791ℎ(119905 minus 119909
119896) 119889119905
=119899
sum119896=0
119892 (119909119896) 1198791ℎ(119909 minus 119909
119896)
(9)
Letting 119909 = 1199091198960 we arrive for 119896
which is a triangular system of 119899 + 1 nonlinear equations onunknowns 119888
0 1198881 119888
119899 Notice that the coefficient matrix for
the system is a triangular matrix which means that we solve119888119894= 119891(119888
119894) + 119882
119894 where 119882
119894is a number not depending on
119888119894 for 119894 = 0 1 119899 For the convergence rate of solution
of the Volterra integral equations (2) we have the followingProposition 2
Proposition 2 Given that 119910(119909) 119892(119909) isin 119862[119886 119887] 11991010158401015840(119909) and11989210158401015840(119909) exist and are bounded in [119886 119887]119870(119909 119910) isin 119862[119886 119887]times[119886 119887](1205972120597119909119904120597119910119905)119870(119909 119910) (119904+119905 = 2) exist and are bounded in [119886 119887]times[119886 119887] Furthermore 119870(119909 119910) satisfies the condition
4 Numerical Method Solving Systems ofIntegral Equations
The system of Volterra integral equations is critical to manyphysical biological and engineering models For instancefor some heat transfer problems in physics the heat equa-tions are usually replaced by a system of Volterra integralequations [8] Many well-known models for neural networksin biomathematics nuclear reactor dynamics problems andthermoelasticity problems are also based on a system ofVolterra integral equations ([9 10]) Our method could beextended to solve the system of Volterra integral equationsGiven
119910119904(119909) = 119892
119904(119909) +
119898
sum119901=0
120582119901int119909
119886
119870119901119904(119909 119905) 119891
119901119904(119910119901(119905)) 119889119905
119909 isin (119886 119887) 119904 = 1 2 119898
(17)
4 ISRN Applied Mathematics
in an interval (119886 119887) we let ℎ = (119887 minus 119886)119899 119909119894= 119886 + 119894ℎ 119894 =
100381610038161003816100381610038161205821199011003816100381610038161003816100381610038161003816100381610038161003816119910119901 (119909) minus 119906119901 (119909)
10038161003816100381610038161003816
(20)
where |119871119872| lt 1 Let 119899 be an integer ℎ = (119887 minus 119886)119899 let 119909119894=
119886 + 119894ℎ 119910119894= 119910(119909
119894) 119894 = 0 1 2 119899 119888
0119901 1198881119901 119888
119899119901satisfies
the linear system (1198781)
119910lowast119901(119909) =
119899
sum119896=0
1198881198941199011198791ℎ(119909 minus 119896ℎ) (21)
The proposed method is a simple and effective procedurefor solving nonlinear Volterra integral equations as well as a
ISRN Applied Mathematics 7
system of nonlinear Volterra integral equationsThemethodscan be adapted easily to the Volterra integral equations of thefirst kind which have the form 119892(119909) = int
119860
119870(119909 119905)119910(119905)119889119905 Theconvergence rate could be higher if we use more complicatedorthonormal or cardinal splines Nevertheless the resultingsystem of coefficients will be more complicated nonlinearsystems which could take more time and effort to solve
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was funded by the Natural Science Foundationof Anhui Province of China under Grant no 1208085MA15and the Major Project of the Nature Science Foundation ofthe Education Department Anhui Province under Grant noKJ2014ZD30
References
[1] I J Schoenberg ldquoOn trigonometric spline interpolationrdquo Jour-nal of Mathematics and Mechanics vol 13 pp 795ndash825 1964
[2] T Lyche L L Schumaker and S Stanley ldquoQuasi-interpolantsbased on trigonometric splinesrdquo Journal of ApproximationTheory vol 95 no 2 pp 280ndash309 1998
[3] T Lyche and R Winther ldquoA stable recurrence relation fortrigonometric 119861-splinesrdquo Journal of ApproximationTheory vol25 no 3 pp 266ndash279 1979
[4] A Sharma and J Tzimbalario ldquoA class of cardinal trigonometricsplinesrdquo SIAM Journal on Mathematical Analysis vol 7 no 6pp 809ndash819 1976
[5] X Liu ldquoBivariate cardinal spline functions for digital signalprocessingrdquo inTrends in ApproximationTheory K Kopotum TLyche andM Neamtu Eds pp 261ndash271 Vanderbilt UniversityNashville Tenn USA 2001
[6] X Liu ldquoUnivariate and bivariate orthonormal splines and car-dinal splines on compact supportsrdquo Journal of Computationaland Applied Mathematics vol 195 no 1-2 pp 93ndash105 2006
[7] X Liu ldquoInterpolation by cardinal trigonometric splinesrdquo Inter-national Journal of Pure and Applied Mathematics vol 40 no 1pp 115ndash122 2007
[8] R Kress Linear Integral Equations vol 82 ofAppliedMathemat-ical Sciences Springer Berlin Germany 1989
[9] B L Moiseiwitsch Integral Equations Dover New York NYUSA 2005
[10] A D Polyanin Handbook of Integral Equations CRC PressBoca Raton Fla USA 1998
[11] A Vahidian Kamyad MMehrabinezhad and J Saberi-NadjafildquoA numerical approach for solving linear and nonlinear volterraintegral equations with controlled errorrdquo IAENG InternationalJournal of Applied Mathematics vol 40 no 2 pp 69ndash75 2010
[12] J Goncerzewicz H Marcinkowska W Okrasinski and KTabisz ldquoOn the percolation of water from a cylindrical reservoirinto the surrounding soilrdquo Zastosowania Matematyki vol 16no 2 pp 249ndash261 1978
[13] D Wei ldquoUniqueness of solutions for a class of non-linearvolterra integral equations without continuityrdquo Applied Math-ematics and Mechanics vol 18 no 12 pp 1191ndash1196 1997
cardinal spline functions Then a simple system of nonlinearequations on the coefficients is deducted It is relatively simpleto solve the linear system since the system is triangular anda good approximation of the original solution is obtainedThe sufficient condition for the existence is discussed and theconvergence rate is investigated In Section 4 the applicationsof trigonometric cardinal splines on solving the systems ofVolterra integral equations are explored In Section 5 numer-ical examples are given on solving the nonlinear Volterraintegral equations and a system of nonlinear Volterra integralequations Section 6 contains the conclusion remarks
2 A Cardinal Trigonometric Spline witha Small Support
To construct cardinal trigonometric splines let
120583ℎ(119909) =
1 minusℎ2lt 119909 le ℎ
2
0 elsewhere(3)
This is the zero degree polynomial or trigonometric B-spline
Let 1198790ℎ(119909) = 120583
ℎ(119909) A continuous univariate cardinal
trigonometric spline with a small support is
1198791ℎ(119909) =
11986811198790(119909)
11986811198790(0)
= 12 sin (ℎ2)
intℎ2
minusℎ2
120583ℎ(119909 + 119905) cos 119905 119889119905
(4)
Explicitly
1198791ℎ(119909) =
12 sin (ℎ2)
(sin(ℎ2) minus sin(119909 minus ℎ
2))
for 0 le 119909 le ℎ1
2 sin (ℎ2)(sin(ℎ
2) + sin(119909 + ℎ
2))
for minus ℎ le 119909 le 00 for 119909 gt ℎ or 119909 lt minusℎ
(5)
The graph of 1198791ℎ(119909) is Figure 1
Proposition 1 If 119910(119909) isin 1198621[119886 119887]11991010158401015840(119909) exists and is boundedon the finite interval [119886 119887] (where 119886 lt 119887) for any 119909 isin [119886 119887] andany integer 119899 such that ℎ = (119887 minus 119886)119899 lt 1 let
119879119871119910 (119909) =119899
sum119895=0
119910 (119886 + 119895ℎ) 1198791ℎ(119909 minus 119886 minus 119895ℎ) (6)
then |119910(119909) minus 119879119871119910((119909))| le 6ℎ2Max119909isin[119886119887]
|11991010158401015840(119909)|If 119910(119909) isin 1198621(minusinfininfin) 11991010158401015840(119909) exists on (minusinfininfin) and both
1199101015840(119909) and 11991010158401015840(119909) are bounded for any 119909 isin (minusinfininfin) and anychosen ℎ lt 1 let
119879119871 (119910 (119909)) =infin
sum119895=minusinfin
119910 (119895ℎ) 1198791ℎ(119909 minus 119895ℎ) (7)
then |119910(119909) minus 119879119871119910((119909))| le 6ℎ2Max119909isin(minusinfininfin)
|11991010158401015840(119909)|
x
y
minush h
1
Figure 1 The graph of 1198791ℎ(119909)
3 Numerical Method SolvingIntegral Equations
To solve the Volterra integral equations (2) in an interval(119886 119887) we let ℎ = (119887 minus 119886)119899 119909
119894= 119886 + 119894ℎ 119894 = 0 1 119899
Furthermore let
119910 (119909) =119899
sum119896=0
1198881198961198791ℎ(119909 minus 119909
119896)
119891 (119910 (119909)) =119899
sum119896=0
119891 (119888119896) 1198791ℎ(119909 minus 119909
119896)
119870 (119909 119905) =119899
sum119894=0
119899
sum119895=0
119870(119909119894 119909119895) 1198791ℎ(119909 minus 119909
119894) 1198791ℎ(119905 minus 119909
119895)
119892 (119909) =119899
sum119896=0
119892 (119909119896) 1198791ℎ(119909 minus 119909
119896)
(8)
plugging in (2) we get119899
sum119896=0
1198881198961198791ℎ(119909 minus 119909
119896)
minus 120582119899
sum119894=0
119899
sum119895=0
119899
sum119896=0
1198791ℎ(119909 minus 119909
119894) int119909
119886
119870(119909119894 119909119895) 1198791ℎ(119905 minus 119909
119895)
119891 (119888119896) 1198791ℎ(119905 minus 119909
119896) 119889119905
=119899
sum119896=0
119892 (119909119896) 1198791ℎ(119909 minus 119909
119896)
(9)
Letting 119909 = 1199091198960 we arrive for 119896
which is a triangular system of 119899 + 1 nonlinear equations onunknowns 119888
0 1198881 119888
119899 Notice that the coefficient matrix for
the system is a triangular matrix which means that we solve119888119894= 119891(119888
119894) + 119882
119894 where 119882
119894is a number not depending on
119888119894 for 119894 = 0 1 119899 For the convergence rate of solution
of the Volterra integral equations (2) we have the followingProposition 2
Proposition 2 Given that 119910(119909) 119892(119909) isin 119862[119886 119887] 11991010158401015840(119909) and11989210158401015840(119909) exist and are bounded in [119886 119887]119870(119909 119910) isin 119862[119886 119887]times[119886 119887](1205972120597119909119904120597119910119905)119870(119909 119910) (119904+119905 = 2) exist and are bounded in [119886 119887]times[119886 119887] Furthermore 119870(119909 119910) satisfies the condition
4 Numerical Method Solving Systems ofIntegral Equations
The system of Volterra integral equations is critical to manyphysical biological and engineering models For instancefor some heat transfer problems in physics the heat equa-tions are usually replaced by a system of Volterra integralequations [8] Many well-known models for neural networksin biomathematics nuclear reactor dynamics problems andthermoelasticity problems are also based on a system ofVolterra integral equations ([9 10]) Our method could beextended to solve the system of Volterra integral equationsGiven
119910119904(119909) = 119892
119904(119909) +
119898
sum119901=0
120582119901int119909
119886
119870119901119904(119909 119905) 119891
119901119904(119910119901(119905)) 119889119905
119909 isin (119886 119887) 119904 = 1 2 119898
(17)
4 ISRN Applied Mathematics
in an interval (119886 119887) we let ℎ = (119887 minus 119886)119899 119909119894= 119886 + 119894ℎ 119894 =
100381610038161003816100381610038161205821199011003816100381610038161003816100381610038161003816100381610038161003816119910119901 (119909) minus 119906119901 (119909)
10038161003816100381610038161003816
(20)
where |119871119872| lt 1 Let 119899 be an integer ℎ = (119887 minus 119886)119899 let 119909119894=
119886 + 119894ℎ 119910119894= 119910(119909
119894) 119894 = 0 1 2 119899 119888
0119901 1198881119901 119888
119899119901satisfies
the linear system (1198781)
119910lowast119901(119909) =
119899
sum119896=0
1198881198941199011198791ℎ(119909 minus 119896ℎ) (21)
The proposed method is a simple and effective procedurefor solving nonlinear Volterra integral equations as well as a
ISRN Applied Mathematics 7
system of nonlinear Volterra integral equationsThemethodscan be adapted easily to the Volterra integral equations of thefirst kind which have the form 119892(119909) = int
119860
119870(119909 119905)119910(119905)119889119905 Theconvergence rate could be higher if we use more complicatedorthonormal or cardinal splines Nevertheless the resultingsystem of coefficients will be more complicated nonlinearsystems which could take more time and effort to solve
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was funded by the Natural Science Foundationof Anhui Province of China under Grant no 1208085MA15and the Major Project of the Nature Science Foundation ofthe Education Department Anhui Province under Grant noKJ2014ZD30
References
[1] I J Schoenberg ldquoOn trigonometric spline interpolationrdquo Jour-nal of Mathematics and Mechanics vol 13 pp 795ndash825 1964
[2] T Lyche L L Schumaker and S Stanley ldquoQuasi-interpolantsbased on trigonometric splinesrdquo Journal of ApproximationTheory vol 95 no 2 pp 280ndash309 1998
[3] T Lyche and R Winther ldquoA stable recurrence relation fortrigonometric 119861-splinesrdquo Journal of ApproximationTheory vol25 no 3 pp 266ndash279 1979
[4] A Sharma and J Tzimbalario ldquoA class of cardinal trigonometricsplinesrdquo SIAM Journal on Mathematical Analysis vol 7 no 6pp 809ndash819 1976
[5] X Liu ldquoBivariate cardinal spline functions for digital signalprocessingrdquo inTrends in ApproximationTheory K Kopotum TLyche andM Neamtu Eds pp 261ndash271 Vanderbilt UniversityNashville Tenn USA 2001
[6] X Liu ldquoUnivariate and bivariate orthonormal splines and car-dinal splines on compact supportsrdquo Journal of Computationaland Applied Mathematics vol 195 no 1-2 pp 93ndash105 2006
[7] X Liu ldquoInterpolation by cardinal trigonometric splinesrdquo Inter-national Journal of Pure and Applied Mathematics vol 40 no 1pp 115ndash122 2007
[8] R Kress Linear Integral Equations vol 82 ofAppliedMathemat-ical Sciences Springer Berlin Germany 1989
[9] B L Moiseiwitsch Integral Equations Dover New York NYUSA 2005
[10] A D Polyanin Handbook of Integral Equations CRC PressBoca Raton Fla USA 1998
[11] A Vahidian Kamyad MMehrabinezhad and J Saberi-NadjafildquoA numerical approach for solving linear and nonlinear volterraintegral equations with controlled errorrdquo IAENG InternationalJournal of Applied Mathematics vol 40 no 2 pp 69ndash75 2010
[12] J Goncerzewicz H Marcinkowska W Okrasinski and KTabisz ldquoOn the percolation of water from a cylindrical reservoirinto the surrounding soilrdquo Zastosowania Matematyki vol 16no 2 pp 249ndash261 1978
[13] D Wei ldquoUniqueness of solutions for a class of non-linearvolterra integral equations without continuityrdquo Applied Math-ematics and Mechanics vol 18 no 12 pp 1191ndash1196 1997
which is a triangular system of 119899 + 1 nonlinear equations onunknowns 119888
0 1198881 119888
119899 Notice that the coefficient matrix for
the system is a triangular matrix which means that we solve119888119894= 119891(119888
119894) + 119882
119894 where 119882
119894is a number not depending on
119888119894 for 119894 = 0 1 119899 For the convergence rate of solution
of the Volterra integral equations (2) we have the followingProposition 2
Proposition 2 Given that 119910(119909) 119892(119909) isin 119862[119886 119887] 11991010158401015840(119909) and11989210158401015840(119909) exist and are bounded in [119886 119887]119870(119909 119910) isin 119862[119886 119887]times[119886 119887](1205972120597119909119904120597119910119905)119870(119909 119910) (119904+119905 = 2) exist and are bounded in [119886 119887]times[119886 119887] Furthermore 119870(119909 119910) satisfies the condition
4 Numerical Method Solving Systems ofIntegral Equations
The system of Volterra integral equations is critical to manyphysical biological and engineering models For instancefor some heat transfer problems in physics the heat equa-tions are usually replaced by a system of Volterra integralequations [8] Many well-known models for neural networksin biomathematics nuclear reactor dynamics problems andthermoelasticity problems are also based on a system ofVolterra integral equations ([9 10]) Our method could beextended to solve the system of Volterra integral equationsGiven
119910119904(119909) = 119892
119904(119909) +
119898
sum119901=0
120582119901int119909
119886
119870119901119904(119909 119905) 119891
119901119904(119910119901(119905)) 119889119905
119909 isin (119886 119887) 119904 = 1 2 119898
(17)
4 ISRN Applied Mathematics
in an interval (119886 119887) we let ℎ = (119887 minus 119886)119899 119909119894= 119886 + 119894ℎ 119894 =
100381610038161003816100381610038161205821199011003816100381610038161003816100381610038161003816100381610038161003816119910119901 (119909) minus 119906119901 (119909)
10038161003816100381610038161003816
(20)
where |119871119872| lt 1 Let 119899 be an integer ℎ = (119887 minus 119886)119899 let 119909119894=
119886 + 119894ℎ 119910119894= 119910(119909
119894) 119894 = 0 1 2 119899 119888
0119901 1198881119901 119888
119899119901satisfies
the linear system (1198781)
119910lowast119901(119909) =
119899
sum119896=0
1198881198941199011198791ℎ(119909 minus 119896ℎ) (21)
The proposed method is a simple and effective procedurefor solving nonlinear Volterra integral equations as well as a
ISRN Applied Mathematics 7
system of nonlinear Volterra integral equationsThemethodscan be adapted easily to the Volterra integral equations of thefirst kind which have the form 119892(119909) = int
119860
119870(119909 119905)119910(119905)119889119905 Theconvergence rate could be higher if we use more complicatedorthonormal or cardinal splines Nevertheless the resultingsystem of coefficients will be more complicated nonlinearsystems which could take more time and effort to solve
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was funded by the Natural Science Foundationof Anhui Province of China under Grant no 1208085MA15and the Major Project of the Nature Science Foundation ofthe Education Department Anhui Province under Grant noKJ2014ZD30
References
[1] I J Schoenberg ldquoOn trigonometric spline interpolationrdquo Jour-nal of Mathematics and Mechanics vol 13 pp 795ndash825 1964
[2] T Lyche L L Schumaker and S Stanley ldquoQuasi-interpolantsbased on trigonometric splinesrdquo Journal of ApproximationTheory vol 95 no 2 pp 280ndash309 1998
[3] T Lyche and R Winther ldquoA stable recurrence relation fortrigonometric 119861-splinesrdquo Journal of ApproximationTheory vol25 no 3 pp 266ndash279 1979
[4] A Sharma and J Tzimbalario ldquoA class of cardinal trigonometricsplinesrdquo SIAM Journal on Mathematical Analysis vol 7 no 6pp 809ndash819 1976
[5] X Liu ldquoBivariate cardinal spline functions for digital signalprocessingrdquo inTrends in ApproximationTheory K Kopotum TLyche andM Neamtu Eds pp 261ndash271 Vanderbilt UniversityNashville Tenn USA 2001
[6] X Liu ldquoUnivariate and bivariate orthonormal splines and car-dinal splines on compact supportsrdquo Journal of Computationaland Applied Mathematics vol 195 no 1-2 pp 93ndash105 2006
[7] X Liu ldquoInterpolation by cardinal trigonometric splinesrdquo Inter-national Journal of Pure and Applied Mathematics vol 40 no 1pp 115ndash122 2007
[8] R Kress Linear Integral Equations vol 82 ofAppliedMathemat-ical Sciences Springer Berlin Germany 1989
[9] B L Moiseiwitsch Integral Equations Dover New York NYUSA 2005
[10] A D Polyanin Handbook of Integral Equations CRC PressBoca Raton Fla USA 1998
[11] A Vahidian Kamyad MMehrabinezhad and J Saberi-NadjafildquoA numerical approach for solving linear and nonlinear volterraintegral equations with controlled errorrdquo IAENG InternationalJournal of Applied Mathematics vol 40 no 2 pp 69ndash75 2010
[12] J Goncerzewicz H Marcinkowska W Okrasinski and KTabisz ldquoOn the percolation of water from a cylindrical reservoirinto the surrounding soilrdquo Zastosowania Matematyki vol 16no 2 pp 249ndash261 1978
[13] D Wei ldquoUniqueness of solutions for a class of non-linearvolterra integral equations without continuityrdquo Applied Math-ematics and Mechanics vol 18 no 12 pp 1191ndash1196 1997
100381610038161003816100381610038161205821199011003816100381610038161003816100381610038161003816100381610038161003816119910119901 (119909) minus 119906119901 (119909)
10038161003816100381610038161003816
(20)
where |119871119872| lt 1 Let 119899 be an integer ℎ = (119887 minus 119886)119899 let 119909119894=
119886 + 119894ℎ 119910119894= 119910(119909
119894) 119894 = 0 1 2 119899 119888
0119901 1198881119901 119888
119899119901satisfies
the linear system (1198781)
119910lowast119901(119909) =
119899
sum119896=0
1198881198941199011198791ℎ(119909 minus 119896ℎ) (21)
The proposed method is a simple and effective procedurefor solving nonlinear Volterra integral equations as well as a
ISRN Applied Mathematics 7
system of nonlinear Volterra integral equationsThemethodscan be adapted easily to the Volterra integral equations of thefirst kind which have the form 119892(119909) = int
119860
119870(119909 119905)119910(119905)119889119905 Theconvergence rate could be higher if we use more complicatedorthonormal or cardinal splines Nevertheless the resultingsystem of coefficients will be more complicated nonlinearsystems which could take more time and effort to solve
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was funded by the Natural Science Foundationof Anhui Province of China under Grant no 1208085MA15and the Major Project of the Nature Science Foundation ofthe Education Department Anhui Province under Grant noKJ2014ZD30
References
[1] I J Schoenberg ldquoOn trigonometric spline interpolationrdquo Jour-nal of Mathematics and Mechanics vol 13 pp 795ndash825 1964
[2] T Lyche L L Schumaker and S Stanley ldquoQuasi-interpolantsbased on trigonometric splinesrdquo Journal of ApproximationTheory vol 95 no 2 pp 280ndash309 1998
[3] T Lyche and R Winther ldquoA stable recurrence relation fortrigonometric 119861-splinesrdquo Journal of ApproximationTheory vol25 no 3 pp 266ndash279 1979
[4] A Sharma and J Tzimbalario ldquoA class of cardinal trigonometricsplinesrdquo SIAM Journal on Mathematical Analysis vol 7 no 6pp 809ndash819 1976
[5] X Liu ldquoBivariate cardinal spline functions for digital signalprocessingrdquo inTrends in ApproximationTheory K Kopotum TLyche andM Neamtu Eds pp 261ndash271 Vanderbilt UniversityNashville Tenn USA 2001
[6] X Liu ldquoUnivariate and bivariate orthonormal splines and car-dinal splines on compact supportsrdquo Journal of Computationaland Applied Mathematics vol 195 no 1-2 pp 93ndash105 2006
[7] X Liu ldquoInterpolation by cardinal trigonometric splinesrdquo Inter-national Journal of Pure and Applied Mathematics vol 40 no 1pp 115ndash122 2007
[8] R Kress Linear Integral Equations vol 82 ofAppliedMathemat-ical Sciences Springer Berlin Germany 1989
[9] B L Moiseiwitsch Integral Equations Dover New York NYUSA 2005
[10] A D Polyanin Handbook of Integral Equations CRC PressBoca Raton Fla USA 1998
[11] A Vahidian Kamyad MMehrabinezhad and J Saberi-NadjafildquoA numerical approach for solving linear and nonlinear volterraintegral equations with controlled errorrdquo IAENG InternationalJournal of Applied Mathematics vol 40 no 2 pp 69ndash75 2010
[12] J Goncerzewicz H Marcinkowska W Okrasinski and KTabisz ldquoOn the percolation of water from a cylindrical reservoirinto the surrounding soilrdquo Zastosowania Matematyki vol 16no 2 pp 249ndash261 1978
[13] D Wei ldquoUniqueness of solutions for a class of non-linearvolterra integral equations without continuityrdquo Applied Math-ematics and Mechanics vol 18 no 12 pp 1191ndash1196 1997
The proposed method is a simple and effective procedurefor solving nonlinear Volterra integral equations as well as a
ISRN Applied Mathematics 7
system of nonlinear Volterra integral equationsThemethodscan be adapted easily to the Volterra integral equations of thefirst kind which have the form 119892(119909) = int
119860
119870(119909 119905)119910(119905)119889119905 Theconvergence rate could be higher if we use more complicatedorthonormal or cardinal splines Nevertheless the resultingsystem of coefficients will be more complicated nonlinearsystems which could take more time and effort to solve
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was funded by the Natural Science Foundationof Anhui Province of China under Grant no 1208085MA15and the Major Project of the Nature Science Foundation ofthe Education Department Anhui Province under Grant noKJ2014ZD30
References
[1] I J Schoenberg ldquoOn trigonometric spline interpolationrdquo Jour-nal of Mathematics and Mechanics vol 13 pp 795ndash825 1964
[2] T Lyche L L Schumaker and S Stanley ldquoQuasi-interpolantsbased on trigonometric splinesrdquo Journal of ApproximationTheory vol 95 no 2 pp 280ndash309 1998
[3] T Lyche and R Winther ldquoA stable recurrence relation fortrigonometric 119861-splinesrdquo Journal of ApproximationTheory vol25 no 3 pp 266ndash279 1979
[4] A Sharma and J Tzimbalario ldquoA class of cardinal trigonometricsplinesrdquo SIAM Journal on Mathematical Analysis vol 7 no 6pp 809ndash819 1976
[5] X Liu ldquoBivariate cardinal spline functions for digital signalprocessingrdquo inTrends in ApproximationTheory K Kopotum TLyche andM Neamtu Eds pp 261ndash271 Vanderbilt UniversityNashville Tenn USA 2001
[6] X Liu ldquoUnivariate and bivariate orthonormal splines and car-dinal splines on compact supportsrdquo Journal of Computationaland Applied Mathematics vol 195 no 1-2 pp 93ndash105 2006
[7] X Liu ldquoInterpolation by cardinal trigonometric splinesrdquo Inter-national Journal of Pure and Applied Mathematics vol 40 no 1pp 115ndash122 2007
[8] R Kress Linear Integral Equations vol 82 ofAppliedMathemat-ical Sciences Springer Berlin Germany 1989
[9] B L Moiseiwitsch Integral Equations Dover New York NYUSA 2005
[10] A D Polyanin Handbook of Integral Equations CRC PressBoca Raton Fla USA 1998
[11] A Vahidian Kamyad MMehrabinezhad and J Saberi-NadjafildquoA numerical approach for solving linear and nonlinear volterraintegral equations with controlled errorrdquo IAENG InternationalJournal of Applied Mathematics vol 40 no 2 pp 69ndash75 2010
[12] J Goncerzewicz H Marcinkowska W Okrasinski and KTabisz ldquoOn the percolation of water from a cylindrical reservoirinto the surrounding soilrdquo Zastosowania Matematyki vol 16no 2 pp 249ndash261 1978
[13] D Wei ldquoUniqueness of solutions for a class of non-linearvolterra integral equations without continuityrdquo Applied Math-ematics and Mechanics vol 18 no 12 pp 1191ndash1196 1997
The proposed method is a simple and effective procedurefor solving nonlinear Volterra integral equations as well as a
ISRN Applied Mathematics 7
system of nonlinear Volterra integral equationsThemethodscan be adapted easily to the Volterra integral equations of thefirst kind which have the form 119892(119909) = int
119860
119870(119909 119905)119910(119905)119889119905 Theconvergence rate could be higher if we use more complicatedorthonormal or cardinal splines Nevertheless the resultingsystem of coefficients will be more complicated nonlinearsystems which could take more time and effort to solve
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was funded by the Natural Science Foundationof Anhui Province of China under Grant no 1208085MA15and the Major Project of the Nature Science Foundation ofthe Education Department Anhui Province under Grant noKJ2014ZD30
References
[1] I J Schoenberg ldquoOn trigonometric spline interpolationrdquo Jour-nal of Mathematics and Mechanics vol 13 pp 795ndash825 1964
[2] T Lyche L L Schumaker and S Stanley ldquoQuasi-interpolantsbased on trigonometric splinesrdquo Journal of ApproximationTheory vol 95 no 2 pp 280ndash309 1998
[3] T Lyche and R Winther ldquoA stable recurrence relation fortrigonometric 119861-splinesrdquo Journal of ApproximationTheory vol25 no 3 pp 266ndash279 1979
[4] A Sharma and J Tzimbalario ldquoA class of cardinal trigonometricsplinesrdquo SIAM Journal on Mathematical Analysis vol 7 no 6pp 809ndash819 1976
[5] X Liu ldquoBivariate cardinal spline functions for digital signalprocessingrdquo inTrends in ApproximationTheory K Kopotum TLyche andM Neamtu Eds pp 261ndash271 Vanderbilt UniversityNashville Tenn USA 2001
[6] X Liu ldquoUnivariate and bivariate orthonormal splines and car-dinal splines on compact supportsrdquo Journal of Computationaland Applied Mathematics vol 195 no 1-2 pp 93ndash105 2006
[7] X Liu ldquoInterpolation by cardinal trigonometric splinesrdquo Inter-national Journal of Pure and Applied Mathematics vol 40 no 1pp 115ndash122 2007
[8] R Kress Linear Integral Equations vol 82 ofAppliedMathemat-ical Sciences Springer Berlin Germany 1989
[9] B L Moiseiwitsch Integral Equations Dover New York NYUSA 2005
[10] A D Polyanin Handbook of Integral Equations CRC PressBoca Raton Fla USA 1998
[11] A Vahidian Kamyad MMehrabinezhad and J Saberi-NadjafildquoA numerical approach for solving linear and nonlinear volterraintegral equations with controlled errorrdquo IAENG InternationalJournal of Applied Mathematics vol 40 no 2 pp 69ndash75 2010
[12] J Goncerzewicz H Marcinkowska W Okrasinski and KTabisz ldquoOn the percolation of water from a cylindrical reservoirinto the surrounding soilrdquo Zastosowania Matematyki vol 16no 2 pp 249ndash261 1978
[13] D Wei ldquoUniqueness of solutions for a class of non-linearvolterra integral equations without continuityrdquo Applied Math-ematics and Mechanics vol 18 no 12 pp 1191ndash1196 1997
system of nonlinear Volterra integral equationsThemethodscan be adapted easily to the Volterra integral equations of thefirst kind which have the form 119892(119909) = int
119860
119870(119909 119905)119910(119905)119889119905 Theconvergence rate could be higher if we use more complicatedorthonormal or cardinal splines Nevertheless the resultingsystem of coefficients will be more complicated nonlinearsystems which could take more time and effort to solve
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was funded by the Natural Science Foundationof Anhui Province of China under Grant no 1208085MA15and the Major Project of the Nature Science Foundation ofthe Education Department Anhui Province under Grant noKJ2014ZD30
References
[1] I J Schoenberg ldquoOn trigonometric spline interpolationrdquo Jour-nal of Mathematics and Mechanics vol 13 pp 795ndash825 1964
[2] T Lyche L L Schumaker and S Stanley ldquoQuasi-interpolantsbased on trigonometric splinesrdquo Journal of ApproximationTheory vol 95 no 2 pp 280ndash309 1998
[3] T Lyche and R Winther ldquoA stable recurrence relation fortrigonometric 119861-splinesrdquo Journal of ApproximationTheory vol25 no 3 pp 266ndash279 1979
[4] A Sharma and J Tzimbalario ldquoA class of cardinal trigonometricsplinesrdquo SIAM Journal on Mathematical Analysis vol 7 no 6pp 809ndash819 1976
[5] X Liu ldquoBivariate cardinal spline functions for digital signalprocessingrdquo inTrends in ApproximationTheory K Kopotum TLyche andM Neamtu Eds pp 261ndash271 Vanderbilt UniversityNashville Tenn USA 2001
[6] X Liu ldquoUnivariate and bivariate orthonormal splines and car-dinal splines on compact supportsrdquo Journal of Computationaland Applied Mathematics vol 195 no 1-2 pp 93ndash105 2006
[7] X Liu ldquoInterpolation by cardinal trigonometric splinesrdquo Inter-national Journal of Pure and Applied Mathematics vol 40 no 1pp 115ndash122 2007
[8] R Kress Linear Integral Equations vol 82 ofAppliedMathemat-ical Sciences Springer Berlin Germany 1989
[9] B L Moiseiwitsch Integral Equations Dover New York NYUSA 2005
[10] A D Polyanin Handbook of Integral Equations CRC PressBoca Raton Fla USA 1998
[11] A Vahidian Kamyad MMehrabinezhad and J Saberi-NadjafildquoA numerical approach for solving linear and nonlinear volterraintegral equations with controlled errorrdquo IAENG InternationalJournal of Applied Mathematics vol 40 no 2 pp 69ndash75 2010
[12] J Goncerzewicz H Marcinkowska W Okrasinski and KTabisz ldquoOn the percolation of water from a cylindrical reservoirinto the surrounding soilrdquo Zastosowania Matematyki vol 16no 2 pp 249ndash261 1978
[13] D Wei ldquoUniqueness of solutions for a class of non-linearvolterra integral equations without continuityrdquo Applied Math-ematics and Mechanics vol 18 no 12 pp 1191ndash1196 1997