Chebyshev Polynomial Based Numerical Inverse Laplace Transform Solutions of Linear Volterra Integral and Integro-Differential Equations

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1 Ameican Reseach Jounal of Mathematics ISSN X Oiginal Aticle Volume, Issue, Feb-5 Chebyshev Polynomial Based Numeical Invese Laplace Tansfom Solutions of Linea Voltea Integal and Intego-Diffeential Equations Vinod Misha, Dimple Rani Depatment of Mathematics, Sant Longowal Institute of Engg. & Tech, Longowal Punjab) Abstact: Thee ae enomous occasions when the methods fo finding solutions of integal and integodiffeential equations lead to failue because of difficulty in inveting Laplace tansfom by standad technique. Numeically inveting Laplace tansfom is cost effective in compaison to athe complicated technique of comple analysis. In the pocess of numeical invesion, an odd cosine seies which is ultimately based on Chebyshev polynomial has been used. The adequacy of method is illustated though numeical eamples of convolution type linea Voltea integal equations of second ind which include wealy singula Abel's integal equation and Voltea intego-diffeential equation. Keywods: Voltea integal equation; Voltea Intego-diffeential equation; Numeical Invesion of Laplace tansfom; Gaussian quadatue; Cosine seies; Chebyshev polynomial. I. INTRODUCTION Linea Voltea Integal and Intego-Diffeential equations ae etensively used in the vaious specialties of science and engineeing. These include mathematical physics, chemical inetic, heat conduction, seismology, fluid dynamics, biological models, population dynamics, metallugy and semi-conductos [, &3]. The Voltea Integal equation with a convolution enel is defined by t) f t) dt,, T f ) y ) ) while Voltea Intego-Diffeential equation by u ) u ) t) u ) d, u ) u, ) Thee have been many etant methods fo solving Voltea Integal and Intego-Diffeential equations with convolution enels. Wazwaz[] and Al-Hayani[8] discussed the Adomian polynomial based Laplace Tansfom methods to solve these equations. Yang[3] poposed a method by which solution is epessed in powe seies and to impove the convegence ate applied the Pade appoimant. Babolian-Shamloo[7], Aznam-Hussin[7] and Misha et al.[9] used opeational matices of piecewise constant othogonal function o Haa wavelet to solve these equations. Homotopy petubation method with finite diffeence technique was used by Raftai[] to solve Voltea Intego-Diffeential equations. Zaebnia[4] solved these equations using Sinc function. A modified Taylo seies method has been applied to appoimate the solution of linea Intego-Diffeential equations in [].. Numeical Invese Laplace Tansfom The Laplace tansfom of function is defined by L st f t) F e f t) dt. 3) Coesponding Autho: vinodmisha.@ediffmail.com

2 Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X As usual, the equations ae fist conveted into algebaic equations using Laplace tansfom. The invese Laplace tansfom is then applied and the numeical solution is finally epessed in tems of Chebyshev seies. Applying Laplace tansfoms on both sides of above equations and then using convolution popety in ) & ), educe the equations in the fom f ) F. L The invesion leads to f ) L F. Compehensive liteatue consists of a numbe of methods fo numeically inveting Laplace tansfom suited fo poblems in paticula situations. Fo a detailed suvey of vaious methods fo Laplace tansfom invesion numeically efes to Cohen[], Davies-Matin[5] and Misha[6]. Bellman et al. 966)[] have outlined a method which they deive fom the consideation of Gauss-Legende's quadatue ule. Substituting t e, in [ & ],,, as F is tansfomed fom the inteval to F s / g ) d, Whee g ) f log. F can be numeically inveted using Legende seies of g ) P ) by consideing g) being even function in [,]. This method has slow convegence as the coefficients decease slowly due to the singulaity of g) at [ & ]. Edelyi 943), Papoulis 956) and Lanczos 957) have poposed Legende's function to find the appoimate value of function f t) [, 5, 6, & 8]. In [3] Dubne and Abate 968) have epessed the f t) in tems Fouie cosine tansfoms. Dubin 974) [4] has poposed the tapezoidal ule by esulting appoimation.. Chebyshev Seies based Invesion Chebyshev polynomial is due to Panfnuty Chebyshev b. 8), a Russian. It is basically a class of othogonal polynomials. In sities emaable development to the theoy lead to computation of function appoimations, integals and solution of diffeential equations using Chebyshev polynomials, temed as Chebyshev seies epansion of a function. The ange of poblems coveed includes singula poblem and netwo synthesis [5]. Hee we popose a technique paallel to Papoulis[6] by maing the substitution sin,. e 4) The inteval, ) tansfomed into, and ) f becomes f logsin g ). 5) Now eq. 3) taes the fom sin cos g ) d. F 6) s By setting s ), =,,, we have 3

3 Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X 4. ) cos sin ) ) d g F 7) Hee we assume that ) f g. In case this does not hold then aange it by subtacting a suitable function fom ) g. The function ) g can be epanded in, as the odd cosine seies ) cos ) g 8) and is valid in the inteval., Now we have to detemine the coefficients. cos sin i i i i e e i e e Maing substitution i e, we find that ) ) ) ) ) ) ) ) cos sin = cos ) cos ) cos 9) Substitution of 8) & 9) and using the esult of the othogonality, 4 ) cos / d eq. 7) gives ) ) F =. 4 )

4 Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X 4 F ) ). ) which is the linea system in,,... It can be conveniently put in the mati fom C AB, ) whee the mati C and the coefficient mati A can be obtained by putting the values of and RHS of ) espectively. A Thus 3 can be obtained by solving ), B B A C and hence g ) can be obtained fom eq. 7). In geneal, we compute fist N ) tems of eq.8), that is, the finite seies g As N N,,, in LHS ) cos ). 3) N, g N ) g ). Fom g ) we can detemine f ) f ) cos ). 4) cos DefiningU ), sin, whee ) cos Then cos And f ) e / / e U e )..3 Numeical Eamples Eample Conside the wealy singula Voltea integal equation of second ind [&3] U is the Chebyshev polynomial of second ind of degee 5

5 Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X u t) u ) dt,, 5) t Taing Laplace tansfom on both sides of eq. 5) and using convolution theoem, we obtain L u) F. s s. Following initial value theo- The necessay condition fo compliance of invese ) em, u ) lim sf lim s. s s s s u of eq.6) is that u ) Theefoe, the solution u) can be obtained and can be computed using elation ). Table.. Coefficients in the Epansion of u ) ) 6

6 Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X Table.. Computed Pade appoimant [4/4] and Appoimation Solution fo Eample X Pade appoimant [4/4] method Pesent method Absolute Eo 5.4E E E E E E E E E E E E-3 Eample. Conside the integal equation u t) u t) dt sin. 7) Laplace tansfom on both sides of eq. 7) gives L u) F. s ) 8) The condition u is not satisfied u) lim sf lim s s s. s As pe povision of condition a possible function need to be subtacted fom u) is. A function which taes the value at is. Theefoe, U ) u ) e as U ). Since F F. Theeby, the e solution u) can be obtained by computing the coefficients the elation ). Table.. Coefficients in the Epansion of u ) s 7

7 u) Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X Table.. Computed Eact and Appoimation Solution fo Eample Eact solution Pesent method Absolute eo 3.E E E E E E E E E E E-3 Fig.. Compaison of eact and pesent appoimate solution Eact solution Pesent method Eample3. Conside the Voltea integal with a convolution enel given by [3] u ) cos t) u t) dt sin. 9) As usual taing Laplace tansfom on both sides of eq.9) yield L u) F. s s ) The condition u ) is satisfied as in Eample, that is u ) lim sf lim s. s s s s 8

8 u) Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X Theefoe, the solution u) is feasible. The coefficients computed ) ae shown below. Table3..Coefficients in the Epansion of u) Fig.3. Comapison of Pade appoimant [4/4] and pesent appoimate solution Pade appoimant [4/4] method Pesent method

9 Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X Table 3.. Computed Pade appoimant [4/4] and Appoimation Solution fo Eample3 X Pade appoimant [4/4] Pesent method Absolute eo 4.E E E E E E E E E E E E- Eample4. Conside the fist ode linea Voltea Intego-diffeential equation of the fom [8] u ) t) u ) e u t) dt, ) u If we choose,,,,, u. u. ) Taing Laplace tansfom on both sides of eq. ) and using deivative popety and convolution theoem of Laplace tansfom, s L u) F. s s ) ) The condition u is not satisfied as s u ) lim sf lim s. s s s s In this case we choose a possible function that will be subtacted fom u) to be. A function which taes the value at is e. Theefoe, U ) u ) e as U ). Since F F, the solution s u) eists. The coefficients can be calculated using the elation ). Table 4.. Coefficients in the Epansion of u )

10 u) Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X -5.86E-5 Table4.. Computed Eact and Appoimation Solution fo Eample4 Eact solution Pesent method Absolute eo.e e E E E E E E E E E-3 Fig.4. Compaison of eact solution and pesent appoimate solution Eact solution Pesent method II. CONCLUSION In this pape we have gone though a new insight into the use of Chebyshev polynomials. The seies solutions in tems of Chebyshev polynomials have been used as numeically inveting Laplace tansfom tool fo finding solutions of Voltea integal and intego-diffeential equations. The outcome of fou test poblems have been compaed with eact o Pade appoimants and found to be numeically efficient. REFERENCES [] Bellman, R.E., H.H. Kagiwada and R.E. Kalba, Numeical Invesion of Laplace Tansfoms and Some Invese Poblems in Radiative Tansfe, Jounal of Atmospheic Sciences 3966), [] Cohen, A.M., Numeical Methods fo Laplace Tansfom Invesion, Spinge, 7. [3] Dubne, H. and J. Abate, Numeical Invesion of Laplace Tansfoms by Relating them to the Finite Fouie Cosine Tansfom, Jou. Assoc. Comput. Math ), 5-3. [4] Dubin, F., Numeical Invesion of Laplace Tansfoms: An efficient impovement to Dubne and Abate's method, Comput. Jou ),

11 Ameican Reseach Jounal of Mathematics, Volume, Issue, Febuay 5 ISSN X [5] Davies, Bian and Bian Matin, Numeical Invesion of Laplace Tansfom: A Suvey and Compaison of Methods, Jounal of Computational Physics 33979), -3. [6] Papoulis, A., A New Method of Invesion of the Laplace Tansfom, Quat. Appl. Math. 4956), [7] Babolian, E. and A. Salimi Shamloo, Numeical Solution of Voltea Integal and Intego-diffeential Equations of Convolution Type by using Opeational Matices of Piecewise Constant Othogonal Functions, Jounal of Computational and Applied Mathematics 4 8), [8] Filiz, A., Numeical Method fo Linea Voltea Intego-diffeential Equation with Cash-ap Method, Asian Jounal of Fuzzy and Applied Modeling 4), -. [9] Misha, Vinod, Hapeet Kau and R.C. Mittal, Haa Wavelet Solutions of Cetain Diffeential, Integal and Integodiffeential Equations, Intenational Jounal of Applied Mathematics and Mechanics 8 ), [] Raftai, B., Numeical Solution of Linea Voltea Intego-diffeential Equations: Homotopy Petubation Method and Finite Diffeence Method, Wold Applied Sciences Jounal 9 ), 7-. [] Rashidinia, J. and Ali Tahmasebi, Appoimate Solution of Linea Intego-diffeential Equation using Modified TayloEpansion Method, Wold Jounal of Modeling and Simulation 9 3), [] Wazwaz, A.M., Linea and Nonlinea Integal Equations: Methods and Applications, Spinge,. [3] Yang, Changqing and Jianhua Hou, Numeical Method fo Solving Voltea Integal Equations with a Convolution Kenel, IAENG Intenational Jounal of Applied Mathematics 43 3), -5. [4] Zaebnia, M. and Z. Nipou, Solution of linea Voltea Intego-diffeential Equation via Sinc Functions, Intenational Jounal of Applied Mathematics and Computation ), -. [5] Piessens, Robet, Computing Integal Tansfoms and Solving Integal Equations using Chebyshev Polynomial Appoimations, Jounal of Computational and Applied Mathematics ) 3-4. [6] Misha Vinod, Review of Numeical Invesion of Laplace Tansfoms using Fouie Analysis, Fast Fouie Tansfom and Othogonal Polynomials, Mathematics in Engineeing, Science and Aeospace USA) 54), [7] Aznam, Suazlan Mt and Aman Hussin, Numeical Method fo Invese Laplace Tansfom with Haa Wavelet Opeational Mati, Malaysian Jounal of Fundamental & Applied Sciences 8 ), [8] Al-Hayani, Waleed, Solving nth-ode Intego-Diffeential Equations Using the Combined Laplace Tansfom-Adomian Decomposition Method, Applied Mathematics, 3, 4,