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УДК 51
SOLUTION TO PARTIAL DIFFERENTIAL EQUATIONS USING CHEBYSHEV POLYNOMIALS
Tormosov Egor Alexandrovich
Graduate student of SAFU "Northern Arctic Federal University named after M.V. Lomonosov"
This paper presents an efficient numerical method based on shifted Chebyshev polynomials for solving Partial Differential Equations (PDEs). In this method, a power series solution in terms of shifted Chebyshev polynomials has been chosen such that it satisfies the given conditions. Substituting this series solution into the given PDE and using appropriate collocation points a system of linear equations with unknown Chebyshev coefficients is obtained. Then, unknowns are found with the help of Gauss elimination. Next, different discretization patterns have also been considered to understand the behavior of the results depending upon the collocation points in the domain. These are the two main modifications and novelty of the procedure and this contribution in the assumption of power series solution in terms of shifted Chebyshev polynomials results in obtaining the approximate solution with less number of terms with good accuracy. Several numerical examples are provided to confirm the reliability and effectiveness of the proposed method.
Keywords: Shifted Chebyshev polynomials, Wave equation, Gauss elimination method, Collocation method.
РЕШЕНИЕ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИИ В ЧАСТНЫХ ПРОИЗВОДНЫХ С ИСПОЛЬЗОВАНИЕМ МНОГОЧЛЕНОВ ЧЕБЫШЕВА
Тормосов Егор Александрович
Аспирант САФУ «Северного Арктического федерального
университета им. М.В. Ломоносова»
В данной статье представлен эффективный численный метод, основанный на сдвинутых полиномах Чебышева для решения уравнений в частных производных. В этом методе решение степенного ряда в терминах сдвинутых полиномов Чебышева было выбрано таким образом, чтобы оно удовлетворяло заданным условиям. Подставляя это решение ряда в заданные дифференциальные уравнения в частные производные и используя соответствующие точки сопоставления, получается система линейных уравнений с неизвестными коэффициентами Чебышева. Затем неизвестные обнаруживаются с помощью исключения Гаусса. Далее, также были рассмотрены различные шаблоны дискретизации, чтобы понять поведение результатов в зависимости от точек коллокации в домене. Это две основные модификации и новизна процедуры, и этот вклад в предположение о решении степенных рядов в терминах сдвинутых полиномов Чебышева приводит к получению приближенного решения с меньшим числом членов с хорошей точностью. Для подтверждения надежности и эффективности предлагаемого метода приведено несколько численных примеров.
Ключевые слова: Сдвинутые полиномы Чебышева, Волновое уравнение, метод исключения Гаусса, Метод коллокации.
Partial differential equations (PDEs) attract remarkable attention as many problems in science and engineering are modeled using PDEs. An analytical solution to PDEs cannot always be found due to the complexity of the differential equations. The method of characteristics, finite difference method [1, 2], finite element method [3], finite volume method [4, 5], and homotopy perturbation method [6]
have been proven to be efficient and suitable to solve partial differential equations. With the help of similarity variables, the coupled partial differential equation can be converted to an ordinary differential equation [7-8]. In recent years, Chebyshev polynomials have attracted phenomenal attention for finding the solution to various types of PDEs, such as linear, non-linear, coupled system of PDEs and fractional differential equations [9]. Akyz-Dascioglu [10]
used Chebyshev polynomials for approximating the solution to high-order partial differential equations. A method based on shifted Chebyshev polynomials was applied to find the numerical solution to fractional differential equations by Bhrawy et al. [11]. Caporale and Cerrato [12] used Chebyshev nodes to approximate partial differential equations. The solution to space fractional order diffusion equations was determined using second kind shifted polynomials by Sweilam et al. [13]. Mall and Chakraverty proposed a new algorithm using Chebyshev polynomials as base functions to solve singular initial value problems of Emden-Fowler type equations [14] and Lane-Emden type equations [15] by Chebyshev neural network method. Different methods to approximate the solution to various differential equations using Chebyshev polynomials were described in [16-17].
The most common concept in the literature discussed above is that, initially a series solution in terms of Chebyshev polynomials is assumed with unknown Chebyshev coefficients. Unknown Chebyshev coefficients have been found using various techniques such as the spectral matrix method, artificial neural network etc. [18-19].
The main purpose of the present paper was to choose a power series solution such that all the terms of the series satisfy the given conditions and then they can be used in the collocation method. Different discretization patterns were also considered to understand the behavior of the results depending upon the collocation points in the domain. These are the main modifications and novelty of the procedure. This contribution in the assumption of power series solution in terms of shifted Chebyshev polynomials results in obtaining the approximate solution with less number of terms with good accuracy.
Methods.
1. Shifted Chebyshev collocation
method
The Russian mathematician Pafnuty Chebyshev introduced an nth degree polynomial with leading coefficient unity in the interval [- 1, 1] in his paper in the year 1854 [16] as below
— cos(ncos Hx)) =^-7Tn(x)
in the variable x = 2t - 1 [17] they are called Shifted Chebyshev polynomials and are defined as T„(t) = Tn(2t — l),n = 0,1,2
2
(1)
The Chebyshev polynomials can be determined using the recurrence formula Tn+1(x) = 2xTn(x) — T„_1(x),n = 1,2,3 (2)
The first four Cheyshev polynomials are
given by
T0(x) = 1 Ti(x) = x T2(x) = 2x2 — 1 T3(x) = 4x3 — 3x The Chebyshev polynomials are defined in the interval [0, 1]. By introducing the change
(3)
T0(t) = 1 Ti(t) = 2t - 1
T2(t) = 1 + 8t2 - 8t T4(t) = 32t3 - 48t2 + 18t - 1
which all are defined in t e [0,1].
To illustrate the shifted Chebyshev collocation method, we consider a general partial differential equation with two independent variables x and y and a dependent variable u as A(u) =f(x,y)
(4)
subject to the conditions: u(x0,y) = g(y) and/or u(x,y0) = h(x)
(5)
where u is an unknown function; f, g and h are known functions and, x0 and y0 are constants. First, we assume a bivariate power series solution in terms of shifted Chebyshev polynomials that satisfies conditions given in equation (5) as below
N N
u(x,y) = ^(x,y) + ^(x,y) ^ ^ ars T;(x)Ts*(y)
r=0 s=0
(6)
It is worth mentioning that ^(x,y) and ^(x,y) are functions of x and y which are to be chosen such that Eq. (6) satisfies the given conditions (5).
Next, substituting the assumed solution (6) into equation (4), we obtain a residual equation (x,y) = 0 , which involves Chebyshev coefficients ars, and using proper collocation points in the domain [0,1] x [0,1], we get a system of equations with unknown Chebyshev coefficients ars as
Av = b
(7)
where A is the known coefficient matrix, v is the column vector with Chebyshev coefficients, and b is the known right-hand side vector. Here A and b have real entries which are obtained using collocation points in the residual equation R(x,y) = 0. Here N2 collocation points are needed if we consider Np polynomials where Np = N + 1 . Finally, by solving the system of equations Chebyshev coefficients ars 's can be obtained and substituting they into equation (6) we get an approximate solution to (4). It can be noted that discretization of the domain is a critical issue. In this connection, various discretization patterns are also tried and the best one is chosen for obtaining acceptable results Results and discussion 1. In this study, two example problems were solved using the Shifted Chebyshev collocation method Example 1
First, we consider the Cauchy problem for the one-dimensional homogenous wave equation
u^ = u^O ^ x ^ I-0 ^ y ^ 1 The exact solution to the above
(8) differential equation is u(x,y) = x3 + 3xy2 + u(x, 0)=x3,u (x, 0)=x xy. Let us assume that the bivariate series
(9) solution to wave equation (8) is as below
u(x,y) = (x3 + xy) + xy2 £2=02s=0ars Tr*(x)Ts*(y) (10)
The proposed solution (10) satisfies the respect to x and y partially, we obtain partial
initial conditions (9). Differentiating (10) with derivatives as
uxx = 6x + 4a10y2 + 4a11(2y3 - y2) + 4a12(8y4 - 8y3 + y2) + a20(48x - 16)y2 + 4a21(48x - 16)(2y3 -
y2) + 4a22 (48x - 16) (8y4 - 8y3 + y2) (11)
And
uyy = 2 a00x + a10x(12y - 2) + a02x(96y2 - 48y + 2) + 2a10(2x2 - x) + a11(2x2 - x)(12y - 2) + a12(2x2 - x)(96y2 - 48y + 2) + 2a20(8x3 - 8x2 + x) + a21(8x3 - 8x2 + x)(12y - 2) + a22(8x3 - 8x2 +
x)(96y2-48y + 2) (12)
Substituting Equations (11) and (12) into (8), we get the residual equations R(x, y) = 0 as
{6x + 4a10y2 + 4a11(2y3 - y2) + 4a12(8y4 - 8y63 + y2) + a20(48x - 16)y2 + 4a21(48 - 16)(2y3 - y2) + 4a22(48x - 16)(8y4 - 8y3 + y2)} - {2a00x + a10x(12y - 2) + a02x(96y2 - 48y + 2) + 2a10(2x2 - x) + a11(2x2 - x)(12y - 2) + a12(2x2 - x)(96y2 - 48y + 2) + 2a20(8x3 - 8x2 + x) + a21(8x3 - 8x2 + x)(12y -2) + a22(8x 3 - 8x 2 + x)(96y2 - 48y + 2)} = 0
Then using nine collocation points, (since given in Figure 1, we can obtain the system of
N = 2, so the number of collocation points nine linear equations as shown below
required here is 9) in the domain [0, 1] x [0, 1] as
Oa00 + 0a01 + 0a02 + la10 + Oa^ - la12 - 4a20 + 0a21 + 16a22 = 0 11 1 31 17 3 3
- 2 a00 — ^ a01 + la02 3l° + - g a12 — ^ a20 + "g" a21 + 2 a22 = — ^
-la00 + la01 - la02 + 0a10 + Oa^ + la12 + 0a20 - 9a21 + 9a22 = -3 17 5 13 9993
- 2 aoo — ^ aoi - ^a02 - — a10 + 2a11 + — a12 - — a20 - — a21 + — a22 = — —
—laQQ — 2a01 — lag2 + la^o + Oa-^ + 0a^2 + 2a2Q + 2a2^ — 7a22 = —3 3 3 1 1 17 5 77 23 9
— 2 aoo — ^ aoi + 3a02 — — a10 — — a11 + — a12 + — a20 — a21 — a22 = — —
—laQQ — 5am — 25a02 + 4a^o + + 5a^2 + 8a2g + 13a2^ + 33a22 = —3 3 21 3 3 63 45 101 5 9
— 2 aoo 4"ao1 — + 2 aio — 2 3l1 s"312 + "iT320 8~ 321 — 2 322 = — 2 -2a00 - 4a01 - 2a02 - la10 - 4a!! - 5a12 + 8a20 - 4a21 - 34a22 = -6
Solving the above system using Gauss elimination, we get
aoo = 3; a01 = a02 = a10 = a11 = a12 = a20 = a21 = a22 = 0
Substituting all ars's into equation (10), as u(x,y) = x3 + 3xy2 + xy , which is the
we obtain the solution to the wave equation (8) exact solution.
(H
Fig. 1. Collocation points in the domain [0, 1
Fig. 2. Solution to equation (8) in the domain 0 < x < 1, 0 < y < 1
It is worth mentioning that choosing a proper discretization scheme of the domain plays an important role in obtaining the best approximate solution to the PDEs. To explain this, we used two different discretization schemes in the next example.
Example 2.
Let us consider another example of a partial differential equation of second order with variable coefficients
(13)
(14)
1
Uyy - - x2uxx = 0;x<x<l,0<y<l
u(x,0) = x and u„(x,0) = x2
The exact solution to (9) is u(x,y) = x + x2sinh(y) [17]. First, we assume a solution to second order hyperbolic partial differential
equation (8) that satisfies the given conditions (9) as below
u(x,y) = (x + x2y) + y2 ^^ ars Tr*(x)Ts*(y)
r=0 s=0
(15)
Substituting proposed series solution (15) into equation (13), we can get the residue as below (2a00 + a01(6y — 2)) + 2a10(2x — 1) + 3ll(2x — a)(12y — 2) — 0,5x2(l + 2y) = 0
(16)
Here N = 1, so using 4 collocation points g;0),(0;i),(l;£),g;l) in the domain [0, 1] x
[0, 1] as shown in figure 3, we obtain the system of four linear equations as below
Fig. 3. Collocation points in the domain
Now, we consider the collocation points (4 '4)' (4 '4)' (4 '4) (4'4) as shown in Figure 4, and substituting these collocation points into the residual equation (16) we can obtain the system of four linear equations as below
The above linear system of equations has no solution.
1
2a00 + Oa01 + Oa10 — 2a11 = —
2a00 + a01 — 2a10 — 4a11 = 0
3
2a00 + a01 + Oa10 + 0a11 = — 2a00 + a01 + 2a10 + 4a11 = 1
Fig. 4.
1 1 3
2a00 — 2aoi — aio — 2an = 64
1 1 27
2a00 — 2aoi + aio + 2an = 64
5 7 5
2a00 + 2aoi — aio — 2an = 64
1 7 5
2a00 + 2aoi + aio + 2an = 64
Solving the above linear system of equations using the Gaussian elimination method, we obtain the unknown coefficients of the series solution (15) as
a00 = 0,1042; a01 = —0,0521; a10 = 0,2188; aw = —0,0625.
Therefore, the solution to the hyperbolic PDE (14) is
u(x,y) = (x + x2y) + y2[0,1042 —0,0521(2y—1) + 0,2188(2x — 1) — 0,0625(2x — l)(2y — 1)].
points in the domain
Figure 5 represents the comparison of the present solution to equation (13) with an analytical solution for fixed y = 1 and N = 1, whereas Figure 6 represents the solution for N = 3. The solution to equation (13) using the present method for N = 1 and N = 3 is compared with an analytical solution for fixed x = 0.5 in figure 7. It can be seen from figures 5 and 6 that the solution obtained by the proposed method is almost the same as the exact solution only considering N = 1 and N = 3 (that is two and four terms, respectively), whereas in [17] N = 7 was taken to find the solution with error 6.0 E-5. From figure 7, it can be observed that for N = 1 the solution to equation (13) by the present method has more error. But for N = 3, the solution coincides with the analytical solution.
Fig. 5. Comparison of the solution to (13) by the proposed method with the analytical solution at y = 1.0 with N = 1
Fig. 6. Comparison of the solution to (13) by the proposed method with the analytical solution at y = 1.0 with N = 3
Fig. 7 Comparison of the solution to (13) by the proposed method with analytical solution
at x = 0.5 with N = 1 and N = 3
This shows that as N increases, the error in the solution by the present method is less, and it can exactly coincide with the analytical solution.
Conclusion.
The proposed modification in the collocation method based on shifted Chebyshev polynomial is successfully applied to solve different linear partial differential equations. In this method a series solution in terms of shifted
Chebyshev polynomials is assumed satisfying the given conditions. This modification in the assumption has succeeded to get the approximate solution with less number of terms comparing with other methods available in literature. Two examples have been solved and results are compared with existing literature. The obtained results confirm that the proposed method is effective and needs less number of terms to get the convergence.
REFERENCES
1. Forsythe GW, Wasow WR (1960) Finite-difference methods for partial differential equations. Applied Mathematical Series Wiley, New York 25-27;
2. Thomas JW (2013) Numerical partial differential equations: finite difference methods. Springer, New York 45-47;
3. Johnson C (2012) Numerical solution of partial differential equations by the finite element method. Courier Corporation, Chelmsford 65-68;
4. Alieldin SS, Alshorbagy AE, Shaat M (2011) A first-order shear deformation finite element model for elastostatic analysis of laminated composite plates and the equivalent functionally graded plates. Ain Shams Eng J 2:53-62;
5. Nayak S, Chakraverty S (2018) Interval finite element method with MATLAB. Academic Press, New York;
6. Eymard R, Gallouet T, Herbin R (2000) Finite volume methods. Handb Numer Anal 7:713-1018;
7. LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University press, Cambridge;
8. He JH (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178:257-262;
9. Singh R, Singh S, Wazwaz AM (2016) A modified homotopy perturbation method for singular time dependent Emden- Fowler equations with boundary conditions. J Math Chem 54(4):918-931;
10. Karunakar P, Chakraverty S (2018) Solution of interval shallow water wave equations using homotopy perturbation method. Eng Comput 34:1610-1624;
11. Karunakar P, Chakraverty S (2017) Comparison of solutions of linear and non-linear shallow water wave equations using homotopy perturbation method. Int J Numer Methods Heat Fluid Flow 27:20152029;
12. Makinde OD, Animasaun IL (2016) Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution. J Mol Liq 221:733-743;
13. Motsa SS, Animasaun IL (2018) Bivariate spectral quasi-linearisation exploration of heat transfer in the boundary layer flow of micropolar fluid with strongly concentrated particles over a surface at absolute zero due to impulsive. Int J Comput Sci Math 9(5):455-473;
14. Makinde OD, Omojola MT, Mahanthesh B, Alao FI, Adegbie KS, Animasaun IL, Wakif A, Sivaraj R, Tshehla MS (2018) Significance of buoyancy, velocity index and thickness of an upper horizontal surface of a paraboloid of revolution: the case of Non-Newtonian carreau fluid. Defect Diffus Forum 387:550-561. https ://doi. org/10.4028/www.scien tific .net/ddf.387.550;
15. Animasaun IL, Mahanthesh B, Koriko OK (2018) On the motion of non-newtonian Eyring-Powell fluid conveying tiny gold particles due to generalized surface slip velocity and Buoyancy. Int J Appl Comput Math 4(6):137;
16. Piessens R (2000) Computing integral transforms and solving integral equations using Chebyshev polynomial approximations. J Comput Appl Math 121(1-2):113-124;
17. Yuksel G, Sezer MA (2013) Chebyshev series approximation for linear second-order partial differential equations with complicated conditions. Gazi Univ J Sci 26(4):515-525;
18. Khalifa AK, Elbarbary EM, Elrazek MA (2003) Chebyshev expansion method for solving second and fourth-order elliptic equations. Appl Math Comput 135(2-3):307-318;
19. Akyüz-Dascioglu A (2009) Chebyshev polynomial approximation for high-order partial differential equations with complicated conditions. Numer Methods Partial. Differ. Equ. 25(3):610-621.
