Solving Linear Fredholm Fuzzy Integral Equations System by Taylor Expansion Method
|
|
- Ralf Gardner
- 5 years ago
- Views:
Transcription
1 Applied Mthemticl Sciences, Vol 6, 212, no 83, 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, Urmi Brnch Islmic Azd University, Urmi, Irn Astrct In this pper we intend to offer numericl scheme to solve liner Fredholm fuzzy integrl equtions system of the second kind For this im, we pply Tylor expnsion method to convert the given fuzzy system into liner system in crisp cse Now the solution of this system yields the unknown Tylor coefficients of the solution functions The proposed method is illustrted y n exmple Results re compred with the exct solution y using computer simultions Keywords: Fredholm fuzzy integrl equtions system; Tylor series; Convergence nlysis; Approximte solutions 1 Introduction Integrl equtions re very useful for solving mny prolems in severl pplied fields like mthemticl economics nd optiml control theory Since these equtions usully cn not e solved explicitly, so it is required to otin pproximte solutions There re numerous numericl methods which hve een focusing on the solution of integrl equtions For exmple, Tricomi, in his ook [19], introduced the clssicl method of successive pproximtions for nonliner integrl equtions Vritionl itertion method [12] nd Adomin decomposition method [4] were effective nd convenient for solving integrl equtions Also the Homotopy nlysis method (HAM) ws proposed y Lio [13] nd then hs een pplied in [1] Moreover, some different vlid methods for solving this kind of equtions hve een developed First time, Tylor expnsion pproch ws presented for solution integrl equtions y Knwl nd Liu in [11] nd then hs een extended in [14, 15, 16] In ddition, Bolin et l [3] y using the orthogonl tringulr sis functions, solved some integrl equtions systems Jfri et l [9] pplied Legendre wvelets method to find 1 jfrin5594@yhoocom
2 414 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel numericl solution system of liner integrl equtions In this pper we wnt to propose new numericl pproch to pproximte the solution of liner Fredholm fuzzy integrl equtions system This method converts the given fuzzy system tht supposedly hs n unique fuzzy solution, into crisp liner system For this scope, first the Tylor expnsions of unknown functions re sustituted in prmetric form of the given fuzzy system Then we differentite oth sides of the resulting integrl equtions of the system N times nd lso pproximte the Tylor expnsion y suitle trunction limit This work yields liner system in crisp cse, such tht the solution of the liner system yields the unknown Tylor coefficients of the solution functions An interesting feture of this method is tht we cn get n pproximte of the Tylor expnsion in ritrry point to ny desired degree of ccurcy Here is n outline of the pper In section 2, the sic nottions nd definitions of the integrl eqution nd the Tylor polynomil method re riefly presented Section 3 descries how to find n pproximte solution of the given Fredholm fuzzy integrl equtions system y using proposed pproch Finlly in section 4, we pply the proposed method y n exmple to show the simplicity nd efficiency of the method 2 Preliminries In this section the most sic used nottions in fuzzy clculus nd nd integrl equtions re riefly introduced We strted y defining the fuzzy numer Definition 1 A fuzzy numer is fuzzy set u : R 1 I =[, 1] such tht: i u is upper semi-continuous, ii u(x) = outside some intervl [, d], iii There re rel numers, c : c d, for which: 1 u(x) is monotoniclly incresing on [, ], 2 u(x) is monotoniclly decresing on [c, d], 3 u(x) =1, x c The set of ll fuzzy numers (s given y definition 1 ) is denoted y E 1 [7, 17] Definition 2 A fuzzy numer v is pir (v, v) of functions v(r) nd v(r) : r 1, which stisfy the following requirements: i v(r) is ounded monotoniclly incresing, left continuous function on (, 1] nd right continuous t,
3 Liner Fredholm fuzzy integrl equtions 415 ii v(r) is ounded monotoniclly decresing, left continuous function on (, 1] nd right continuous t, iii v(r) v(r): r 1 A populr fuzzy numer is the tringulr fuzzy numer v =(v m,v l,v u ) where v m denotes the modl vlue nd the rel vlues v l nd v u represent the left nd right fuzziness, respectively The memership function of tringulr fuzzy numer is defined s follows: μ v (x) = x v m v l +1, v m v l x v m, v u +1, v m x v m + v u,, otherwise v m x Its prmetric form is: v(r) =v m + v l (r 1), v(r) =v m + v u (1 r), r 1 Tringulr fuzzy numers re fuzzy numers in LR representtion where the reference functions L nd R re liner 21 Opertion on fuzzy numers We riefly mention fuzzy numer opertions defined y the Zdeh extension principle [21, 22] The following ddition, multipliction nd nonliner mpping of fuzzy numers re necessry to define when deling with fuzzy integrl equtions: μ A+B (z) =mx{μ A (x) μ B (y) z = x + y}, μ f(net) (z) =mx{μ A (x) μ B (y) z = xy}, where A nd B re fuzzy numers, μ () denotes the memership function of ech fuzzy numer, is the minimum opertor, nd f is continuous ctivtion function (such s f(x) = x) of output unit of our fuzzy neurl network The ove opertions on fuzzy numers re numericlly performed on level sets (ie α-cuts) For <α 1, α-level set of fuzzy numer A is defined s: [A] α = {x μ A (x) α, x R}, nd [A] = αɛ(,1] [A]α Since level sets of fuzzy numers ecome closed intervls, we denote [A] α y [A] α =[[A] α l, [A]α u],
4 416 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel where [A] α l nd [A] α u re the lower nd the upper limits of the α-level set [A] α, respectively From intervl rithmetic [2], the ove opertions on fuzzy numers re written for the α-level sets s follows: [A] α +[B] α =[[A] α l, [A]α u ]+[[B]α l, [B]α u ]=[[A]α l +[B] α l, [A]α u +[B]α u ], (1) f([net] α )=f([net] α l, [Net]α u ]) = [f([net]α l ),f([net]α u )], k[a] α = k[[a] α l, [A] α u]=[k[a] α l,k[a] α u], if k, (2) k[a] α = k[[a] α l, [A] α u]=[k[a] α u,k[a] α l ], if k < For ritrry u =(u, u) nd v =(v, v) we define ddition (u + v) nd multipliction y k s [7, 17]: (u + v)(r) =u(r)+v(r), (u + v)(r) =u(r)+v(r), (ku)(r) =ku(r), (kv)(r) =ku(r), if k, (ku)(r) =ku(r), (kv)(r) =ku(r), if k < Definition 3 For ritrry fuzzy numers u, v ɛ E 1 the quntity D(u, v) = sup {mx[ u(r) v(r), u(r) v(r) ]} r 1 is the distnce etween u nd v It is shown tht (E 1,D) is complete metric spce [18] Definition 4 Let f :[, ] E 1 For ech prtition P = {t,t 1,, t n } of [, ] nd for ritrry ξ i ɛ [t i 1,t i ](1 i n), suppose R P = n i=1 f(ξ i )(t i t i 1 ), Δ:=mx{ t i t i 1,i=1,, n} The definite integrl of f(t) over [, ] is f(t)dt = lim Δ R P
5 Liner Fredholm fuzzy integrl equtions 417 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 [7] Also, ( ( f(t, r) dt) = f(t, r) dt) = f(t, r)dt, f(t, r)dt More detils out properties of the fuzzy integrl re given in [7, 1] 22 System of integrl equtions The sic definition of integrl eqution is given in [8] Definition 5 The Fredholm integrl eqution of the second kind is where F (t) =f(t)+λ(ku)(t), (3) (ku)(t) = K(s, t)f (s)ds In Eq (3), K(s, t) is n ritrry kernel function over the squre s, t nd f(t) is function of t : t If the kernel function stisfies K(s, t) =, s>t,we otin the Volterr integrl eqution F (t) =f(t)+λ t K(s, t)f (s)ds (4) In ddition, if f(t) e crisp function then the solution of ove eqution is crisp s well Also if f(t) e fuzzy function we hve Fredholm fuzzy integrl eqution of the second kind which my only process fuzzy solutions Sufficient conditions for the existence eqution of the second kind where f(t) is fuzzy function, re given in [5, 6] Definition 6 The second kind fuzzy Fredhoolm integrl equtions system is in the form F 1 (t) =f 1 (t)+ ( m ) j=1 λ 1j K 1j(s, t)f j (s)ds F i (t) =f i (t)+ ( m ) j=1 λ ij K ij(s, t)f j (s)ds, (5) F m (t) =f m (t)+ m j=1 ( ) λ mj K mj(s, t)f j (s)ds
6 418 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel where t, s nd λ ij (for i,j =1,, m) re rel constnts Moreover, in system (5), the fuzzy function f i (t) nd kernel K i,j (s, t) re given nd ssumed to e sufficiently differentile with respect to ll their rguments on the intervl t, s Also F (t) =[F 1 (t),, F m (t)] T is the solution to e determined Now let (f i (t, r), f i (t, r)) nd (F i (t, r), F i (t, r)) ( r 1; t ) e prmetric form of f i (t) nd F i (t), respectively In order to design numericl scheme for solving (5), we write the prmetric form of the given fuzzy integrl equtions system s follows: F i (t, r) =f i (t, r)+ j=1 ( ) λ ij U i,j(s, r)ds, (6) F i (t, r) =f i (t, r)+ j=1 ( ) λ ij U i,j(s, r)ds,i=1,, m, where nd K i,j (s, t)f j (s, r),k i,j (s, t) U i,j (s, r) = K i,j (s, t)f j (s, r),k i,j (s, t) < K i,j (s, t)f j (s, r),k i,j (s, t) U i,j (s, r) = K i,j (s, t)f j (s, r),k i,j (s, t) <, 23 Tylor series Let us first recll the sic principles of the Tylor polynomil method for solving Fredholm fuzzy integrl equtions system (5) Becuse these results re the key for our prolems therefore we explin them Without loss of generlity, we ssume tht λ i,j K i,j (s, t) λ i,j K i,j (s, t) <, s c i,j,c i,j s
7 Liner Fredholm fuzzy integrl equtions 419 With ove supposition, the system (6) is trnsformed to following form: F i (t, r) =f i (t, r)+ ( m j=1 λ ci,j ij K i,j (s, t)f j (s, r)ds + ) c i,j K i,j (s, t)f j (s, r)ds,i=1,, m F i (t, r) =f i (t, r)+ ( m j=1 λ ci,j ij K i,j (s, t)f j (s, r)ds + ) c i,j K i,j (s, t)f j (s, r)ds (7) Now we wnt to otin the solution of the ove system in the form of F j,n (t, r) = N i= ( 1 i! (i) F j (t, r) t i t=z (t z) i ), t, z, r 1, (8) F j,n (t, r) = N i= ( 1 i! (i) F j (t, r) t i t=z (t z) i ), t, z, r 1, (for j =1,, m) which re the Tylor expnsions of degree N t t = z for the unknown functions F j (t, r) nd F j (t, r), respectively For this scope we differentite ech eqution of system (7), (N +1) times (for p =,, N) with respect to t nd get (p) F i (t,r) = (p) f i (t,r) + t p t ( p m j=1 λ ci,j (p) K i,j (s,t) ij t p (p) F i (t,r) = (p) f i (t,r) + t p t ( p m j=1 λ ci,j (p) K i,j (s,t) ij t p F j (s, r)ds + c i,j (p) K i,j (s,t) t p F j (s, r)ds + c i,j (p) K i,j (s,t) t p where (i =1,, m) For revity, we define elow symols s: F (p) ) F j (s, r)ds ) F j (s, r)ds, (9) j (z, r) := (p) F j (t, r) t p t=z nd F (p) j (z, r) := (p) F j (t, r) t p t=z,j=1,, m The im of this study is determining of the coefficients F (p) j (z, r) nd F (p) j (z, r), (for p =,, N; j =1,, m) in system (9) For this intent, we expnded F j (s, r) nd F j (s, r) in Tylor series t ritrry point z : z for
8 411 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel exmple, z = nd sustituted it s N-th trunction in (9) Now we cn write: F (p) i (, r) = (p) f i (t,r) t p t= + ( m N j=1 q= w(i,j) p,q F (q) j (, r)+ ) N q= w (i,j) p,q F (q) j (, r), (1) F (p) i (, r) = (p) f i (t,r) t p t= + ( m N j=1 q= w(i,j) p,q F (q) j (, r)+ ) N q= w (i,j) p,q F (q) j (, r) where p,q = λ i,j q! ci,j (p) K i,j (s, t) t p t= (s ) q ds, p, q =,, N, nd w (i,j) p,q = λ i,j q! c i,j (p) K i,j(s, t) t p t= (s ) q ds, i, j =1,, m Consequently, the mtrix form of expression (1) cn e written s follows: where Y =[F 1 (, r),, F (N) 1 (, r), F 1 (, r),, F (N) 1 (, r), WY = E, (11),F m (, r),, F m (N) (, r), F m (, r),, F (N) m (, r)], E =[ f 1 (, r),, (N) f 1 (t, r) t N t=, f 1 (, r),, (N) f 1 (t, r) t N t=, f m (, r),, (N) f m (t, r) t N t=, f m (, r),, (N) f m (t, r) t N t= ] nd W (1,1) W (1,m) W = W (m,1) W (m,m)
9 Liner Fredholm fuzzy integrl equtions 4111 Prochil mtrices W (i,j), (for i,j =1,, m) re defined with following elements: W (i,j) 1,1 W (i,j) 1,2 W (i,j) =, W (i,j) 2,1 W (i,j) 2,2 where W (i,j) 1,1 = W (i,j) 2,2 =, 1,1,N 1 1, 1,1 1 1,N 1 N 1,1 N 1,N 1 1 N 1, N, N,1 N,N 1,N 1,N w(i,j) N 1,N N,N 1, W (i,j) 1,2 = W (i,j) 2,1 =,,1,N 1 1, 1,1 N 1, 1,N 1 N 1,1 N 1,N 1 N, N,1 N,N 1,N 1,N N 1,N N,N 3 Convergence nlysis In this section we proved tht the ove numericl method convergence to the exct solution of fuzzy system (5) Theorem 1 Let F j,n (t) nd F j,n (t) (for j =1,, m) e Tylor polynomils of degree n tht their coefficients hve een produced y solving the liner system (11) Then these polynomils converge to the exct solution of the fuzzy Fredholm integrl equtions system (5), when N + Proof Consider the system (5) Since, the series (8) converge to F j (t, r) nd F j (t, r) (for j =1,, m) respectively, then we conclude tht:
10 4112 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel F in (t, r) =f i (t, r)+ ( m j=1 λ ci,j ij K i,j (s, t)f jn (s, r)ds + ) c i,j K i,j (s, t)f jn (s, r)ds F in (t, r) =f i (t, r)+ ( m j=1 λ ci,j ij K i,j (s, t)f jn (s, r)ds + ) c i,j K i,j (s, t)f jn (s, r)ds, (12) where (i =1,, m) nd it holds tht F j (t, r) = lim F jn(t, r), nd F j (t) = lim F jn(t, r) N N We defined the error function e N (t, r) y sutrcting Eqs (12)-(7) s follows: where e N (t, r) = e in (t, r) = ( F i (t, r) F in (t, r) ) + e i,n (t, r), (13) i=1 e i,n (t, r) =e i,n (t, r)+e i,n (t, r), j=1 λ ij ( ci,j K i,j (s, t)(f j (s, r) F jn (s, r))ds ) nd + j=1 e in (t, r) = ( F i (t, r) F in (t, r) ) + ( ) λ ij c i,j K i,j (s, t)(f j (s, r) F jn (s, r))ds, j=1 λ ij ( ci,j K i,j (s, t)(f j (s, r) F jn (s, r))ds ) + j=1 ( ) λ ij c i,j K i,j (s, t)(f j (s, r) F jn (s, r))ds, We must prove when N +, the error function e N (t) ecomes to zero Hence we proceed s follows: ( e N e in = e in + e in ein + e in ) i=1 i=1 i=1
11 Liner Fredholm fuzzy integrl equtions 4113 ( (F i (t, r) F in (t, r)) + (F i (t, r) F in (t, r)) ) + i=1 ( λ i,j i=1 j=1 k i,j ( F j (s, r) F jn (s, r) + F j (s, r) F jn (s, r) )ds) Since k i,j is ounded, therefore (F j (s, r) F jn (s, r)) nd (F j (s, r) F jn (s, r)) imply tht e N nd proof is completed 4 Numericl exmples In this section, we present n exmple of liner Fredholm fuzzy integrl equtions system nd results will e compred with the exct solutions Exmple 41 Consider the system of Fredholm fuzzy integrl equtions with: f 1 (t, r) = 14t2 (r 2) 3 + 3t2 (r 3 2) 4 t(r 2) + 9rt2 (r 4 +2), 4 f 1 (t, r) =rt 27t2 (r 3 2) 4 14rt2 3 rt2 (r 4 +2), 4 f 2 (t, r) = 8(t2 + 1)(r 2) 3 t(3r 3 6) + 9(r3 2)(t 2) r(r4 + 2)(t 2) 2, 1 f 2 (t, r) =t(r 5 +2r) 141(r3 2)(t 2) 2 1 8r(t2 +1) 3 3r(r4 + 2)(t 2) 2, 1 kernel functions K 1,1 (s, t) =t 2 (1 + s), K 1,2 (s, t) =t 2 (1 s 2 ), K 2,1 (s, t) =s(1 + t 2 ) nd K 2,2 (s, t) =(t 2) 2 (1 s 3 ), s, t 2,
12 4114 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel nd =, =2, λ i,j =1(for i,j =1, 2) The exct solution in this cse is given y F 1 (t, r) =t(2 r), F 1 (t, r) =tr, F 2 (t, r) =t(6 3r 3 ) nd F 2 (t, r) =t(r 5 +2r) In this exmple we ssume tht z = Using Eqs (1)-(11), the coefficients mtrix W is clculted s following: W 1,1 W 1,2 W =, W 2,1 W 2,2 where 1 W 1,1 = 1 1,W1,2 =, W 2,1 = 2 8 nd W 2, = With using of ove mtrices, we cn rewrite the liner system (11) s follows: F 1 (,r) F 1(,r) F 1 (,r) r F 1 W (,r) = r 2 F 2 (,r) 6 F 2 (,r) 5 r r3 + 76r F 2 (,r) r r3 22r F 5 r r3 64 r (,r) 94 5 r r r
13 Liner Fredholm fuzzy integrl equtions 4115 The vector solution of ove liner system is: F 1 (,r) F 1 (,r) F 1 (,r) r F 1(,r) = 2 r F 2 (,r) F 2 (,r) r 5 +2r F 2 (,r) F 2 (,r) 6 3r 3 As showing in Figs 1 nd 2, fter propgting this solution in Eq (8) the clculted solution is equl to exct solution In other words, with using of this method we cn find the nlyticl solution for this kind of equtions system, if the exct solution of given prolem e polynomil r 1 8 F 1 (s,r) F 1 (s,r) t Fig 1 F 1 (s, r) nd F 1 (s, r) for Exmple F 2 (s,r) F 2 (s,r) r t Fig 2 F 2 (s, r) nd F 2 (s, r) for Exmple 41
14 4116 A Jfrin, S Mesoomy Ni, S Tvn nd M Bnifzel 5 Conclusions In some cses, n nlyticl solution cn not e found for integrl equtions system Therefore, numericl methods hve een pplied In this pper we hve worked out computtionl method to pproximting solution of Fredholm fuzzy integrl equtions system of the second kind In this study the present course is method for computing unknown Tylor coefficients of the solution functions Consider tht to get the est pproximting solutions of the given fuzzy equtions, the trunction limit N must e chosen lrge enough An interesting feture of this method is finding the nlyticl solution for given system, if the exct solution e polynomils of degree N or less thn N The nlyzed exmple illustrted the ility nd reliility of the present method References [1] S Asndy, Numericl solution of integrl eqution: Homotopy perturtion method nd Adomin s decomposition method, Applied Mthemtics nd Computtion, 173 (26), [2] G Alefeld nd J Herzerger, Introduction to Intervl Computtions, Acdemic Press, New York, 1983 [3] E Bolin, Z Msouri nd S Htmzdeh-Vrmzyr, A direct method for numericlly solving integrl equtions system using orthogonl tringulr functions, Int J Industril Mth 2 (29), [4] E Bolin, H Sdeghi Goghry nd S Asndy, Numericl solution of liner Fredholm fuzzy integrl equtions of the second kind y Adomin method, Applied Mthemtics nd Computtion, 161 (25), [5] W Congxin nd M Ming, On the integrls, series nd integrl equtions of fuzzy set-vlued functions J Hrin Inst Technol 21 (199), 9-11 [6] M Friedmn, M M nd A Kndel, Numericl solutions of fuzzy differentil nd integrl equtions, Fuzzy Sets nd Systems 16 (1999), [7] R Goetschel nd W Voxmn, Elementry clculus, Fuzzy Sets nd Systems, 18 (1986), [8] H Hochstdt, Integrl equtions, New York, Wiley, 1973 [9] H Jfri, H Hosseinzdeh nd S Mohmdzdeh, Numericl solution of system of liner integrl equtions y using Legendre wvelets, Int J Open Prolems Compt Mth 5 (21), 63-71
15 Liner Fredholm fuzzy integrl equtions 4117 [1] O Klev, Fuzzy differentil equtions, Fuzzy Sets nd Systems, 24 (1987), [11] RP Knwl nd KC Liu, A Tylor expnsion pproch for solving integrl equtions, Int J Mth Educ Sci Technol 2 (1989), [12] X Ln, Vritionl itertion method for solving integrl equtions, Comput Mth Appl 54 (27), [13] SJ Lio, Beyond Perturtion: Introduction to the Homotopy Anlysis Method, Chpmn Hll/CRC Press, Boc Rton, 23 [14] K Mleknejd nd N Aghzdeh, Numericl solution of Volterr integrl equtions of the second kind with convolution kernel y using Tylor-series expnsion method, Applied Mthemtics nd Computtion, 161 (25), [15] S Ns, S Ylcins nd M Sezer, A Tylor polynomil pproch for solving high-order liner Fredholm integrodifferentil equtions, Int J Mth Educ Sci Technol 31 (2), [16] S Ns, S Ylcins nd M Sezer, A Tylor polynomil pproch for solving high-order liner Fredholm integrodifferentil equtions, Int J Mth Educ Sci Technol 31 (2), [17] HT Nguyen, A note on the extension principle for fuzzy sets, J Mth Anl Appl 64 (1978), [18] ML Puri nd D Rlescu, Fuzzy rndom vriles, J Mth Anl Appl 114 (1986), [19] FG Tricomi, Integrl equtions, Dover Pulictions, New York, 1982 [2] S Ylcins nd M Sezer, The pproximte solution of high-order liner Volterr Fredholm integro-differentil equtions in terms of Tylor polynomils, Applied Mthemtics nd Computtion, 112 (2), [21] LA Zdeh, Towrd generlized theory of uncertinty (GTU) n outline, Informtion Sciences, 172 (25), 1-4 [22] LA Zdeh, The concept of liguistic vrile nd its ppliction to pproximte resoning: Prts 1-3, Informtion Sciences, 8 (1975), , ; 9 (1975) 43-8 Received: Mrch, 212
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 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 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 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 informationFuzzy Fredholm integro-differential equations with artificial neural networks
Avilble online t www.ispcs.com/cn 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,
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 informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
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 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 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 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 informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
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 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 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 informationCOSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature
COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III - Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting
More informationSolution of First kind Fredholm Integral Equation by Sinc Function
World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Solution of First kind Fredholm Integrl Eqution y Sinc Function Khosrow Mleknejd, Rez Mollpoursl,Prvin
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 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 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 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 informationSection 6.1 Definite Integral
Section 6.1 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 e determined
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 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 informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
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 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 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 informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
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 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 informationResearch Article Analytical Solution of the Fractional Fredholm Integrodifferential Equation Using the Fractional Residual Power Series Method
Hindwi Compleity Volume 7, Article ID 457589, 6 pges https://doi.org/.55/7/457589 Reserch Article Anlyticl Solution of the Frctionl Fredholm Integrodifferentil Eqution Using the Frctionl Residul Power
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 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 informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
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 informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
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 informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl
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 informationDiscrete Least-squares Approximations
Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationApplication Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem
Applied nd Computtionl Mthemtics 5; 4(5): 369-373 Pulished online Septemer, 5 (http://www.sciencepulishinggroup.com//cm) doi:.648/.cm.545.6 ISSN: 38-565 (Print); ISSN: 38-563 (Online) Appliction Cheyshev
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 informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationAn Alternative Approach to Estimating the Bounds of the Denominators of Egyptian Fractions
Leonrdo Journl of Sciences ISSN 58-0 Issue, Jnury-June 0 p. -0 An Alterntive Approch to Estimting the Bounds of the Denomintors of Egyptin Frctions School of Humn Life Sciences, University of Tsmni, Locked
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 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 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 information10 D. Chakraborty, D. Guha / IJIM Vol. 2, No. 1 (2010) 9-20
Aville online t http://ijim.sriu.c.ir Int. J. Industril Mthemtics Vol., No. (00) 9-0 Addition of Two Generlized Fuzzy Numers D. Chkrorty, D. Guh Deprtment of Mthemtics, IIT-Khrgpur Khrgpur-730, Indi Received
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 informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
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 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 informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
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 informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationWENJUN LIU AND QUÔ C ANH NGÔ
AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous
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 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 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 informationSOME INTEGRAL INEQUALITIES OF GRÜSS TYPE
RGMIA Reserch Report Collection, Vol., No., 998 http://sci.vut.edu.u/ rgmi SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE S.S. DRAGOMIR Astrct. Some clssicl nd new integrl inequlities of Grüss type re presented.
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 informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
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 informationA Companion of Ostrowski Type Integral Inequality Using a 5-Step Kernel with Some Applications
Filomt 30:3 06, 360 36 DOI 0.9/FIL6360Q Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: http://www.pmf.ni.c.rs/filomt A Compnion of Ostrowski Type Integrl Inequlity Using
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationarxiv: v2 [math.nt] 2 Feb 2015
rxiv:407666v [mthnt] Fe 05 Integer Powers of Complex Tridigonl Anti-Tridigonl Mtrices Htice Kür Duru &Durmuş Bozkurt Deprtment of Mthemtics, Science Fculty of Selçuk University Jnury, 08 Astrct In this
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 informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationDEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
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 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 informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationHarmonic Mean Derivative - Based Closed Newton Cotes Quadrature
IOSR Journl of Mthemtics (IOSR-JM) e-issn: - p-issn: 9-X. Volume Issue Ver. IV (My. - Jun. 0) PP - www.iosrjournls.org Hrmonic Men Derivtive - Bsed Closed Newton Cotes Qudrture T. Rmchndrn D.Udykumr nd
More informationMeixner s Polynomial Method for Solving Integral Equations
Article Interntionl Journl of Modern Mthemticl Sciences, 15, 13(3): 91-36 Interntionl Journl of Modern Mthemticl Sciences Journl homepge:www.modernscientificpress.com/journls/ijmms.spx ISSN:166-86X Florid,
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationAMATH 731: Applied Functional Analysis Fall Some basics of integral equations
AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)
More informationGreen function and Eigenfunctions
Green function nd Eigenfunctions Let L e regulr Sturm-Liouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationAn Overview of Integration
An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is
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 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 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 informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More information221B Lecture Notes WKB Method
Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationOrthogonal Polynomials and Least-Squares Approximations to Functions
Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
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 informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
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 information1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More information