Microsoft Word - 126-134 Mathematics | 126 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 The Approximation Solution of a Nonlinear Parabolic Boundary Value Problem Via Galerkin Finite Elements Method with Crank-Nicolson Jamil Amir Al-Hawasy Jhawassy17@uomustansiriyah.edu.iq Marwa Ahmed Jawad hawasy20@yahoo.com Department of Mathematics, College of Science, Mustansiriyah University, Bahgdad, Iraq. Article history: Received 16 July 2018, Accepted 3 September 2018, Published December 2018 Abstract This paper deals with finding the approximation solution of a nonlinear parabolic boundary value problem (NLPBVP) by using the Galekin finite element method (GFEM) in space and Crank Nicolson (CN) scheme in time, the problem then reduces to solve a Galerkin nonlinear algebraic system(GNLAS). The predictor and the corrector technique (PCT) is applied here to solve the GNLAS, by transforms it to a Galerkin linear algebraic system (GLAS). This GLAS is solved once using the Cholesky method (CHM) as it appears in the matlab package and once again using the Cholesky reduction order technique (CHROT) which we employ it here to save a massive time. The results, for CHROT are given by tables and figures and show the efficiency of this method, from other sides we conclude that the both methods are given the same results, but the CHROT is very fast than the CHM. Keywords: nonlinear parabolic boundary value problem, Galerkin finite element methods, Crank-Nicolson. 1. Introduction In the last decades many researchers interested to study the solution of boundary value problems (bvps) in general and the solution of NLPBVP in particular, there are many different methods for solving the NLPBVP, f.g. in 2000, Karlsen and Riserbo used a corrected operator Splitting method [1], Pao in 2001 used the time period solutions [2], in 2006, Alam and etc used the simultaneous space–time adaptive wavelet method [3]. Timothy in 2010 studied the explicit and implicit difference method [4], in 2011 Ghoreishi and Ismail are used the Homotopy Perturbation Method (HPM) [ 5], and many others. The study of the solution for the parabolic bvp using the finite element method (FEM) back to the beginning of the 17th century, and are studied from many researchers so as Douglas and Dupont [6], in 1993 Reddy introduced in his book an introduction to the FEM was applied to linear, one and two-dimensional problems of engineering and applied sciences [7]. In 1997-2006 Thomee [8] studied the GFEM with backward Euler method for nonlinear parabolic bvp. According to these studied it was important in this paper to study the approximate solution for NLPBVP using the GFEM method for the space variable and the CN scheme for the time variable. This paper starts with give a description of proposed NLPBVP and its weak form (wf). The approximation solution of the problem is obtained by discretize the wf by using the GFEM for the space variable and the CN Scheme for the time variable, the problem then reduces to solve a GNLAS which transforms it to a LAS which is solved once using the CHM and once again Mathematics | 127 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 using that we give it the name CHROT to save a massive time, which is explained in a two steps formula. Finally, illustrative examples are given to solve different problems using MATLAB R2013a software, the results are given by tables and by figures and are show the efficiency of this method, and are show that the CHROT is very fast to solve the linear algebraic system than the CHM. Definition 1, [9]: A point π‘₯βˆ— ∈ 𝑆 βŠ‚ ℝ is said to be fixed point of a given function π‘ž: 𝑆 β†’ ℝ , if π‘ž π‘₯ βˆ— π‘₯βˆ—. Definition 2, [9]: A function π‘ž: 𝑆 βŠ‚ ℝ β†’ ℝ is said to be contractive on 𝑆, if for each π‘₯, 𝑦 ∈ 𝑆: β€–π‘ž π‘₯ π‘ž 𝑦 β€– 𝛽‖π‘₯ 𝑦‖ , where 0 𝛽 1 is a constant. Theorem 3, [9]: A contractive function π‘ž on a complete normed space 𝑆 has a unique fixed point π‘₯ βˆ— in . Theorem 4, [9]: Let β€–βˆ™β€– is a norm in ℝ and 𝑆 βŠ‚ ℝ . If π‘ž: 𝑆 β†’ ℝ is contractive function on 𝑆, and one of the following is satisfied: (1) π‘ž π‘₯ ∈ 𝑆, βˆ€ π‘₯ ∈ 𝑆 . (2) 𝑆 π‘₯|β€–π‘₯ 𝑦‖ πœ‡ and β€–π‘ž 𝑦 𝑦‖ 1 𝛽 πœ‡ . (3) 𝑆 π‘₯|β€–π‘₯ π‘₯ βˆ— β€– πœ‡ , where π‘₯ βˆ— is a fixed point of π‘ž Then π‘₯ ∈ 𝑆 , where π‘₯ is the 𝑙 th iterative value of π‘₯ . Theorem 5, [9]: Let β€–βˆ™β€– is a norm in ℝ and 𝑆 be a closed subset of ℝ . If π‘ž: 𝑆 β†’ ℝ is contractive function on, and π‘₯ ∈ 𝑆, then (1) π‘₯ is converge to a fixed point π‘₯βˆ— ∈ 𝑆 (2) π‘₯ βˆ— is a unique in 𝑆 3. Description of the (NLPBVPCC) Let W = π‘₯ π‘₯ , π‘₯ ∈ ℝ : 0 π‘₯ , π‘₯ 1 , with Lipischitz boundary βˆ‚W , and let I 0, T , 0 T ∞ , and P W I. The nonlinear parabolic equation is given by: - 𝑒 βˆ†π‘’ 𝐻 οΏ½βƒ—οΏ½, 𝑑, 𝑒 , in P (1) with the boundary condition (b.c) 𝑒 οΏ½βƒ—οΏ½, 𝑑 0 , on βˆ‚π‘Š I (2) and the initial condition (i.c) 𝑒 οΏ½βƒ—οΏ½, 0 𝑒 οΏ½βƒ—οΏ½ , in W (3) where 𝑒 𝑒 οΏ½βƒ—οΏ½, 𝑑 , βˆ†π‘’ βˆ‘ is the Laplace operator and 𝐻 ∈ 𝐢 π‘Š . In this work the inner product and norm in L W will be denoted by βˆ™ ,βˆ™ and β€–βˆ™β€– , the inner product and norm in Sobolev space V H W will be denoted by βˆ™ , βˆ™ and β€–βˆ™β€– , the duality bracket between V and its dual Vβˆ— will be denoted by βŒ©βˆ™ ,βˆ™βŒͺ and β€–βˆ™β€– be the norm in L P . Now, the wf of (1-3) is given by: βŒ©π‘’ , πœ‰ βŒͺ βŒ©π›»π‘’ , π›»πœ‰βŒͺ = 𝐻 𝑒 , πœ‰ , βˆ€ πœ‰ ∈ 𝑉 a.e on I (4) 𝑒 0 , πœ‰ = 𝑒 , πœ‰ , in W (5) with 𝑒 belongs to 𝑉 and to 𝐿 W since 𝑉 βŠ‚ 𝐿 W . Assumptions: (1) for some positive constants 𝛾 , 𝛾 and for each πœ‚ , πœ‚ ∈ 𝑉 & 𝑑 ∈ I Μ… , the following inequality are satisfies | π›»πœ‚ , π›»πœ‚ | 𝛾 β€–π›»πœ‚ β€– β€–π›»πœ‚ β€– π›»πœ‚ , π›»πœ‚ 𝛾 β€–π›»πœ‚ β€– Mathematics | 128 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 (2) the function H is of a Carathéodory type on P ℝ , satisfies the following sub linearity and Lipischitz conditions |𝐻 οΏ½βƒ—οΏ½, 𝑑, 𝑒 | 𝛿 οΏ½βƒ—οΏ½, 𝑑 𝑐 |𝑒|, where 𝑐 0 , (οΏ½βƒ—οΏ½, 𝑑 ∈ 𝑃 , , 𝑒 ∈ ℝ and 𝛿 ∈ 𝐿 𝑃, ℝ |𝐻 οΏ½βƒ—οΏ½, 𝑑, 𝑒 𝐻 οΏ½βƒ—οΏ½, 𝑑, 𝑒 | 𝐿|𝑒 𝑒 |, where (π‘₯, 𝑑 ∈ 𝑃, 𝑒 , 𝑒 ∈ ℝ, L is a Lipischitz constant. 4. Discretization of the Continuous Equation: The wf of (4-5) is discretized by using the GFEM, as follows, let the domain W is a polyhedron. For every integer 𝑛, let π‘Š be an admissible regular traingulation of W into closed disimplices [8], 𝐼 be a subdivision of the interval IΜ… into 𝑁𝑇 𝑛 intervals, where 𝑑 𝑑 , 𝑑 , of equal length βˆ†π‘‘ 𝑇 𝑁𝑇⁄ , let 𝑂 π‘Š 𝐼 and 𝑉 βŠ‚ 𝑉 𝐻 W be the space of continuous piecewise affine in W. The Discrete state equation (DSEq) of the wf (4-5) is obtained after using the CN formula and is given by 𝑒 𝑒 , πœ‰ + βˆ†π‘‘ 𝛻𝑒 , π›»πœ‰ = βˆ†π‘‘ 𝐻 𝑑 , 𝑒 , πœ‰ , j=0,1,...,NT-1 (6) 𝑒 , πœ‰ 𝑒 , πœ‰ (7) where πœ‰ ∈ 𝑉 , 𝑒 𝑒 𝑒 , 𝑑 𝑑 𝑑 , j=0,1 ...,NT, 𝑒 ∈ 𝑉 π‘Žπ‘›π‘‘ 𝑒 𝑒 𝑑 ∈ 𝑉 , βˆ€ 𝑗 0,1, … , 𝑁𝑇. 5. The Approximation Solution of the Nonlinear Parabolic Equation: To find the approximation solution (app.sol) 𝑒 𝑒 , 𝑒 , … , 𝑒 of (6-7) by using the GFEM, the following procedure can be used: (1) For fixed any 𝑗 , 0 𝑗 𝑁𝑇 1 , let πœ‰ , 𝑖 1,2, … , 𝑁 , π‘€π‘–π‘‘β„Ž πœ‰ οΏ½βƒ—οΏ½ 0 , π‘œπ‘› πœ•π‘Š be a continuous piecewise affine finite basis of 𝑉 in W , then for any i=1,2,...,N and 𝑒 , 𝑒 ∈ 𝑉 (6-7) can be rewritten as: 𝑒 𝑒 , πœ‰ βˆ†π‘‘ 𝛻𝑒 , π›»πœ‰ βˆ†π‘‘ 𝐻 𝑑 , 𝑒 𝑒 , πœ‰ (8) 𝑒 , πœ‰ 𝑒 , πœ‰ , πœ‰ ∈ 𝑉 , (9) (2) Using the Galerkin method [8], with the basis πœ‰ , πœ‰ , … , πœ‰ of 𝑉 , one has 𝑒 βˆ‘ 𝑋 πœ‰ , , 𝑒 βˆ‘ 𝑋 πœ‰ , and 𝑒 βˆ‘ 𝑋 πœ‰ where , 𝑋 𝑋 𝑑 , for each 𝑗 0,1, … , 𝑁𝑇 are unknown constants to be determine. (3) Substitute 𝑒 π‘Žπ‘›π‘‘ 𝑒 in (8) to get the following nonlinear algebraic system π‘Œ βˆ†π‘‘π‘ 𝑋 π‘Œ βˆ†π‘‘π‘ 𝑋 𝑏 𝑑 , 𝑗 0,1, … , 𝑁𝑇 1 (10) and substituting 𝑒 in (9) to get the following linear algebraic system π‘Œπ‘‹ 𝑏 (11) where π‘Œ 𝑦 , 𝑦 πœ‰ , πœ‰ , 𝑍 𝑧 , 𝑧 π›»πœ‰ , π›»πœ‰ , 𝑋 π‘₯ , π‘₯ , … , π‘₯ , 𝑉 πœ‰ , πœ‰ , … , πœ‰ , οΏ½βƒ—οΏ½ 𝑏 , 𝑏 βˆ†π‘‘ 𝐻 𝑉 𝑋 𝑉 𝑋 , πœ‰ and 𝑏 𝑏 , 𝑏 𝑒 , πœ‰ , βˆ€ 𝑖, π‘˜ 1,2, … , 𝑁. System (10-11) has a unique solution [10]. To solve it, the linear algebraic system (11) is solved at first to get 𝑋 , then to solve the nonlinear system (10) the PCT is used here [8], as follows : For each 𝑗 0 𝑗 𝑁𝑇 1 we predict at first the value 𝑋 by using the Mathematics | 129 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 explicit form (just the value of 𝑋 ) in the component of 𝑏 in the right hand side (RHS) of (10) , then by substitute 𝑋 𝑋 , in the component of 𝑏 in the RHS of (10), makes system (10) linear, solving it w.r.t 𝑋 to get the corrector solution , this procedure can be repeated (more than one time if we need) by substitute the corrector solution 𝑋 𝑋 in the RHS of the linear system (10) and solve it again w.r.t 𝑋 , for fixed 𝑗 , to get a new corrector solution). Hence the corrector equation described as follows: 𝑒 𝑒 , πœ‰ βˆ†π‘‘ 𝛻𝑒 𝛻𝑒 , π›»πœ‰ βˆ†π‘‘ 𝐻 𝑒 𝑒 , πœ‰ (12) where 𝑒 ∢ 𝑒 is the predictor solution at the iteration 𝑙 1, 𝑒 ∢ 𝑒 is its corresponding corrector solution at the iteration 𝑙 and 𝑒 𝑒 is the known corrector solution for the previous step j, i.e. (12) can be written as : 𝑒 π‘ž 𝑒 (13) Theorem 6: The discrete state Equation (6-7) with fixed point and for βˆ†π‘‘ sufficiently small has a unique solution 𝑒 𝑒 , 𝑒 , … , 𝑒 , and the sequence of corrector solutions is convergence in ℝ. Proof: Let 𝑒 𝑒 , … , 𝑒 , … , 𝑒 and 𝑣 𝑣 , … , 𝑣 , … , 𝑣 are two solutions of (12) , i.e. (𝑒 𝑒 , πœ‰ βˆ†π‘‘ βˆ‡π‘’ βˆ‡π‘’ , βˆ‡πœ‰ βˆ†π‘‘ 𝐻 𝑒 𝑒 , πœ‰ (14) ( 𝑣 𝑒 , πœ‰ βˆ†π‘‘ βˆ‡π‘£ βˆ‡π‘’ , βˆ‡πœ‰ βˆ†π‘‘ 𝐻 𝑣 𝑒 , πœ‰ (15) By subtracting (15) from (14), setting πœ‰ 𝑒 𝑣 in the obtained equation and using Lipschitz condition on 𝐻 with respect to for 𝑒, once get that 𝑒 𝑣 + βˆ†π‘‘ βˆ‡π‘’ βˆ‡π‘£ βˆ†π‘‘πΏ 𝑒 𝑣 , 𝑒 𝑣 . Keep in mind that the 2 term in the left hand side (LHS) is positive and then using Cauchy Schwarz (CS) inequality on the RHS of above inequality, once get that 𝑒 𝑣 𝛽 𝑒 𝑣 , where 𝛽 βˆ†π‘‘πΏ , using (13), to get that π‘ž 𝑒 π‘ž 𝑣 𝛽 𝑒 𝑣 Since βˆ†π‘‘ is sufficiently small and 𝛽 1, then q is contractive, and by theorem (1) we get 𝑒 𝑣 , hence the DSEq has a unique solution, also since 𝑒 ∈ ℝ , βˆ€ 𝑙 then Mathematics | 130 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 π‘ž 𝑒 𝑒 ∈ ℝ , βˆ€ 𝑙 implies that π‘ž 𝑒 ∈ 𝑅 , βˆ€ 𝑒 ∈ ℝ , and by using Theorem (3) with S=ℝ , we get that 𝑒 is converged to a point in ℝ . 6. Cholesky Reduction Order Technique This technique in fact is based on an idea which is introduce first in [11] about reducing the diagonal elements of the Galerkin matrix into columns, we formulate it by the following steps hence we called it by the Cholesky reduction order technique (CHROT): First, the 𝑁 𝑁 obtained matrix is reduced to 𝑁 𝑀1 matrix 𝐴 by transform the lower diagonals (𝑀1 of 𝑁 𝑁 matrix to columns, second the reduction of matrix 𝐴 is a new 𝑁 𝑀1 matrix 𝑅 which is computed by using the following formula : for 1,2, … , 𝑁 , 𝑗 𝑖 1 , … , min 𝑖 𝑀, 𝑁 β€’ if 𝑖 1, then 𝑅 𝐴 and 𝑅 , 𝑙 𝑖 𝑗 𝑀1 β€’ if 𝑖 1, then 𝑅 𝐴 βˆ‘ 𝑅 , 𝐾 max 𝑖 𝑀, 1 : 𝑖 1 𝑅 𝐴 βˆ‘ 𝑅 𝑅 , 𝑠 π‘Ÿ 𝑖 𝑗 , with 𝑗 𝐾 𝑀 . Example 7: Consider the following nonlinear parabolic b.v.p.: 𝑒 βˆ†π‘’ 𝐻 οΏ½βƒ—οΏ½, 𝑑, 𝑒 , where οΏ½βƒ—οΏ½ π‘₯ , π‘₯ 𝑒 οΏ½βƒ—οΏ½, 𝑑 0 , on βˆ‚π‘Š I 𝑒 οΏ½βƒ—οΏ½, 0 π‘₯ π‘₯ 1 π‘₯ 1 π‘₯ , on π‘Š where, 𝐻 οΏ½βƒ—οΏ½, 𝑑, 𝑒 𝑒 π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ 1 sin 𝑒 π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ 𝑒 sin 𝑒 The exact solution of this problem is: 𝑒 οΏ½βƒ—οΏ½, 𝑑 π‘₯ π‘₯ 1 π‘₯ 1 π‘₯ 𝑒 This problem solved using the GFEM for M=9 and NT=20, the results are shown in Table 1. and Figure 1. at time οΏ½Μ‚οΏ½ 0.5 , the table shows the approximate solution 𝑒 π‘₯ , π‘₯ , 𝑑 , the exact solution 𝑒 π‘₯ , π‘₯ , 𝑑 and the absolute error at π‘₯ & π‘₯ . The Mat lap Software is used to solve this problem, it takes 5-hours when we use the CHM, while takes 1-hour and 7- minutes when we use the CHROT. Table 1. Comparison between exact and approximation solutions π‘₯ Β  π‘₯ Β  𝑒 π‘₯ , π‘₯ , 𝑑 𝑒 π‘₯ , π‘₯ , 𝑑 absolute error π‘₯ π‘₯ 𝑒 π‘₯ , π‘₯ , 𝑑 𝑒 π‘₯ , π‘₯ , 𝑑 absolute error 0.1 0.1 0.0137 0.0137 0.0000 0.6 0.5 0.1014 0.1029 0.0015 0.2 0.1 0.0243 0.0247 0.0004 0.7 0.5 0.0887 0.0901 0.0014 0.3 0.1Β  0.0319 0.0323 0.0004 0.8 0.5 0.0676 0.0687 0.0011 0.4 0.1 0.0365 0.0370 0.0005 0.9 0.5 0.0380 0.0386 0.0006 0.5 0.1 0.0380 0.0386 0.0006 0.1 0.6 0.0365 0.0370 0.0005 0.6 0.1 0.0365 0.0370 0.0005 0.2 0.6 0.0649 0.0659 0.0010 0.7 0.1 0.0319 0.0324 0.0005 0.3 0.6 0.0852 0.0865 0.0013 0.8 0.1 0.0243 0.0247 0.0004 0.4 0.6 0.0974 0.0988 0.0014 0.9 0.1 0.0137 0.0140 0.0003 0.5 0.6 0.1014 0.1029 0.0015 0.1 0.2 0.0243 0.0247 0.0004 0.6 0.6 0.0974 0.0988 0.0014 0.2 0.2 0.0433 0.0440 0.0007 0.7 0.6 0.0852 0.0865 0.0013 0.3 0.2 0.0568 0.0577 0.0009 0.8 0.6 0.0649 0.0659 0.0010 0.4 0.2 0.0649 0.0659 0.0010 0.9 0.6 0.0365 0.0370 0.0005 0.5 0.2 0.0676 0.0687 0.0011 0.1 0.7 0.0319 0.0324 0.0005 0.6 0.2 0.0649 0.0659 0.0010 0.2 0.7 0.0568 0.0576 0.0008 Mathematics | 131 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 0.7 0.2 0.0568 0.0576 0.0008 0.3 0.7 0.0745 0.0757 0.0012 0.8 0.2 0.0433 0.0438 0.0005 0.4 0.7 0.0852 0.0865 0.0013 0.9 0.2 0.0243 0.0247 0.0004 0.5 0.7 0.0887 0.0901 0.0014 0.1 0.3 0.0319 0.0323 0.0004 0.6 0.7 0.0852 0.0865 0.0013 0.2 0.3 0.0568 0.0577 0.0009 0.7 0.7 0.0745 0.0756 0.0011 0.3 0.3 0.0745 0.0756 0.0011 0.8 0.7 0.0568 0.0577 0.0009 0.4 0.3 0.0852 0.0865 0.0013 0.9 0.7 0.0319 0.0323 0.0004 0.5 0.3 0.0887 0.0901 0.0014 0.1 0.8 0.0243 0.0247 0.0004 0.6 0.3 0.0852 0.0865 0.0013 0.2 0.8 0.0433 0.0438 0.0005 0.7 0.3 0.0745 0.0757 0.0012 0.3 0.8 0.0568 0.0576 0.0008 0.8 0.3 0.0568 0.0576 0.0008 0.4 0.8 0.0649 0.0659 0.0010 0.9 0.3 0.0319 0.0324 0.0005 0.5 0.8 0.0676 0.0687 0.0011 0.1 0.4 0.0365 0.0370 0.0005 0.6 0.8 0.0649 0.0659 0.0010 0.2 0.4 0.0649 0.0659 0.0010 0.7 0.8 0.0568 0.0577 0.0009 0.3 0.4 0.0852 0.0865 0.0013 0.8 0.8 0.0433 0.0440 0.0007 0.4 0.4 0.0974 0.0988 0.0014 0.9 0.8 0.0243 0.0247 0.0004 0.5 0.4 0.1014 0.1029 0.0015 0.1 0.9 0.0137 0.0140 0.0003 0.6 0.4 0.0974 0.0988 0.0014 0.2 0.9 0.0243 0.0247 0.0004 0.7 0.4 0.0852 0.0865 0.0013 0.3 0.9 0.0319 0.0324 0.0005 0.8 0.4 0.0649 0.0659 0.0010 0.4 0.9 0.0365 0.0370 0.0005 0.9 0.4 0.0365 0.0370 0.0005 0.5 0.9 0.0380 0.0386 0.0006 0.1 0.5 0.0380 0.0386 0.0006 0.6 0.9 0.0365 0.0370 0.0005 0.2 0.5 0.0676 0.0687 0.0011 0.7 0.9 0.0319 0.0323 0.0004 0.3 0.5 0.0887 0.0901 0.0014 0.8 0.9 0.0243 0.0247 0.0004 0.4 0.5 0.1014 0.1029 0.0015 0.9 0.9 0.0137 0.0137 0.0000 0.5 0.5 0.1057 0.1072 0.0015 Figure 1. (a) shows the approximation solution and (b) shows the exact solution Example 8: Consider the following nonlinear b.v.p.: 𝑒 βˆ†π‘’ 𝐻 οΏ½βƒ—οΏ½, 𝑑, 𝑒 , where οΏ½βƒ—οΏ½ π‘₯ , π‘₯ Associated with the i.c and b.c 𝑒 οΏ½βƒ—οΏ½, 𝑑 0 , on βˆ‚π‘Š I 𝑒 οΏ½βƒ—οΏ½, 0 0 , on π‘Š where 𝐻 οΏ½βƒ—οΏ½, 𝑑, 𝑒 π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ 1 𝑑 sin 𝑑 π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ π‘₯ 2𝑑 π‘₯ π‘₯ π‘₯ π‘₯ 𝑒 sin 𝑒 The exact solution of this problem is: 𝑒 οΏ½βƒ—οΏ½, 𝑑 π‘₯ π‘₯ 𝑑 1 π‘₯ 1 π‘₯ 0 0.5 1 0 0.5 1 0 0.02 0.04 0.06 0.08 0.1 0.12 a 0 0.5 1 0 0.5 1 0 0.02 0.04 0.06 0.08 0.1 0.12 b Mathematics | 132 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 This problem is solved using the GFEM for M=9 and NT=20, the results are shown in Table 2. and Figure 2. at οΏ½Μ‚οΏ½ 0.5 , the table shows the approximate solution 𝑒 π‘₯ , π‘₯ , 𝑑 , the exact solution 𝑒 π‘₯ , π‘₯ , 𝑑 and the absolute error at π‘₯ & π‘₯ . This problem is take 13 hours when we use the CHM to solve the PCT, while takes 3 hours and 27 minute when we use the CHROT. Table 2. Comparison between exact and approximation solutions π‘₯ π‘₯ 𝑒 π‘₯ , π‘₯ , 𝑑 𝑒 π‘₯ , π‘₯ , 𝑑 absolute error π‘₯ π‘₯ 𝑒 π‘₯ , π‘₯ , 𝑑 𝑒 π‘₯ , π‘₯ , 𝑑 absolute error 0.1 0.1 -0.0043 -0.0044 0.0001 0.6 0.5 -0.0315 -0.0328 0.0013 0.2 0.1 -0.0076 -0.0079 0.0003 0.7 0.5 -0.0276 -0.0287 0.0011 0.3 0.1 -0.0099 -0.0103 0.0004 0.8 0.5 -0.0210 -0.0219 0.0009 0.4 0.1 -0.0113 -0.0118 0.0005 0.9 0.5 -0.0118 -0.0123 0.0005 0.5 0.1 -0.0118 -0.0123 0.0005 0.1 0.6 -0.0113 -0.0118 0.0005 0.6 0.1 -0.0113 -0.0118 0.0005 0.2 0.6 -0.0202 -0.0210 0.0008 0.7 0.1 -0.0099 -0.0103 0.0004 0.3 0.6 -0.0265 -0.0276 0.0011 0.8 0.1 -0.0076 -0.0079 0.0003 0.4 0.6 -0.0302 -0.0315 0.0013 0.9 0.1 -0.0043 -0.0044 0.0001 0.5 0.6 -0.0315 -0.0328 0.0013 0.1 0.2 -0.0076 -0.0079 0.0003 0.6 0.6 -0.0302 -0.0315 0.0013 0.2 0.2 -0.0134 -0.0140 0.0006 0.7 0.6 -0.0265 -0.0276 0.0011 0.3 0.2 -0.0176 -0.0184 0.0008 0.8 0.6 -0.0202 -0.0210 0.0008 0.4 0.2 -0.0202 -0.0210 0.0008 0.9 0.6 -0.0113 -0.0118 0.0005 0.5 0.2 -0.0210 -0.0219 0.0009 0.1 0.7 -0.0099 -0.0103 0.0004 0.6 0.2 -0.0202 -0.0210 0.0008 0.2 0.7 -0.0176 -0.0184 0.0008 0.7 0.2 -0.0176 -0.0184 0.0008 0.3 0.7 -0.0232 -0.0241 0.0009 0.8 0.2 -0.0134 -0.0140 0.0006 0.4 0.7 -0.0265 -0.0276 0.0011 0.9 0.2 -0.0076 -0.0079 0.0003 0.5 0.7 -0.0276 -0.0287 0.0011 0.1 0.3 -0.0099 -0.0103 0.0004 0.6 0.7 -0.0265 -0.0276 0.0011 0.2 0.3 -0.0176 -0.0184 0.0008 0.7 0.7 -0.0232 -0.0241 0.0009 0.3 0.3 -0.0232 -0.0241 0.0009 0.8 0.7 -0.0176 -0.0184 0.0008 0.4 0.3 -0.0265 -0.0276 0.0011 0.9 0.7 -0.0099 -0.0103 0.0004 0.5 0.3 -0.0276 -0.0287 0.0011 0.1 0.8 -0.0076 -0.0079 0.0003 0.6 0.3 -0.0265 -0.0276 0.0011 0.2 0.8 -0.0134 -0.0140 0.0006 0.7 0.3 -0.0232 -0.0241 0.0009 0.3 0.8 -0.0176 -0.0184 0.0008 0.8 0.3 -0.0176 -0.0184 0.0008 0.4 0.8 -0.0202 -0.0210 0.0008 0.9 0.3 -0.0099 -0.0103 0.0004 0.5 0.8 -0.0210 -0.0219 0.0009 0.1 0.4 -0.0113 -0.0118 0.0005 0.6 0.8 -0.0202 -0.0210 0.0008 0.2 0.4 -0.0202 -0.0210 0.0008 0.7 0.8 -0.0176 -0.0184 0.0008 0.3 0.4 -0.0265 -0.0276 0.0011 0.8 0.8 -0.0134 -0.0140 0.0006 0.4 0.4 -0.0302 -0.0315 0.0013 0.9 0.8 -0.0076 -0.0079 0.0003 0.5 0.4 -0.0315 -0.0328 0.0013 0.1 0.9 -0.0043 -0.0044 0.0001 0.6 0.4 -0.0302 -0.0315 0.0013 0.2 0.9 -0.0076 -0.0079 0.0003 0.7 0.4 -0.0265 -0.0276 0.0011 0.3 0.9 -0.0099 -0.0103 0.0004 0.8 0.4 -0.0202 -0.0210 0.0008 0.4 0.9 -0.0113 -0.0118 0.0005 0.9 0.4 -0.0113 -0.0118 0.0005 0.5 0.9 -0.0118 -0.0123 0.0005 0.1 0.5 -0.0118 -0.0123 0.0005 0.6 0.9 -0.0113 -0.0118 0.0005 0.2 0.5 -0.0210 -0.0219 0.0009 0.7 0.9 -0.0099 -0.0103 0.0004 0.3 0.5 -0.0276 -0.0287 0.0011 0.8 0.9 -0.0076 -0.0079 0.0003 0.4 0.5 -0.0315 -0.0328 0.0013 0.9 0.9 -0.0043 -0.0044 0.0001 0.5 0.5 -0.0328 -0.0342 0.0014 Mathematics | 133 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 Figure 2. (a) shows the approximation solution and (b) shows the exact solution 7. Conclusion β€’ The GFEM associated with the PCT is suitable, efficient and very fast to solve the nonlinear parabolic boundary value problems. β€’ The CHROT is very fast than the CHM with same results and this is important when we have problems gives very large algebraic systems which take a long time in the classical CHM. β€’ The value of οΏ½Μ‚οΏ½ is chose arbitral in the interval I , same results with same accuracy will obtained if we can take any other value of οΏ½Μ‚οΏ½ provided this value belong to I. Acknowledgement: The authors thank Prof. Dr. I. Chryssoverghi for fruitful discussion. References 1. Karlsen K. H.; Riserbo, N. H. Corrected Operator Splitting for Nonlinear Parabolic Equations. Siam J. Numr. Anal. 2000, 37, 3, 980–1003. 2. Pao, C. V. Numerical Methods for Time-Periodic Solutions of Nonlinear Parabolic Boundary Value Problems. SIAM Journal on Numerical Analysis. 2001, 39, 2, 647- 667. 3. Alam, J. M.; Kevlahan, N. K.-R.; Vasilyev, O. V. Simultaneous space–time Adaptive Wavelet Solution of Nonlinear Parabolic Differential Equations. Journal of Computational Physics. 2006, 214, 829–857. 4. Timothy, B. Explicit and Implicit Methods In Solving Differential Equations. Honors Scholarheses. University of Connecticut – Storrs. 2010. 5. Ghoreishi, M. Ismail, A. I. B, The Homotopy Perturbation Method (HPM) for Nonlinear Parabolic Equation with Nonlocal Boundary Conditions. Applied Mathematical Sciences, 2011, 5, 3, 113-123. 6. Douglas, J.; Dupont, T. Galerkin Methods for Parabolic Problems, SIAM J. Numerical Analysis. 1970, 7, 4, 575-626. 7. Reddy, J.N. An Introduction to the Finite Element Method,Β  and Ed., McGraw-Hill, 1993.Β  0 0.5 1 0 0.5 1 -0.04 -0.03 -0.02 -0.01 0 a 0 0.5 1 0 0.5 1 -0.04 -0.03 -0.02 -0.01 0 b Mathematics | 134 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2002 Vol. 31 (3) 2018 8. Thomee, V. Galerkin Finite Element Methods For Parabolic Problems, Springer Verlag Berlin Heidelberg, 2006. 9. Suhas, P.; Uttam, D. Random Fixed Point Theorems for Contraction Mappings in Metric Space. International Journal of Science and Research (IJSR). 2015, 23,19- 7064. 10. Burden, R. L.; Douglas Faires, J. Numerical Analysis, seven ed. Unite state, 2001. 11. Bacopoulos, A.; Chryssoverghi, I. Numerical Solutions of Partial Differential Equations by Finite Elements Methods. Symeon Publishing Co. Athens. 2003.