Fuzzy Fredholm integro-differential equations with artificial neural networks
|
|
- Edmund Miller
- 5 years ago
- Views:
Transcription
1 Avilble online t Volume 202, Yer 202 Article ID cn-0028, 3 pges doi:0.5899/202/cn-0028 Reserch Article Fuzzy Fredholm integro-differentil equtions with rtificil neurl networks Mrym Mosleh, Mhmood Otdi () Deprtment of Mthemtics, Firoozkooh Brnch, Islmic Azd University, Firoozkooh, Irn Copyright 202 c Mrym Mosleh nd Mhmood Otdi. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct In this pper, we use prmetric form of fuzzy number, then feed-forwrd neurl network is presented for obtining pproximte solution for fuzzy Fredholm integro-differentil eqution of the second kind. This pper presents method bsed on neurl networks nd Newton-Cotes methods with positive coefficient. The bility of neurl networks in function pproximtion is our min objective. The proposed method is illustrted by solving some numericl exmples. Keywords: Fuzzy integro-differentil equtions; Artificil neurl networks Introduction The solutions of integrl equtions hve mjor role in the field of science nd engineering. A physicl even cn be modelled by the differentil eqution, n integrl eqution. Since few of these equtions cnnot be solved explicitly, it is often necessry to resort to numericl techniques which re pproprite combintions of numericl integrtion nd interpoltion [2, 32]. There re severl numericl methods for solving liner Volterr integrl eqution [8, 4] nd system of nonliner Volterr integrl equtions [4]. Kuthen in [28] used colloction method to solve the Volterr- Fredholm integrl eqution numericlly. Borzbdi nd Frd in [6] obtined numericl solution of nonliner Fredholm integrl equtions of the second kind. The concept of fuzzy numbers nd fuzzy rithmetic opertions were first introduced by Zdeh [43], Dubois nd Prde [20]. We refer the reder to [26] for more informtion on fuzzy numbers nd fuzzy rithmetic. The topics of fuzzy integrl equtions (FIE) Corresponding uthor. E-mil: mosleh@iufb.c.ir; Tel:
2 which growing interest for some time, in prticulr in reltion to fuzzy control, hve been rpidly developed in recent yers. The fuzzy mpping function ws introduced by Chng nd Zdeh [7]. Lter, Dubois nd Prde [2] presented n elementry fuzzy clculus bsed on the extension principle lso the concept of integrtion of fuzzy functions ws first introduced by Dubois nd Prde [2]. Bbolin et l. nd Abbsbndy et l. in [3, ] obtined numericl solution of liner Fredholm fuzzy integrl equtions of the second kind. Allhvirnloo et l. in [6] presented new method for solving fuzzy integrodifferentil eqution under generlized differentibility. Another group of reserchers tried to extend some numericl methods to solve fuzzy differentil equtins (FDEs) [, 8, 38] such s Runge-Kutt method [2], Adomin method [0], predictor-corrector method nd multi-step methods [7]. Fuzzy neurl network hve been extensively studied [5, 9] nd recently, successfully used for solving fuzzy polynomil eqution nd systems of fuzzy polynomils [4, 5], pproximte fuzzy coefficients of fuzzy regression models [33, 34, 35], pproximte solution of fuzzy liner systems nd fully fuzzy liner systems [36, 39]. Lgris et l. in [3] used multilyer perceptron to estimte the solution of differentil eqution. Their neurl network model ws trined over n intervl (over which the differentil eqution must be solved), so the inputs of the neurl network model were the trining points. The comprison of their method with the existing numericl method shows tht their method ws more ccurte nd the solution hd lso more generliztions. Recently Effti et l. in [22] nd Mosleh nd Otdi [37] used rtificil neurl networks to solve fuzzy ordinry differentil equtions. But in this pper, we extend the rtificil neurl networks to solve integro-differentil equtions. The bility of neurl networks in function pproximtion is our min objective. In this pper, we present novel nd very simple numericl method bsed upon neurl networks for solving fuzzy liner Fredholm integro-differentil equtions of the second kind 2 Preliminries X (s) = y(s) + λ b k(s, t)x(t)dt. In this section the bsic nottions used in fuzzy opertions re introduced. We strt by defining the fuzzy number. Definition 2.. [30] A fuzzy number is fuzzy set u : R I = [0, ] such tht i. u is upper semi-continuous; ii. u(x) = 0 outside some intervl [, d]; iii. There re rel numbers b nd c, b c d, for which. u(x) is monotoniclly incresing on [, b], 2. u(x) is monotoniclly decresing on [c, d], 3. u(x) =, b x c. The set of ll the fuzzy numbers (s given in Definition (2.) is denoted by E. An lterntive definition which yields the sme E is given by Klev [27]. Definition 2.2. A fuzzy number u is pir (u, u) of functions u(r) nd u(r), 0 r, which stisfy the following requirements: i. u(r) is bounded monotoniclly incresing, left continuous function on (0, ] nd right continuous t 0; 2 ISPACS GmbH
3 ii. u(r) is bounded monotoniclly decresing, left continuous function on (0, ] nd right continuous t 0; iii. u(r) u(r), 0 r. A crisp number r is simply represented by u(α) = u(α) = r, 0 α. The set of ll the fuzzy numbers is denoted by E. This fuzzy number spce s shown in [42], cn be embedded into the Bnch spce B = C[0, ] C[0, ]. Definition 2.3. [27] For rbitrry u = (u(r), u(r)), v = (v(r), v(r)), we sy tht u = v if nd only if u = v nd u = v. For rbitrry u = (u(r), u(r)), v = (v(r), v(r)) nd k R we define ddition nd multipliction by k s (u + v)(r) = (u(r) + v(r)), (u + v)(r) = (u(r) + v(r)), ku(r) = ku(r), ku(r) = ku(r), if k 0, ku(r) = ku(r), ku(r) = ku(r), if k < 0. Definition 2.4. [24] For rbitrry fuzzy numbers u, v, we use the distnce D(u, v) = sup 0 r mx{ u(r) v(r), u(r) v(r) } nd it is shown tht (E, D) is complete metric spce [40]. Definition 2.5. [23, 24] Let f : [, b] E, for ech prtition P = {t 0, t,..., t n } of [, b] nd for rbitrry ξ i [t i, t i ], i n suppose The definite integrl of f(t) over [, b] is R p = n i= f(ξ i)(t i t i ), := mx{ t i t i, i =, 2,..., n}. b f(t)dt = lim 0 R p provided tht this limit exists in the metric D. If the fuzzy function f(t) is continuous in the metric D, its definite integrl exists [24] nd lso, ( b f(t; r)dt) = b f(t; r)dt, ( b f(t; r)dt) = b f(t; r)dt. Definition 2.6. Let u, v E. If there exists w E such tht u = v +w then w is clled the H-difference of u, v nd it is denoted by u v. 3 ISPACS GmbH
4 Definition 2.7. A function f : (, b) E is clled H-differentible t ˆt (, b) if, for h > 0 sufficiently smll, there exist the H-differences f(ˆt + h) f(ˆt), f(ˆt) f(ˆt h), nd n element f (ˆt) E such tht: lim h 0 +D( f(ˆt + h) f(ˆt) h Then f (ˆt) is clled the fuzzy derivtive of f t ˆt. 3 Fuzzy integro-differentil eqution The liner Fredholm integro-differentil equtions [25] X (s) = y(s) + λ, f (ˆt)) = lim h 0 +D( f(ˆt) f(ˆt h), f (ˆt)) = 0. h b k(s, t)x(t)dt, X(s 0 ) = X 0, (3.) where λ > 0, k is n rbitrry given kernel function over the squre s, t b nd y(s) is given function of s [, b]. If X is fuzzy function, y(s) is given fuzzy function of s [, b] nd X is the fuzzy derivtive (ccording to Definition (2.7)) of X, this eqution my only possess fuzzy solution. Sufficient for the existence eqution of the second kind, re given in [3]. Let X(s) = (X(s; r), X(s; r)) is fuzzy solution of Eq.(3.), therefore by Definition (2.3), Definition (2.5) nd Definition (2.7) we hve the equivlent system X (s) = y(s) + λ b k(s, t)x(t)dt, X(s 0) = X 0, X (s) = y(s) + λ b k(s, t)x(t)dt, X(s 0) = X 0 (3.2) which possesses unique solution (X, X) B which is fuzzy function, i.e. for ech s, the pir (X(s; r), X(s; r)) is fuzzy number, therefore ech solution of Eq.(3.) is solution of system (3.2) nd conversely lso Eq.(3.) nd system (3.2) re equivlent. The prmetric form of Eqs.(3.2) is given by X (s, r) = y(s, r) + λ b k(s, t)x(t, r)dt, X(s X (s, r) = y(s, r) + λ b k(s, t)x(t, r)dt, X(s 0) = X 0 (r), 0) = X 0 (r) (3.3) for r [0, ]. suppose k(s, t) be continuous in s b nd for fix t, k(s, t) chnges its sing in finite points s s i where x i [, s ]. For exmple, let k(s, t) be nonnegtive over [, s ] nd negtive over [s, b], therefore we hve X (s, r) = y(s, r) + λ s X (s, r) = y(s, r) + λ s k(s, t)x(t, r)dt + λ b s k(s, t)x(t, r)dt, k(s, t)x(t, r)dt + λ b s k(s, t)x(t, r)dt, X(s 0 ) = X 0 (r), X(s 0 ) = X 0 (r). In most cses, however, nlyticl solution to Eq.(3.3) my not be found nd numericl pproch must be considered. 4 ISPACS GmbH
5 4 Function pproximtion The use of neurl networks provides solutions with very good generlizbility (such s differentibility). On the other hnd, n importnt feture of multi-lyer perceptrons is their utility to pproximte functions, which leds to wide pplicbility in most problems. In this pper, the function pproximtion cpbilities of feed-forwrd neurl networks is used by expressing the tril solutions for system (3.3) s the sum of two terms. The first term stisfies the intil conditions nd contins no djustble prmeters. The second term involves feed-forwrd neurl network to be trined so s to stisfy the integrodifferentil equtions. Since it is known tht multilyer perceptron with one hidden lyer cn pproximte ny function to rbitrry ccurcy, the multilyer perceptron is used s the type of the network rchitecture. If X T (s, r, p) is tril solution for the first eqution in system (3.3) nd X T (s, r, p) is tril solution for the second eqution in the system (3.3) where p nd p re djustble prmeters, (indeed X T (s, r, p) nd X T (s, r, p) re pproximtions of X(s, r) nd X(s, r) respectively) thus the problem of finding the pproximted solutions for (3.3) over some colloction point in [, b] is equivlent to clculte the functionls X T nd X T tht stisfies the following constrined optimiztion problem [29]: M min p i= {(X T (s i, r, p) y(s i, r) F (s i, r, p)) 2 + (X T (s i, r, p) y(s i, r) F (s i, r, p)) 2 }, (4.4) X T (s 0, r, p) = X 0 (r), X T (s 0, r, p) = X 0 (r) where p = (p, p) contin ll djustble prmeters (weights of input nd output lyers nd bises) nd F (s, r, p) = λ b k(s, t)x T (t, r, p)dt, F (s, r, p) = λ b k(s, t)x T (t, r, p)dt. In generl we cnnot be ble to crry out nlyticlly the integrtions, involved. In this cse we nturlly turn to numericl qudrture. We introduce qudrture rule R for the intervl [, b] with positive weights w j nd N nodes t j, i.e., Rf = N w j f(t j ) = If Ef = j= b f(t)dt Ef, where Ef is the error. If we first ignore the error of this qudrture rule then the Eq. (4.4) is replced by the pproximte eqution M min p i= {(X T (s i, r, p) y(s i, r) λ N j= w jk(s i, t j )X T (t j, r, p)) 2 + (X T (s i, r, p) y(s i, r) λ N j= w jk(s i, t j )X T (t j, r, p)) 2 }, X T (s 0, r, p) = X 0 (r), X T (s 0, r, p) = X 0 (r). (4.5) Ech tril solution X T nd X T employs one feed-forwrd neurl network for wich the corresponding networks re denoted by N nd N, with djustble prmeters p nd p, respectively. The relted tril functions will be in the form [22]: X T (s, r, p) = X(s 0, r) + (s s 0 )N(s, r, p), X T (s, r, p) = X(s 0, r) + (s s 0 )N(s, r, p), (4.6) 5 ISPACS GmbH
6 where N nd N re single-output feed-forwrd neurl networks with djustble prmeters p nd p, respectively. Here s nd r re the network inputs. This solutions by intention stisfies the initil condition in (4.6). According to (4.6) it is stright forwrd to show tht: X T (s, r, p) = N(s, r, p) + (s s 0) N s, X T (s, r, p) = N(s, r, p) + (s s 0 ) N s. (4.7) Now consider multilyer perceptron hving one hidden lyer with H sigmoid units nd liner output unit (Fig., Fig. 2). b s r w w 2 Input w 2 Input w 22 w H 2 b 2 z 2 z z H N Liner w H2 H Fig.. Three lyered perceptron with two input nd N output. b H b s r w w 2 Input w 2 Input w 22 w H 2 b 2 z 2 z H z Liner N w H2 H Fig. 2. Three lyered perceptron with two input nd N output. b H Here we hve: N = H i= v iσ(z i ), z i = w i s + w i2 r + b i, N = H i= v iσ(z i ), z i = w i s + w i2 r + b i, where σ(z) is the sigmoid trnsfer function. The following is obtined: (4.8) N s N s = H i= v iw i σ (z i ), = H i= v iw i σ (z i ), 6 ISPACS GmbH
7 where σ ( z i ) is the first derivtive of the sigmoid function. Also there re mny choices for the sigmoid function, here we choose σ(z) = /( + e z ) since it is possible to derive ll the derivtives of σ(z) in terms of the sigmoid function itself. i.e. σ (z) = σ 2 (z) + σ(z). 5 Exmple To illustrte the technique proposed in this pper, consider the following exmple. For ech fuzzy numbers, we use r = 0, 0.,...,, where we clculte the ccurcy of the method by Eq. (4.5). In the computer simultion of this section, we use the H = 0 sigmoid units in the hidden lyer. Exmple 5.. Consider the following fuzzy liner Fredholm integro-differentil eqution X (s) = ( r, 2 r)(e s s) + 0 tsx(t)dt, X(0) = ( r, 2 r); 0 r, 0 s, t. The exct solution in this cse is given by The tril functions for this problem re X = ( r, 2 r)e s. X T (s; r) = ( r) + s H i= X T (s; r) = (2 r) + s H i= v i +e w i s w i2 r b i, v i +e w i s w i2 r b i. The exct nd obtined solution of fuzzy liner Fredholm integro-differentil eqution in this exmple t s = re shown in Figure 3, lso the error by Eq. (4.5) is.2433e Exct solution Approximte solution Fig. 3. The exct nd pproximte solution for exmple ISPACS GmbH
8 Figs. 4-7 show the convergence property of the computed vlues of the weights Fig. 4. Convergence of the weights w i for exmple Fig. 5. Convergence of the weights w i2 for exmple ISPACS GmbH
9 Fig. 6. Convergence of the weights v i for exmple Fig. 7. Convergence of the weights b i for exmple Conclusion Solving fuzzy integro-differentil eqution (FIDE) by using universl pproximtors (UA), tht is, neurl network model (NNM) is presented in this pper. In this pper, the originl fuzzy integro-differentil eqution is replced by two prmetric liner Fredholm integro-differentil equtions which re then solved numericlly using UAM. The min reson for using neurl networks ws their pplicbility in function pproximtion. Our computer simultion in this pper were performed for three-lyer feedforwrd neurl networks. Since we hd good simultion result even from three-lyer 9 ISPACS GmbH
10 neurl networks, we do not think tht the extension of our NNM to neurl networks with more thn three lyers is n ttrctive reserch direction. References [] S. Abbsbndy, T. A. Virnloo, Ó. López, J. J. Nieto, Numericl methods for fuzzy differentil inclusions, Computers nd Mthemtics with Applictions, 48 (2004) [2] S. Abbsbndy nd T. Allhvirnloo, Numericl solution of fuzzy differentil equtions by Runge-Kutt method, Nonliner Studies, () (2004) [3] S. Abbsbndy, E. Bbolin, M. Alvi, Numericl method for solving liner Fredholm fuzzy integrl equtions of the second kind, Chos Solitons & Frctls, 3 (2007) [4] S. Abbsbndy nd M. Otdi, Numericl solution of fuzzy polynomils by fuzzy neurl network, Applied Mthemtics nd Computtion, 8 (2006) [5] S. Abbsbndy, M. Otdi nd M. Mosleh, Numericl solution of system of fuzzy polynomils by fuzzy neurl network, Informtion Sciences, 78 (2008) [6] T. Allhvirnloo, S. Abbsbndy, O. Sedghtfr nd P. Drbi, A new method for solving fuzzy integro-differentil eqution under generlized differentibility, Neurl Computing nd Applictions, 2 (202) [7] T. Allhvirnloo, N. Ahmdi nd E. Ahmdi, Numericl solution of fuzzy differentil equtions by predictor-corrector method, Informtions Sciences, 77 (2007) [8] T. Allhvirnloo, N.A. Kini nd M. Brkhordri, Towrd the existence nd uniqueness of solutions of second-order fuzzy differentil equtions, Informtions Sciences, 77 (2009) [9] T. Allhvirnloo, M. Khezerloo, O. Sedghtfr nd S. Slhshour, Towrd the existence nd uniqueness of solutions of second-order fuzzy volterr integro-differentil equtions with fuzzy kernel, Neurl Computing nd Applictions. [0] E. Bbolin, H. Sdeghi nd Sh. Jvdi, Numericlly solution of fuzzy differentil equtions by Adomin method, Applied Mthemtics nd Computtion, 49 (2004) ISPACS GmbH
11 [] E. Bbolin, H.S. Goghry, S. Abbsbndy, Numericl solution of liner Fredholm fuzzy integrl equtions of the second kind by Adomin method, Applied Mthemtics nd Computtion, 6 (2005) [2] C.T.H. Bker, A perspective on the numericl tretment of Volterr equtions, J. Comput. Appl. Mth., 25 (2000) [3] P. Blsubrmnim, S. Murlisnkr, Existence nd uniqueness of fuzzy solution for the nonliner fuzzy integro-differentil equtions, Applied mthemtics letters, 4 (200) [4] M.I. Berenguer, D.Gmez, A.I. Grrld-Guillem, M. Ruiz Gln, M.C. Serrno Perez, Biorthogonl systems for solving Volterr integrl eqution systems of the second kind, J. Comput. Appl. Mth., 235 (20) [5] J.F. Bernrd, Use of rule-bsed system for process control, IEEE Contr. System Mg., 8 (988) [6] A.H. Borzbdi nd O.S. Frd, A numericl scheme for clss of nonliner Fredholm integrl equtions of the second kind, Journl of Computtionl nd Applied Mthemtics, 232 (2009) [7] S.S.L. Chng, L. Zdeh, On fuzzy mpping nd control, IEEE Trns. System Mn Cybernet, 2 (972) [8] Y. Chen, T. Tng, Spectrl methods for wekly singulr Volterr integrl equtions with smooth solutions, J. Comput. Appl. Mth., 233 (2009) [9] Y.C. Chen, C.C. Teng, A model reference control structure using fuzzy neurl network, Fuzzy Sets nd Systems, 73 (995) [20] D. Dubois nd H. Prde, Opertions on fuzzy numbers, J. Systems Sci. 9 (978) [2] D. Dubois, H. Prde, Towrds fuzzdifferentil clculus, Fuzzy Sets Systems, 8 (982) [22] S. Effti nd M. Pkdmn, Artificil neurl network pproch for solving fuzzy differentil equtions, Informtion Sciences, 80 (200) ISPACS GmbH
12 [23] M. Friedmn, M. M, A. Kndel, Numericl solutions of fuzzy differentil nd integrl equtions, Fuzzy Sets nd Systems 06 (999) [24] R. Goetschel, W. Vxmn, Elementry clculus, Fuzzy sets Syst., 8 (986) [25] H. Hochstdt, Integrl equtions, New York: Wiley; 973. [26] A. Kufmnn nd M.M. Gupt, Introduction Fuzzy Arithmetic, Vn Nostrnd Reinhold, New York, 985. [27] O. Klev, Fuzzy differentil equtions, Fuzzy Sets Syst. 24 (987) [28] J.P. Kuthen, Continuous time colloction method for Volterr-Fredholm integrl equtions, Numer. Mth. 56 (989) [29] D. R. Kincid, E.W. Cheney, Numericl nlysis: Mthemtics of scientific comuting, third ed., Brooks/Cole, Pcific Grove, CA, [30] G.J. Klir, U.S. Clir, B. Yun, Fuzzy set theory: foundtions nd pplictions, Prentice-Hll Inc.;997. [3] I. E. Lgris nd A. Liks, Artificil neurl networks for solving ordinry nd prtil differentil equtions, IEEE Trnsctions on Neurl Networks 9 (5) (998), September. [32] P. Linz, Anlyticl nd numericl methods for Volterr equtions, SIAM, Phildelphi, PA, [33] M. Mosleh, M. Otdi nd S. Abbsbndy, Evlution of fuzzy regression models by fuzzy neurl network, Journl of Computtionl nd Applied Mthemtics, 234 (200) [34] M. Mosleh, M. Otdi nd S. Abbsbndy, Fuzzy polynomil regression with fuzzy neurl networks, Applied Mthemticl Modelling, 35 (20) [35] M. Mosleh, T. Allhvirnloo nd M. Otdi, Evlution of fully fuzzy regression models by fuzzy neurl network, Neurl Comput nd Applictions, 2 (202) [36] M. Otdi nd M. Mosleh, Simultion nd evlution of dul fully fuzzy liner systems by fuzzy neurl network, Applied Mthemticl Modelling, 35 (20) [37] M. Mosleh nd M. Otdi, Simultion nd evlution of fuzzy differentil equtions by fuzzy neurl network, Applied Soft Computing, 2 (202) ISPACS GmbH
13 [38] M. Mosleh nd M. Otdi, Miniml solution of fuzzy liner system of differentil equtions, Neurl Computing nd Applictions, 2 (202) [39] M. Otdi, M. Mosleh nd S. Abbsbndy, Numericl solution of fully fuzzy liner systems by fuzzy neurl network, Soft Computing, 5 (20) [40] M.L. Puri, D. Rlescu, Fuzzy rndom vribles, J Mth. Anl. Appl., 4 (986) [4] H.H. Sorkun, S. Ylcinbs, Approximte solutions of liner Volterr integrl eqution systems with vrible coefficients, Applied Mthemticl Modelling, 34 (200) [42] Wu Congxin, M Ming, On embedding problem of fuzzy number spces, Prt, Fuzzy Sets nd Systems, 44 (99) [43] L.A. Zdeh, The concept of linguistic vrible nd its ppliction to pproximte resoning, Inform. Sci. 8 (975) ISPACS GmbH
Solution to Fredholm Fuzzy Integral Equations with Degenerate Kernel
Int. J. Contemp. Mth. Sciences, Vol. 6, 2011, no. 11, 535-543 Solution to Fredholm Fuzzy Integrl Equtions with Degenerte Kernel M. M. Shmivnd, A. Shhsvrn nd S. M. Tri Fculty of Science, Islmic Azd University
More informationNUMERICAL SOLUTIONS OF NONLINEAR FUZZY FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF THE SECOND KIND
Irnin Journl of Fuzzy Systems Vol. 12, No. 2, (2015). 117-127 117 NUMERICAL SOLUTIONS OF NONLINEAR FUZZY FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF THE SECOND KIND M. MOSLEH AND M. OTADI Abstrct. In this
More informationSolving Linear Fredholm Fuzzy Integral Equations System by Taylor Expansion Method
Applied Mthemticl Sciences, Vol 6, 212, no 83, 413-4117 Solving Liner Fredholm Fuzzy Integrl Equtions System y Tylor Expnsion Method A Jfrin 1, S Mesoomy Ni, S Tvn nd M Bnifzel Deprtment of Mthemtics,
More informationA Modified ADM for Solving Systems of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 6, 2012, no. 26, 1267-1273 A Modified ADM for Solving Systems of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi nd T. Dmercheli Deprtment of Mthemtics,
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationFredholm Integral Equations of the First Kind Solved by Using the Homotopy Perturbation Method
Int. Journl of Mth. Anlysis, Vol. 5, 211, no. 19, 935-94 Fredholm Integrl Equtions of the First Kind Solved by Using the Homotopy Perturbtion Method Seyyed Mhmood Mirzei Deprtment of Mthemtics, Fculty
More informationArithmetic Mean Derivative Based Midpoint Rule
Applied Mthemticl Sciences, Vol. 1, 018, no. 13, 65-633 HIKARI Ltd www.m-hikri.com https://doi.org/10.1988/ms.018.858 Arithmetic Men Derivtive Bsed Midpoint Rule Rike Mrjulis 1, M. Imrn, Symsudhuh Numericl
More informationFUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS
VOL NO 6 AUGUST 6 ISSN 89-668 6-6 Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom FUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS Muhmmd Zini Ahmd Nor
More informationResearch Article On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation
Journl of Applied Mthemtics Volume 2011, Article ID 743923, 7 pges doi:10.1155/2011/743923 Reserch Article On Existence nd Uniqueness of Solutions of Nonliner Integrl Eqution M. Eshghi Gordji, 1 H. Bghni,
More informationOn the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 2, 28, no. 2, 57-62 On the Decomposition Method for System of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi 1 nd M. Mokhtri Deprtment of Mthemtics, Shhr-e-Rey
More informationAN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
More informationA Bernstein polynomial approach for solution of nonlinear integral equations
Avilble online t wwwisr-publictionscom/jns J Nonliner Sci Appl, 10 (2017), 4638 4647 Reserch Article Journl Homepge: wwwtjnscom - wwwisr-publictionscom/jns A Bernstein polynomil pproch for solution of
More informationRealistic Method for Solving Fully Intuitionistic Fuzzy. Transportation Problems
Applied Mthemticl Sciences, Vol 8, 201, no 11, 6-69 HKAR Ltd, wwwm-hikricom http://dxdoiorg/10988/ms20176 Relistic Method for Solving Fully ntuitionistic Fuzzy Trnsporttion Problems P Pndin Deprtment of
More informationAn improvement to the homotopy perturbation method for solving integro-differential equations
Avilble online t http://ijimsrbiucir Int J Industril Mthemtics (ISSN 28-5621) Vol 4, No 4, Yer 212 Article ID IJIM-241, 12 pges Reserch Article An improvement to the homotopy perturbtion method for solving
More informationAn iterative method for solving nonlinear functional equations
J. Mth. Anl. Appl. 316 (26) 753 763 www.elsevier.com/locte/jm An itertive method for solving nonliner functionl equtions Vrsh Dftrdr-Gejji, Hossein Jfri Deprtment of Mthemtics, University of Pune, Gneshkhind,
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationNew implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations
014 (014) 1-7 Avilble online t www.ispcs.com/cn Volume 014, Yer 014 Article ID cn-0005, 7 Pges doi:10.5899/014/cn-0005 Reserch Article ew implementtion of reproducing kernel Hilbert spce method for solving
More informationQUADRATURE is an old-fashioned word that refers to
World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd
More informationGeneration of Lyapunov Functions by Neural Networks
WCE 28, July 2-4, 28, London, U.K. Genertion of Lypunov Functions by Neurl Networks Nvid Noroozi, Pknoosh Krimghee, Ftemeh Sfei, nd Hmed Jvdi Abstrct Lypunov function is generlly obtined bsed on tril nd
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationJordan Journal of Mathematics and Statistics (JJMS) 11(1), 2018, pp 1-12
Jordn Journl of Mthemtics nd Sttistics (JJMS) 11(1), 218, pp 1-12 HOMOTOPY REGULARIZATION METHOD TO SOLVE THE SINGULAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND MOHAMMAD ALI FARIBORZI ARAGHI (1) AND
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationUndergraduate Research
Undergrdute Reserch A Trigonometric Simpson s Rule By Ctherine Cusimno Kirby nd Sony Stnley Biogrphicl Sketch Ctherine Cusimno Kirby is the dughter of Donn nd Sm Cusimno. Originlly from Vestvi Hills, Albm,
More informationarxiv: v1 [math.na] 23 Apr 2018
rxiv:804.0857v mth.na] 23 Apr 208 Solving generlized Abel s integrl equtions of the first nd second kinds vi Tylor-colloction method Eis Zrei, nd Smd Noeighdm b, Deprtment of Mthemtics, Hmedn Brnch, Islmic
More informationSolutions of Klein - Gordan equations, using Finite Fourier Sine Transform
IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 6 Ver. IV (Nov. - Dec. 2017), PP 19-24 www.iosrjournls.org Solutions of Klein - Gordn equtions, using Finite Fourier
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function
More informationModification Adomian Decomposition Method for solving Seventh OrderIntegro-Differential Equations
IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volume, Issue 5 Ver. V (Sep-Oct. 24), PP 72-77 www.iosrjournls.org Modifiction Adomin Decomposition Method for solving Seventh OrderIntegro-Differentil
More information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More informationApplication of Exp-Function Method to. a Huxley Equation with Variable Coefficient *
Interntionl Mthemticl Forum, 4, 9, no., 7-3 Appliction of Exp-Function Method to Huxley Eqution with Vrible Coefficient * Li Yo, Lin Wng nd Xin-Wei Zhou. Deprtment of Mthemtics, Kunming College Kunming,Yunnn,
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationConstruction of Gauss Quadrature Rules
Jim Lmbers MAT 772 Fll Semester 2010-11 Lecture 15 Notes These notes correspond to Sections 10.2 nd 10.3 in the text. Construction of Guss Qudrture Rules Previously, we lerned tht Newton-Cotes qudrture
More informationTravelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge
More informationComposite Mendeleev s Quadratures for Solving a Linear Fredholm Integral Equation of The Second Kind
Globl Journl of Pure nd Applied Mthemtics. ISSN 0973-1768 Volume 12, Number (2016), pp. 393 398 Reserch Indi Publictions http://www.ripubliction.com/gjpm.htm Composite Mendeleev s Qudrtures for Solving
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More informationA Nonclassical Collocation Method For Solving Two-Point Boundary Value Problems Over Infinite Intervals
Austrlin Journl of Bsic nd Applied Sciences 59: 45-5 ISS 99-878 A onclssicl Colloction Method For Solving o-oint Boundry Vlue roblems Over Infinite Intervls M Mlei nd M vssoli Kni Deprtment of Mthemtics
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationThe Solution of Volterra Integral Equation of the Second Kind by Using the Elzaki Transform
Applied Mthemticl Sciences, Vol. 8, 214, no. 11, 525-53 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/1.12988/ms.214.312715 The Solution of Volterr Integrl Eqution of the Second Kind by Using the Elzki
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationA Numerical Method for Solving Nonlinear Integral Equations
Interntionl Mthemticl Forum, 4, 29, no. 17, 85-817 A Numericl Method for Solving Nonliner Integrl Equtions F. Awwdeh nd A. Adwi Deprtment of Mthemtics, Hshemite University, Jordn wwdeh@hu.edu.jo, dwi@hu.edu.jo
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationResearch Article Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method
Discrete Dynmics in Nture nd Society Volume 202, Article ID 57943, 0 pges doi:0.55/202/57943 Reserch Article Numericl Tretment of Singulrly Perturbed Two-Point Boundry Vlue Problems by Using Differentil
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More informationCLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES
Filomt 27:4 (2013) 649 658 DOI 10.2298/FIL1304649M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED
More informationA Modified Homotopy Perturbation Method for Solving Linear and Nonlinear Integral Equations. 1 Introduction
ISSN 1749-3889 (print), 1749-3897 (online) Interntionl Journl of Nonliner Science Vol.13(212) No.3,pp.38-316 A Modified Homotopy Perturbtion Method for Solving Liner nd Nonliner Integrl Equtions N. Aghzdeh,
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationResearch Article Moment Inequalities and Complete Moment Convergence
Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 2009, Article ID 271265, 14 pges doi:10.1155/2009/271265 Reserch Article Moment Inequlities nd Complete Moment Convergence Soo Hk
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationOn the Generalized Weighted Quasi-Arithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 2039-2048 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School
More informationNUMERICAL METHODS FOR SOLVING FUZZY FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND. Muna Amawi 1, Naji Qatanani 2
Interntionl Journl of Applied Mthemtics Volume 28 No 3 2015, 177-195 ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi: http://dxdoiorg/1012732/ijmv28i31 NUMERICAL METHODS FOR SOLVING
More informationTHIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions
THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, Hui-Hsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No.
More informationExact solutions for nonlinear partial fractional differential equations
Chin. Phys. B Vol., No. (0) 004 Exct solutions for nonliner prtil frctionl differentil equtions Khled A. epreel )b) nd Sleh Omrn b)c) ) Mthemtics Deprtment, Fculty of Science, Zgzig University, Egypt b)
More informationExpected Value of Function of Uncertain Variables
Journl of Uncertin Systems Vol.4, No.3, pp.8-86, 2 Online t: www.jus.org.uk Expected Vlue of Function of Uncertin Vribles Yuhn Liu, Minghu H College of Mthemtics nd Computer Sciences, Hebei University,
More informationPOSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY-VALUE PROBLEMS
Electronic Journl of Differentil Equtions, Vol. 27(27), No. 156, pp. 1 8. ISSN: 172-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu (login: ftp) POSITIVE SOLUTIONS
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationChapter 1. Basic Concepts
Socrtes Dilecticl Process: The Þrst step is the seprtion of subject into its elements. After this, by deþning nd discovering more bout its prts, one better comprehends the entire subject Socrtes (469-399)
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationParametrized inequality of Hermite Hadamard type for functions whose third derivative absolute values are quasi convex
Wu et l. SpringerPlus (5) 4:83 DOI.8/s44-5-33-z RESEARCH Prmetrized inequlity of Hermite Hdmrd type for functions whose third derivtive bsolute vlues re qusi convex Shn He Wu, Bnyt Sroysng, Jin Shn Xie
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 3 Polynomial Interpolation: Numerical Differentiation and Integration.
Advnced Computtionl Fluid Dynmics AA215A Lecture 3 Polynomil Interpoltion: Numericl Differentition nd Integrtion Antony Jmeson Winter Qurter, 2016, Stnford, CA Lst revised on Jnury 7, 2016 Contents 3 Polynomil
More informationAn optimal 3-point quadrature formula of closed type and error bounds
Revist Colombin de Mtemátics Volumen 8), págins 9- An optiml 3-point qudrture formul of closed type nd error bounds Un fórmul de cudrtur óptim de 3 puntos de tipo cerrdo y error de fronter Nend Ujević,
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationLYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN
Electronic Journl of Differentil Equtions, Vol. 203 (203), No. 28, pp. 0. ISSN: 072-669. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LYAPUNOV-TYPE INEQUALITIES FOR
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationAN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir
RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationQuadrature Rules for Evaluation of Hyper Singular Integrals
Applied Mthemticl Sciences, Vol., 01, no. 117, 539-55 HIKARI Ltd, www.m-hikri.com http://d.doi.org/10.19/ms.01.75 Qudrture Rules or Evlution o Hyper Singulr Integrls Prsnt Kumr Mohnty Deprtment o Mthemtics
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationTangent Line and Tangent Plane Approximations of Definite Integral
Rose-Hulmn Undergrdute Mthemtics Journl Volume 16 Issue 2 Article 8 Tngent Line nd Tngent Plne Approximtions of Definite Integrl Meghn Peer Sginw Vlley Stte University Follow this nd dditionl works t:
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationA General Dynamic Inequality of Opial Type
Appl Mth Inf Sci No 3-5 (26) Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dxdoiorg/2785/mis/bos7-mis A Generl Dynmic Inequlity of Opil Type Rvi Agrwl Mrtin Bohner 2 Donl O Regn
More informationResearch Article Composite Gauss-Legendre Formulas for Solving Fuzzy Integration
Hindwi Pulishing Corportion Mthemticl Prolems in Engineering, Article ID 873498, 7 pges http://dx.doi.org/0.55/04/873498 Reserch Article Composite Guss-Legendre Formuls for Solving Fuzzy Integrtion Xioin
More informationThe Hadamard s inequality for quasi-convex functions via fractional integrals
Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges 67 73 ISSN: 5-563 The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz
More informationA Generalized Inequality of Ostrowski Type for Twice Differentiable Bounded Mappings and Applications
Applied Mthemticl Sciences, Vol. 8, 04, no. 38, 889-90 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.988/ms.04.4 A Generlized Inequlity of Ostrowski Type for Twice Differentile Bounded Mppings nd Applictions
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationLinear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System
Pure nd Applied Mthemtics Journl 017; 6(1): 5-13 http://www.sciencepublishinggroup.com/j/pmj doi: 10.11648/j.pmj.0170601.1 ISSN: 36-9790 (Print); ISSN: 36-981 (Online) Liner nd Non-liner Feedbck Control
More informationOrthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationA New Grey-rough Set Model Based on Interval-Valued Grey Sets
Proceedings of the 009 IEEE Interntionl Conference on Systems Mn nd Cybernetics Sn ntonio TX US - October 009 New Grey-rough Set Model sed on Intervl-Vlued Grey Sets Wu Shunxing Deprtment of utomtion Ximen
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More informationAdomian Decomposition Method with Green s. Function for Solving Twelfth-Order Boundary. Value Problems
Applied Mthemticl Sciences, Vol. 9, 25, no. 8, 353-368 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/.2988/ms.25.486 Adomin Decomposition Method with Green s Function for Solving Twelfth-Order Boundry
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationx = b a N. (13-1) The set of points used to subdivide the range [a, b] (see Fig. 13.1) is
Jnury 28, 2002 13. The Integrl The concept of integrtion, nd the motivtion for developing this concept, were described in the previous chpter. Now we must define the integrl, crefully nd completely. According
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More information