British Journal of Mathematics & Computer Science 12(6): 1-10, 2016, Article no.BJMCS.21984 ISSN: 2231-0851 SCIENCEDOMAIN international www.sciencedomain.org Legendre and Chebyshev Polynomials for Solving Mixed Integral Equation M. A. Abdou 1 and M. A. Elsayed 2* 1Department of Mathematics, Faculty of Education, Alexandria University, Egypt. 2Department of Basic Science, High Institute for Engineering, Elshorouk Academy, Egypt. Article Information DOI: 10.9734/BJMCS/2016/21984 Editor(s): (1) Feliz Manuel Minhós, Professor, Department of Mathematics, School of Sciences and Technology, University of Évora, Portugal. Reviewers: (1) Ana Magnolia Marin Ramirez, Cartagena University, Colombia. (2) Grienggrai Rajchakit, Maejo University, Thailand. (3) Hammad Khalil, University of Malakand, Pakistan. Complete Peer review History: http://sciencedomain.org/review-history/12182 Original Research Article Received: 12 September 2015 Accepted: 13 October 2015 Published: 09 November 2015 _______________________________________________________________________________ Abstract In this paper, the solution of mixed integral equation (MIE) of the first and second kind in time and position is discussed and obtained in the space . The kernel of position is established in the logarithmic form, while the kernels͆ͦʞƎ 1of,1 timeʟ Ɛ ̽ areʞ0, ͎continuousʟ, ͎ Ɨ 1 and positive functions in C[0,T]. A numerical method is used to obtain a linear system of Fredholm integral equations (SFIEs). In addition, the solution FIE of the second kind, with singular kernel, is solved, using Legendre polynomials. Moreover, Orthogonal polynomials methods are used to obtain the solution of singular FIE of the first kind. Keywords: Mixed integral equation; contact problem; Legendre polynomial; Krein's method; Chebyshev polynomial. AMS: 45E-65R. 1 Introduction The mathematical physics and contact problems in the theory of elasticity lead to an integral equation of the first or second kind, see [1,2,3]. Mkhitarian and Abdou [4,5] discussed some different methods for solving the FIE of the first kind , with logarithmic kernel [4], and Carleman kernel [5] respectively. More information for solving the integral equations and the fractional partial differential equations with its applications can be found in [6-9]. _____________________________________ *Corresponding author: E-mail: [email protected], [email protected]; Abdou and Elsayed; BJMCS, 12(6): 1-10, 2016; Article no.BJMCS.21984 In this work, we consider the MIE / ͥ 3ͯ4 / (1.1) ȄȄͤ ͯͥ ̀ʚͨ, ʛ͟ ʠ Z ʡ ʚͭ, ʛͭ͘ ͘ + Ȅͤ ́ʚͨ, ʛʚͬ, ʛ͘ = ʞ ʚͨʛ Ǝ ͚∗ʚͬʛʟ = ͚ʚͬ, ͨʛ, Where ∞ 1 tanh ͩ $0\ ͟ʚʛ = ǹ ͙ ͩ͘ , ͝ = √Ǝ1, 2 ∞ ͩ ͯ under the condition ͥ (1.2) Ȅͯͥ ʚͬ, ͨʛͬ͘ = ͊ʚͨʛ The two given functions τ and τ , for are positive, continuous with its derivatives ̀ʚͨ, ʛ ́ʚͨ, ʛ ͨ ∈ ʞ0, ͎ʟ, ͎ Ɨ 1 in the class , and represent the two kernels of Volterra integral term. The bad function 3ͯ4 ʞ ʟ ∞ is called̽ 0, the ͎ kernel of Fredholm integral term, in the domain . The given continuous͟ ʠ functionsZ ʡ , ∈ ʚ0, ʛand belongs, respectively to the class , and the spaceʞƎ1,1 ʟ . The given function is ʚ ͨcontinuousʛ ͚ʚͬʛ with its partial derivatives. The ̽unknownʞ0, ͎ʟ function ͆ͦʞƎ1,1 willʟ be obtained in the͚ spaceʚͬ, ͨʛ The integral equation (1.1), under the conditionʚͬ, ͨʛ (1.2), is investigated from the contact͆ͦʞƎ1,1 problemʟ Ɛ ̽ʞ0, of ͎ʟ ,a ͎ rigid Ɨ 1 surface having an elastic material, where is the displacement magnitude and is Poisson’s coefficient. If a stampʚ́, ʛ of length 2 unit and its surface is describinǵ by the formula , is impressed into an elastic layer surface of a strip by a variable force ͚whose∗ʚͬʛ eccentricity of application , that cases rigid displacement . Here the function͊ʚͨʛ, 0 ≤ ͨ ≤τ ͎represents Ɨ 1, the resistance force of material͙ ʚinͨʛ the domain of contact , throughʚͨʛ the time ̀ʚͨ, andʛ is the external force that supplied through the domain of contacʞƎ1,1t problemʟ to increase theͨ ∈ resistanceʞ0, ͎ʟ of́ theʚͨ, domain.ʛ As in Ref. [10], the kernel of position of (1.1) can be written in the form ∞ ͥ ∞ ΢ΏΜΖ 0 $0\ _\ 3ͯ4 ∞ (1.3) ͟ʚʛ = Ȅͯ ͙ ͩ͘ = Ǝ ln ʺtanh ʺ , = ʠ ʡ , ∈ ʚ0, ʛ. ͦ 0 ͨ Z If ∞ and is very small, so that the condition then we have → ʚͬ Ǝ ͭʛ tanh ͮ ≈ ͮ, _\ ͨZ ͨZ (1.4) ͢͠ ʺ͕ͨ͢ℎ ͨ ʺ = ͢͠ || Ǝ ͘ ,͘ = ͢͠ _ ,͘ = ͢͠ _ Hence, the formula (1.1) becomes / ͥ / (1.5) Ǝ ȄȄͤ ͯͥ ̀ʚͨ, ʛʞln |ͬ Ǝ ͭ| Ǝ ͘ʟʚͭ, ʛͭ͘ ͘ + Ȅͤ ́ʚͨ, ʛʚͬ, ʛ͘ = ͚ʚͬ, ͨʛ, In order to guarantee the existence of a unique solution of Eq. (1.1) or (1.5), under the condition (1.2), we assume the following conditions: (i) The kernel of position satisfies the discontinuous condition ͟ʚ|ͬ Ǝ ͭ|ʛ ͥƟ ͥ ͥ ͦ 3ͯ4 ͦ is a constant. ʤ ͟ ʠʺ ʺʡ ͬ͘ ͭ͘ ʥ Ɨ ̻ , ̻ (ii) ForȄȄͯͥ allͯͥ values Zof the two functions with its derivatives belong to For all values of ͨ, ∈ ʞ0, ͎theʟ two functions ̀ʚͨ, ʛ , ́ʚͨ, ʛ ; for all ͨ, and ∈ ʞ0, satisfy ͎ʟ the following conditions.ʚͨ, ʛ , ́ʚͨ, values ʛ |̀ ʚofͨ, ʛ| Ɨ ̼ |,́ whereʚͨ, ʛ| Ɨ ̾,and are constants.̽ʚʞ0, ͎ʟ Ɛ ʞ0, ͎ʟʛ ∈ ʞ0, ͎ʟ ̼ ̾ (iii) The function and its normality in is defined as ͚ʚͬ, ͨʛ ∈ ͆ͦʞƎ1,1ʟ Ɛ ̽ʞ0, ͎ʟ ͆ͦʞƎ1,1ʟ Ɛ ̽ʞ0, ͎ʟ ͥƟ / ͥ ͦ ͦ ‖∅ʚͬ, ͨʛ‖ = ͕ͬ͡ ͤ/ ʤ ͚ ʚͬ, ʛͬ͘ ʥ ͘ (iv) The unknown functionȄ ͤ Ȅͯͥ satisfies Hölder condition with respect to time and Lipschitz condition with respect to position.ʚͬ, ͨʛ 2 Abdou and Elsayed; BJMCS, 12(6): 1-10, 2016; Article no.BJMCS.21984 In this work, a numerical method is used to obtain SFIEs of the first kind or of the second kind according to on the relation between the derivatives of the two functions and for all the values of , with respect to the time . In section 3, we represent̀ʚͨ, the ʛ unknowńʚͨ, ʛfunction, in the Legendre ∈ polynomialʞ0, ͎ʟ form. In addition, in sectionͨ 4, orthogonal polynomials method is used to discuss the solution of FIE of the first kind in the form of spectral relationships. The stability of the solution is discussed. In section 5, numerical results and general conclusions with many important cases are considered and discussed. 2 Numerical Methods To discuss the solution of (1.5), under (1.2), we divide the interval as , to get ʞ0, ͎ʟ, 0 ≤ ͨ ≤ ͎ Ɨ 1 0 = ͨͤ Ɨ ͨͥ Ɨ & ⋯ Ɨ ͨ = ͎ ; 0 =Ɨ ͨ = ͨ , ͟ = 1,2,3, … , ͈ /Ğ ͥ /Ğ (2.1) ʚ & ʛʚ | | ʛ ʚ ʛ ʚ & ʛ ʚ ʛ ʚ &ʛ Ǝ ȄȄͤ ͯͥ ̀ ͨ , ln ͬ Ǝ ͭ Ǝ ͘ ͭ, ͭ͘ ͘ + Ȅͤ ́ ͨ , ͬ, ͘ = ͚ ͬ, ͨ , ͥ (2.2) Ȅ ͯͥ ʚ ͬ, ͨ & ʛ ͬ͘ = ͊ ʚ ͨ & ʛ Hence, we have & ͥ & ͮͥ (2.3) Ǝ ∑ %Ͱͤ ͩ % ̀ Ƴ ͨ & , ͨ % Ʒ Ȅ ͯͥ ʚ ln | ͬ Ǝ ͭ | Ǝ ͘ ʛ Ƴ ͭ, ͨ % Ʒ ͭ͘ + ∑%Ͱͤ ͩ%́Ƴͨ&, ͨ%ƷƳͬ, ͨ%Ʒ + ͉Ƴℏ% Ʒ = ͚ʚͬ, ͨ&ʛ Where . ℏ % = max ͤ%& ℎ % , ℏ $ = max ͤ$& ℎ$ , ℎͥ = ͨ'ͮͥ Ǝ ͨ'. The values of and are depending on the number of the derivatives of and , for ͩ%, ͊, $ ͊Ȱ ̀ʚͨ, ʛ ́ʚͨ, ʛ , with respect to . For example, if ͨ then we have in the first integral ∈ ʞ0,͎ termʟ of Eq. (2.3), ͨwhere we get ̀ʚͨ, ʛ ∈ ̽ ʞ0, ͎ʟ, ͊ and = 4, ͟ ≅ 4, for #t #x , . ͩͤ = ͦ , ͩͨ = ͦ ; ͩ) = ℎ); ͢ = 1,2,3. ͩ) = 0 ͢ > 4 While, if ͧ then we have for the second term of (2.3), hence #t ́ʚͨ, ʛ ∈ ̽ ʞ0, ͎ʟ ͊Ȱ = 3, ͟ ≅ 3 ͤ = ͦ , ͪͧ = #w and for . More information for the characteristic points and theͦ ; quadratureͩ) = ℎ); ͢ coefficients = 1,2 , ͡ = are 1,2 found in) [11,12].= 0 Using͡ > 3the following notations (2.4) ̀Ƴͨ&, ͨ%Ʒ = ̀&,% , ́ʚͨ&, ͨ$ʛ = ́&,$ , ʚͭ, ͨ)ʛ = )ʚͭʛ , ͚ʚͬ, ͨ(ʛ = ͚(ʚͬʛ The formula (2.4) rewrite in the form ʚ͞,͝,͢,͡=0,1,…,͟ ;0Ɨ͟ ≤͈ʛ. & ͥ & (2.5) Ǝ ∑%Ͱͤ ͩ%̀&,% Ȅͯͥʚln |ͬ Ǝ ͭ| Ǝ ͘ʛ%ʚͭʛͭ͘ + ∑%Ͱͤ ͩ%́&,%%ʚͬʛ = ͚&ʚͬʛ In addition, the boundary condition (2.2) becomes ͥ (2.6) Ȅͯͥ &ʚͬʛͬ͘ = ͊& , 0 Ɨ ͟ ≤ ͈ ; ʚ͊& ̽ͣͧ͢. ʛ. Now, we have the following discussion (1) The formula (2.5) represents a linear system of FIE of the second kind, for all cases when the two functions and have the same derivatives with respect to time . (2) When the function G(t,̀ʚͨ,τ) hasʛ n derivativeśʚͨ, ʛ such that the formula (2.5), in this case,ͨ ∈ takesʞ0, ͎ ʟthe forms ͢ Ɨ ͟ 3 Abdou and Elsayed; BJMCS, 12(6): 1-10, 2016; Article no.BJMCS.21984 ) ͥ ) Ǝ ȕ ͩ%̀),% ǹʚln |ͬ Ǝ ͭ| Ǝ ͘ʛ%ʚͭʛͭ͘ + ȕ ͩ%́),%%ʚͬʛ = ͚)ʚͬʛ, %Ͱͤ ͯͥ %Ͱͤ (2.7) ʚ͢ Ɨ ͟, ͟ = 1,2, … ͈ʛ, and & ͥ & (2.8) Ǝ ∑%Ͱ)ͮͥ ͩ%̀&,% Ȅͯͥʚln |ͬ Ǝ ͭ| Ǝ ͘ʛ%ʚͭʛͭ͘ = ͚&ʚͬʛ Ǝ ∑%Ͱͤ %Ƴͩ%, ́),%, ̀),%Ʒ%ʚͬʛ Hence, the formula (2.7) represents linear S FIEs of the second kind, which can solve using the recurrence relations. After obtaining the solution of the system (2.7), we can obtain the solution of Eq. (2.8), which represents linear SFIEs of the first kind. (2) When the function has n derivatives such that , hence the formula (2.5) leads to Eq.