第 55 卷 第 2 期
2020 年 4 月
JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY
Vol. 55 No. 2 Apr. 2020
ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.2.2 Research article
Mathematics
B
OUBAKERW
AVELETF
UNCTIONS FORS
OLVINGH
IGHERO
RDERI
NTEGRO-D
IFFERENTIALE
QUATIONS求解高阶积分微分方程的布巴克小波函数
Eman Hassan Ouda, Suha Shihab, Mohammed Rasheed Applied Science Department, University of Technology
Baghdad, Iraq, [email protected],[email protected], [email protected],
[email protected], [email protected], [email protected]
Received: January 20, 2020 ▪ Review: February 28, 2020 ▪ Accepted: April 14, 2020
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
Abstract
In the present paper, the properties of Boubaker orthonormal polynomials are used to construct new Boubaker wavelet orthonormal functions which are continuous on the interval [0, 1). Then, a Boubaker wavelet orthonormal operational matrix of the derivative is obtained with the new general procedure. The matrix elements can be expressed in a simple form that reduces the computational complexity. The collocation method of the Boubaker orthonormal wavelet functions together with the application of the derived operational matrix of the derivative are then utilized to transform the higher-order integro-differential equation into a solution of linear algebraic equations. As a result, the solution of the original problem reduces to the solution of a linear system of algebraic equations and can be sufficiently solved by an approximate technique. The main advantage of the suggested method is that the orthonormality property greatly simplifies the original problem and leads to easy calculation of the coefficients of expansion. Special attention is needed to perform the convergence analysis. The error is analyzed when a sufficiently smooth function is expanded in terms of the Boubaker orthonormal wavelet functions, then an estimation of the upper bound of the error is calculated. The results obtained by the technique in the current work are reported by solving some numerical examples and the accuracy is checked by comparing the results with the exact solution.
Keywords: Integro-Differential Equation, Wavelet Polynomial, Orthonormal Boubaker Polynomial, Operational Matrix, Convergence Analysis
摘要 在本文中，将布贝克正交多项式的性质用于构造在间隔[0，1）上连续的新布贝克小波正交 函数。然后，使用新的一般程序获得导数的布贝克小波正交运算矩阵。矩阵元素可以简化形式表 达，从而降低了计算复杂度。布贝克正交小波函数的搭配方法以及派生运算矩阵的应用随后被用
于将高阶积分微分方程转换为线性代数方程的解。结果，原始问题的解简化为代数方程的线性系 统的解，并且可以通过近似技术来充分解决。所建议方法的主要优点是正交性极大地简化了原始 问题，并易于计算膨胀系数。需要特别注意以弱假设进行收敛分析。当根据布贝克正交小波函数 展开足够平滑的函数时，分析误差，然后计算误差的上限。通过求解一些数值示例，报告了该技 术在当前工作中获得的结果，并通过将结果与精确解进行比较来检查准确性。 关键词: 积分微分方程，小波多项式，正交布贝克多项式，运算矩阵，收敛性分析
I. I
NTRODUCTIONIntegro-differential equations of ordinary or fractional order arise from a mathematical modeling of the development of an epidemic case and various physical and biological models [39]. They have found applications in many important phenomena, for example, viscoelasticity in material science and electromagnetics are defined by: nth order integro-differential equations [1], nonlinear integro-differential equations [2], [3], linear Volterra integro-differential equations [4], [5], linear Fredholm integro-differential equations [6], [7], mixed Fredholm-Volterra integro-differential equations [8], [9], singular differential equations [10], fuzzy integro-differential equations [11], fractional delay integro-differential equations [12], fractional Fredholm-Volterra integro-differential equations [13], fractional stochastic integro-differential equations [14], nonlinear fractional integro-differential equations [15], [16], [17], [18], and integro partial differential equations [19], [20].
There are various numerical techniques to find the solutions for such problems, for example; the approximation of wavelet functions has played a role in the numerical solution of different kinds integro-differential equations, such as quadratic spline wavelets for solving integro-differential equations [21], wavelet operational matrices for singular Volterra partial integro-differential equations [22], quasi wavelets for nonlinear integro reaction-diffusion equations [23], Legendre wavelets for fractional delay integro-differential equations [24], rational Haar wavelets for nonlinear integro-differential equations [25], multiwavelets for Volterra integro-differential equations [26], special wavelets for partial integro-differential equations and a system of nonlinear fractional equations [27], [28], [29] and Chebyshev wavelets of the second kind for treating nonlinear Fredholm integro-differential equations [30].
In the present work, orthonormal Boubaker wavelets are applied to the approximate solution of higher second kind Volterra integro-differential equations of the form
(1) where , , and are known functions, while is an unknown function, and with the initial conditions
,
(2) The non-orthogonal Boubaker polynomial is used in [31] for solving the variational problem. The application of an orthonormal Boubaker wavelet can be used in pure and applied physics such as solid state physics in order to solve some optical constants of different materials [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53] .
The work in this paper is arranged as follows. Section II gives a fundamental idea about Boubaker polynomials and the definition of an orthonormal Boubaker polynomial. The error analysis is given in Section III, while the orthonormal Boubaker polynomial operation matrix of a derivative is constructed in Section IV. Section V presents the application of the obtained operation matrix together with the collocation method for solving a higher second kind Volterra integro-differential equation. Section VI illustrates the efficiency of the proposed algorithm together with some test examples. Lastly, Section VII provides concluding remarks.
II. B
OUBAKERO
RTHONORMALW
AVELETF
UNCTIONThe Boubaker orthonormal wavelet functions have four arguments,
with representing the degree of the Boubaker orthonormal functions while is the normalized time.
The Boubaker orthonormal functions are given as
(4) Here are the Boubaker orthonormal polynomials of order . They can be defined as follows:
)
(5)
Figure 1. First five Boubaker orthonormal polynomials
III. F
UNCTIONA
PPROXIMATION ANDE
RRORA
NALYSISA square integrable function is defined over the interval [0,1) and is expanded as a Boubaker orthonormal wavelet series in the form
(6) (6) where
Truncating the series in Eq. (4) yields
(7) (7) where U and are the following
matrices
(8) (8)
and
(9) Suppose that is a -times differentiable function on , then the upper-bound error is defined through the following theorem.
Theorem 1. Let be a continuous function on and , where is a positive constant. Let be the approximated solution of using a Boubaker orthonormal wavelet function. Then, the upper-bound error can be obtained as
(10) (10)
Proof. The Boubaker orthonormal wavelets
series of U(t), Ua (t) is given by
(11) (11) Let
(12) (12) Using Eqs. 9 and 10, one can obtain
U (t) (13) (13)
Both the third and fourth terms in (11) become zero because vanishes outside the interval [0, 1).
Then
Using the orthonormality of , yields
Here , and this leads to
(taking
, and we have . This leads to
which is the required result.
IV. B
OUBAKERO
RTHONORMALW
AVELETO
PERATIONM
ATRIXOF THE
D
ERIVATIVELet be the Boubaker orthonormal wavelet vector presented in Eq. 6, and then can be written as
(15) (15) where is the operational matrix of
the derivative expressed as
where is an matrix and expressed as follows:
Related to the definition of the component, we have
V. O
PERATIONM
ATRIXM
ETHODFOR
S
OLVING NTHO
RDERI
NTEGRO-D
IFFERENTIALE
QUATIONSIn the present section, the Boubaker orthonormal wavelet together with its operational matrix of a derivative is applied to find the approximate solution for the integro-differential equation of the following:
(16) Together with the conditions
(17)
where are real and the constant p are positive integers, g(t), and k(t, x) are giving functions whereas u(t) is determined. Eqns. 16 and 17 occur in various fields of physics engineering, astrophysics, etc., whose that
(18) where U and are in Eq. 8 and Eq. 9.
Using Boubaker orthonormal matrix of derivative Eq. 16 can be written as
(19) Residual for Eq.19 can be written as
(20) using collocation method which leads to a system of linear algebraic equations and we can obtain the unknown in vector U which be used to find the solution u(t), after that the initial conditions in Eq. 17 also used to form the system of algebraic equations and are given by:
(21) where denotes matrix power.
VI. I
LLUSTRATIVEE
XAMPLESExample 1. Consider the following Volterra
integro-differential equation:
with exact solution
Table 1 shows the numerical results for this example, with 5 compared with the
exact solution, which is graphically illustrated in Figure 2.
Table 1.
Numerical results of Example 1
t Error 0 0 0 0 0.1 0.0053461737 0.0051961260 1.5e-4 0.2 0.0228777934 0.0228601100 1.4e-5 0.3 0.0550988346 0.0552035379 1.0.e-4 0.4 0.1049051814 0.105016901 1.1e-4 0.5 0.1756393646 0.175669600 3.0e-5 0.6 0.2711524798 0.2711099386 4.2e-5 0.7 0.3958741877 0.3958651274 9.0.e-6 0.8 0.5548918143 0.5550412845 1.4e-5 0.9 0.7540396888 0.7543234336 2.8e-4 1.0 1.0000000000 0.9999755049 2.4e-5
Figure 2. Graphical illustration of Example 1
Example 2. The second test example is
= 0, (23) with the exact solution
The numerical results for this example are listed in Table 2, with compared with the exact solution, which is graphically illustrated in Figure 3.
Table 2.
Numerical results of Example 2
t Error 0 1 1 0 0.1 0.9950041652 0.9950056513 1.4e-6 0.2 0.9800665778 0.9800758838 9.3e-6 0.3 0.9553364891 0.9553596292 2.3e-5 0.4 0.9210609940 0.9210978384 3.6e-5 0.5 0.8775825618 0.8776234813 4.0e-5 0.6 0.8253356149 0.8253615467 2.5e-5 0.7 0.7648421872 0.7648290423 1.3e-5 0.8 0.6967067093 0.6966349950 7.1e-5 0.9 0.6216099682 0.6214804507 1.2e-4 1.0 0.5403023058 0.5401584740 1.4e-4
Figure 3. Graphical illustration of Example 2
Example 3. The third test Volterra
integro-differential equation is
The exact solution of this equation is
(25) Table 3 shows the numerical results for this example, with , compared with the exact solution and is graphically illustrated in Figure 4.
= [-0.186481 843328393 0.177110658 0.00497555 0.00027155129
0.0000000049].
Table 3.
Numerical results of Example 3
t Error 0 -1 -1 0 0.1 -0.8998374166 -0.9008797206 1.0e-3 0.2 -0.7987306641 -0.8001684964 1.4e-3 0.3 -0.6958172081 -0.6972896130 1.4e-3 0.4 -0.5903143571 -0.5916663157 1.3e-3 0.5 -0.4815089580 -0.4827218095 1.2e-3 0.6 -0.3687468344 -0.3698792591 1.1e-3 0.7 -0.2514218946 -0.2525617890 1.1e-3 0.8 -0.1289648456 -0.1301924833 1.2e-3 0.9 -0.0008314511 -0.0021943858 1.3e-3 1.0 0.1335097233 0.13200949978 1.5e-3
Figure 4. Graphical illustration of Example 3
Example 4. Consider the fourth Volterra
integro-differential equation
(26) Table 4 shows the numerical results for this example with , compared with the exact solution and graphically illustrated in Figure 5.
] = [0.75 0.144337567 0 0]
Table 4.
Numerical results of Example 4
t Error 0 0 0 0 0.1 0.1 0.1 0 0.2 0.2 0.2 0 0.3 0.3 0.3 0 0.4 0.4 0.4 0 0.5 0.5 0.5 0 0.6 0.6 0.6 0 0.7 0.7 0.7 0 0.8 0.8 0.8 0 0.9 0.9 0.9 0 1.0 1.0 1.0 0
Figure 5. Graphical illustration of Example 4
VII. C
ONCLUSIONHigher-order integro-differential equations are usually difficult to solve analytically. Therefore, a numerical solution is required. Boubaker orthonormal wavelets function is presented and defined a general formulation for its derivative operational matrix. Then, the collocation method differential equation is applied, which is useful as demonstrated in examples. The error is analyzed and an upper bound error is found. The obtained results are reported in both tables and graphs, and the presented method can be shown to obtain an excellent agreement-approximate solution.
R
EFERENCES[1] XUFENG, S. and DANFU, H. (2010) Application of the variation aliteration method for solving nth-order integro-differential equations. Journal of
Computational and Applied Mathematics,
234 (51), pp. 1442-1447.
[2] GHASEMI, M., TAVASSOLI, K.M., and BABOLIAN, E. (2007) Application of He’s homotopy perturbation method to nonlinear integro-differential equations. Applied Mathematics and Computation, 188 (11), pp.
538-548.
[3] TEMIMI, H., ANSARI, A.R., and SIDDIQUI, A.M. (2011) An approximate solution for the static beam problem and nonlinear integro-differential equations.
Computers & Mathematics with Applications,
62 (8), pp. 3132-3139.
[4] ŞUAYIP, Y. (2016) Improved Bessel collocation method for linear Volterra integro-differential equations with piecewise intervals and application of a Volterra population model. Applied Mathematical
Modelling, 40 (9-10), pp. 5349-5363.
[5] SHAYANFARD, F., DASTJERDI, H.L., and GHAINI, F.M.M. (2020) Collocation method for approximate solution of Volterra integro-differential equations of the third-kind. Applied Numerical Mathematics, 150, pp. 139-148.
[6] DULAT, S.D. (2018) New general solutions to linear Fredholm integro-differential equations and their applications on solving the boundary value problems.
Journal of Computational and Applied Mathematics, 327, pp. 79-108.
[7] ABIGAIL, I.F. and MARK, A.K. (2019) A priori Nyström-method error bounds in approximate solutions of 1-D Fredholm integro-differential equations. International
Journal of Mechanical Sciences, 150, pp.
755-766.
[8] BERENGUER, M.I., GÁMEZ, D., and LÓPEZ LINARES, A.J. (2013) Fixed-point techniques and Schauder bases to approximate the solution of the first order nonlinear mixed Fredholm–Volterra integro-differential equation. Journal of Computational and Applied Mathematics,
252, pp. 52-61.
[9] SALİH, Y. and MEHMET, S. (2000) The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials.
Applied Mathematics and Computation, 112
(2-315), pp. 291-308.
[10] NUGZAR, S., SERGO, K., and OTAR, J. (2016) An approximate solution of one class of singular integro-differential equations. Transactions of A. Razmadze
Mathematical Institute, 170 (3), pp. 420-426.
[11] MAHMOOD, O. and MARYAM, M. (2016) Iterative method for approximate solution of fuzzy integro-differential equations. Beni-Suef University Journal of
Basic and Applied Sciences, 5 (4), pp.
369-376.
[12] SEDAGHAT, S., OSTADZAD, M.H., and BALEANU, D. (2020) A Tau–like numerical method for solving fractional delay integro–differential equations. Applied
Numerical Mathematics, 151, pp. 322-336.
[13] PARISA, R., YADOLLAH, O., and ESMAIL, B. (2017) Fractional-order Bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations. Applied Numerical Mathematics, 122, pp. 66-81.
[14] FARSHID, M. and NASRIN, S. (2017)
Application of
orthonormalBernsteinpolynomialstoconstruct a efficient scheme for solving fractional stochastic integro-differential equation. Optik, 132, pp. 262-273.
[15] PARISA, R. and YADOLLAH, O. (2020) Approximate solution of nonlinear fractional integro-differential equations using fractional alternative Legendre functions.
Journal of Computational and Applied Mathematics, 365, 112365.
[16] HENDI, F.A., WAFA, S., and AL-BADRANI, H. (2019) Existence result and approximate solutions for quadratic integro-differential equations of fractional order.
Journal of King Saud University – Science,
31 (3), pp. 314-321.
[17] LI, H., XIAN-FANG, L., YULIN, Z., and XIANG, Y.D. (2011) Approximate solution of fractional integro-differential equations by Taylor expansion method.
Computers & Mathematics with Applications,
62 (3), pp. 1127-1134.
[18] KHADER, M.M. and SWEILAM, N.H. (2013) On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method. Applied Mathematical Modelling, 37 (2415), pp. 9819-9828.
[19] LEI, Z. and BIN, L. (2019) The obstacle problem of integro-partial differential equations with applications to stochastic optimal control/stopping problem. Journal of
the Franklin Institute, 356 (3), pp. 1396-1423.
[20] KONSTANTINOS, D. (2020) On finite difference schemes for partial integro-differential equations of Lévy type. Journal
of Computational and Applied Mathematics,
368, 112587.
[21] DANA, Č. and VÁCLAV, F. (2020) Galerkin method with new quadratic spline wavelets for integral and integro-differential equations. Journal of Computational and
Applied Mathematics, 363, pp. 426-443.
[22] SAHA, R.S. and BEHERA, S. (2020) Two-dimensional wavelets operational method for solving Volterra weakly singular partial integro-differential equations. Journal
of Computational and Applied Mathematics,
366, 112411.
[23] SACHIN, K. and PRASHANT, P. (2020) Quasi wavelet numerical approach of non-linear reaction diffusion and integro reaction-diffusion equation with Atangana– Baleanu time fractional derivative. Chaos,
Solitons & Fractals, 130, 109456.
[24] NEMATI, S., LIMA, P.M., and SEDAGHAT, S. (2020) Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations. Applied Numerical Mathematics, 149, pp. 99-112.
[25] MAJID, E. and AMIN, M. (2019) Solving the nonlinear integro-differential equation in complex plane with rationalized Haar wavelet. Mathematics and Computers
in Simulation, 165, pp. 223-237.
[26] BEHZAD, N.S. (2019) An efficient algorithm for solving Volterra integro-differential equations based on Alpert’s multi-wavelets Galerkin method. Journal of
Computational and Applied Mathematics,
348, pp. 453-465.
[27] HARISH, K.K. and ANTONY, V.V. (2018) Wavelet based iterative methods for a class of 2D-partial integro differential equations. Computers & Mathematics with
Applications, 75 (11), pp. 187-198.
[28] JIAO, W., TIAN-ZHOU, X., YAN-QIAO, W., and JIA-QUAN, X. (2018)
Numerical simulation for coupled systems of nonlinear fractional order integro-differential equations via wavelets method. Applied
Mathematics and Computation, 324, pp.
36-50.
[29] BABAAGHAIE, A. and MALEKNEJAD, K. (2017) Numerical solution of integro-differential equations of high order by wavelet basis, its algorithm and convergence analysis. Journal of Computational and Applied Mathematics,
325, pp. 125-133.
[30] LI, Z. and QIBIN, F. (2012) Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Communications in
Nonlinear Science and Numerical Simulation,
17 (6), pp. 2333-2341.
[31] OUDA, E.H. (2018) An approximate solution of some variational problems using Boubaker polynomials. Baghdad Science
Journal, 15 (1), pp. 106-109.
[32] RASHEED, M. and BARILLÉ, R. (2017) Optical constants of DC sputtering derived ITO, TiO2 and TiO2:Nb thin films
characterized by spectrophotometry and spectroscopic ellipsometry for optoelectronic devices. Journal of Non-Crystalline Solids, 476, pp. 1-14.
[33] RASHEED, M. and BARILLÉ, R. (2017) Room temperature deposition of ZnO and Al:ZnO ultrathin films on glass and PET substrates by DC sputtering technique.
Optical and Quantum Electronics, 49 (5), pp.
1-14.
[34] RASHEED, M. and BARILLÉ, R. (2017) Comparison the optical properties for Bi2O3 and NiO ultrathin films deposited on
different substrates by DC sputtering technique for transparent electronics. Journal
of Alloys and Compounds, 728, pp.
1186-1198.
[35] AUKŠTUOLIS, A., GIRTAN, M., MOUSDIS, G.A., MALLET, R., SOCOL, M., RASHEED, M., and STANCULESCU, A. (2017) Measurement of charge carrier mobility in perovskite nanowire films by photo-CELIV method. Proceedings of the
Romanian Academy - Series A: Mathematics, Physics, Technical Sciences, Information Science, 18 (1), pp. 34-41.
[36] BOURAS, D., MECIF, A., BARILLÉ, R., HARABI, A., RASHEED, M., MAHDJOUB, A., and ZAABAT, M. (2018) Cu:ZnO deposited on porous ceramic substrates by a simple thermal method for photocatalytic application. Ceramics International, 44 (17), pp. 21546-21555.
[37] KADRI, E., DHAHRI, K., ZAAFOURI, A., KRICHEN, M., RASHEED, M., KHIROUNI, K., and BARILLÉ, R. (2017) Ac conductivity and dielectric behavior of a−Si:H/c−Si1−yGey/p−Si thin films
synthesized by molecular beam epitaxial method. Journal of Alloys and Compounds, 705, pp. 708-713.
[38] DKHILALLI, F., MEGDICHE, S., GUIDARA, K., RASHEED, M., BARILLÉ, R., and MEGDICHE, M. (2018) AC conductivity evolution in bulk and grain boundary response of sodium tungstate Na2WO4. Ionics, 24 (1), pp. 169-180.
[39] AL-NASRAWY, H.H., AL-JUBORY, A.K.O., and HUSSAINA, K.A. (2019) An Efficient Algorithm for Solving Integro-Differential Equation. Journal of Southwest
Jiaotong University, 54 (6). Available from
http://jsju.org/index.php/journal/article/view/ 453.
[40] SARHAN, A.M., SHIHAB, S., and RASHEED, M. (2020). A NEW
BOUBAKER WAVELETS
OPERATIONAL MATRIX OF INTEGRATION. Journal of Southwest
Jiaotong University, 55(2), to appear.
[41] SARHAN, A.M., SHIHAB, S., and RASHEED, M. (2020). On the Properties of Two Dimensional Normalized Boubaker Polynomials. Journal of Southwest Jiaotong
University, 55(2), to appear.
[42] ASMAA, A.A. SHIHAB, S., and RASHEED, M. (2020). Discrete Hermite Wavelet Filters (DHWF) with Prove Mathematical Aspects. Journal of Southwest
Jiaotong University, 55(2).
[43] ASMAA, A.A. SHIHAB, S., and RASHEED, M. (2020). DISCRETE
CHEBYSHEV WAVELET
TRANSFORMATION WITH IMAGE PROCESSING. Journal of Southwest Jiaotong University, 55(2).
[44] DKHILALI, F., MEGDICHE, B. S., RASHEED, M., BARILLE, R., SHIHAB, S.,
GUIDARA, K. and MEGDICHE, M. (2018). Characterizations and morphology of sodium tungstate particles. Royal Society open
science, 5(18), 1-16.
[45] SAIDANI, T., ZAABAT, M., AIDA, M. S., BARILLE, R., RASHEED M. and ALMOHAMMED, Y. (2017). Influence of precursor source on sol–gel deposited ZnO thin films properties. Journal of Materials Science. Materials in Electronics, 28(13), 9252-9257.
[46] BOUMEZOUED, A., GUERGOURI, K., BARILLE, R., RECHEM, D., ZAABAT, M. and RASHEED, M. (2019). ZnO nanopowders doped with bismuth oxide, from synthesis to electrical application.
Journal of Alloys and Compounds, 791,
550-558.
[47] SAIDAI, W., HFAIDH, N., RASHEED, M., GIRTAN, M., MEGRICHE, A. and MAAOUI, M. E. (2016). Effect of B2O3
addition on optical and structural properties of TiO2 as a new blocking layer for multiple
dye sensitive solar cell application (DSSC).
RSC Advances, 6(73), 68819-68826.
[48] DkhilallI, F., BORCHANI, S. M., RASHEED, M., BARILLE, R., GUIDARA, K. and MEGDICHE, M. (2018). Structural, dielectric, and optical properties of the zinc tungstate ZnWO4 compound. Journal of
Materials Science: Materials in Electronics,
29 (8), 6297-6307.
[49] ENNEFFATI, M., LOUATI, B., GUIDARA, K., RASHEED, M. and BARILLE, R. (2018). Crystal structure characterization and AC electrical conduction behavior of sodium cadmium orthophosphate. Journal of Materials Science: Materials in Electronics, 29(1),
171-179.
[50] KADRI, E., KRICHEN, M., MOHAMMED, R., ZOUARI, A. and Khirouni, K. (2016). Electrical transport mechanisms in amorphous silicon/crystalline silicon germanium heterojunction solar cell: impact of passivation layer in conversion efficiency. Optical and Quantum Electronics, 48(12), 546.
[51] KADRI, E., MESSAOUDI, O., KRICHEN, M., DHAHRI, K., RASHEED, M., DHAHRI, E., ZOUARI, A., KHIROUNI, K. and BARILLE, R. (2017).
Optical and electrical properties of SiGe/Si solar cell heterostructures: Ellipsometric study. Journal of Alloys and Compounds, 721, 779-783.
[52] AZAZA, N. B., ELLEUCH, S., RASHEED, M., GINDRE, D., ABID, S., BARILLE, R., ABID, Y. and AMMAR, H. (2019). 3-(p-nitrophenyl) Coumarin derivatives: Synthesis, linear and nonlinear optical properties. Optical Materials, 96, 109328.
[53] ENNEFFATI, M., RASHEED, M., LOUATI, B., GUIDARA, K. and BARILLE, R. (2019). Morphology, UV–visible and ellipsometric studies of sodium lithium orthovanadate. Optical and Quantum Electronics, 51(9), 299.
参考文:
[1] XUFENG，S. 和 DANFU，H.（2010） 变异启发法在求解 n 阶积分微分方程中的 应用。计算与应用数学杂志，234（51）， 第 1442-1447 页。 [2] GHASEMI，M.， TAVASSOLI，K.M。 和 BABOLIAN，E。（2007）他的同伦扰 动方法在非线性积分微分方程中的应用。 应用数学与计算, 188（11），第 538-548 页。 [3] H. TEMIMI，A.R。ANSARI 和 A.M. SIDDIQUI。（2011）的静态梁问题和非 线性积分微分方程的近似解。计算机与数 学与应用，62（8），第 3132-3139 页。 [4] ŞUAYIP，Y.（2016）改进的具有分段 间隔的线性沃尔泰拉积分-微分方程的贝 塞尔配置方法和沃尔泰拉人口模型的应用。 应 用 数 学 建 模 ， 40 （ 9-10 ） ， 第 5349-5363 页。 [5] SHAYANFARD ， F. ， DASTJERDI ， H.L。和 GHAINI，F.M.M。（2020）第三 类沃尔泰拉积分微分方程近似解的配置方 法。应用数值数学，150，第 139-148 页。 [6] DULAT，S.D。（2018）线性弗雷德 霍尔姆积分微分方程的新通用解及其在解 决边值问题上的应用。计算与应用数学学 报，327，第 79-108 页。 [7] 艾比盖尔（I.F.）和 马克（2019）一维 弗雷德霍尔姆积分微分方程的近似解的先 验内斯特罗姆方法误差范围。国际机械科 学杂志，150，第 755-766 页。[8] M.I. BERENGUER ， D 。 GÁMEZ 和 A.J. LÓPEZLINARES。（2013）定点技术 和肖德基近似一阶非线性混合弗雷德霍尔 姆-沃尔泰拉积分微分方程的解。计算与 应用数学杂志，252，第 52-61 页。 [9] SALİH，Y. 和 MEHMET，S.（2000） 用泰勒多项式近似解高阶线性沃尔泰拉· 弗雷德霍尔姆积分微分方程。应用数学与 计算，112（2-315），第 291-308 页。 [10] NUGZAR ， S. ， SERGO ， K. 和 OTAR，J.（2016）一类奇异积分微分方 程的近似解。拉兹马兹数学学院学报， 170（3），第 420-426 页。 [11] MAHMOOD，O. 和 MARYAM，M. （2016）一种模糊积分微分方程近似解的 迭代方法。贝尼·苏埃夫大学基础与应用 科学学报，5 (4)，第 369-376 页。 [12] SEDAGHAT ， S. ， OSTADZAD ， M.H。和 BALEANU，D。（2020）一种 类似于头的数值方法，用于求解分数延迟 积分微分方程。应用数值数学，151，第 322-336 页。 [13] PARISA ， R. ， YADOLLAH ， O. 和 ESMAIL，B.（2017）分数阶伯努利函数 及其在求解分数弗雷德霍姆-沃尔泰拉积 分微分方程中的应用。应用数值数学， 122，第 66-81 页。 [14] FARSHID ， M. 和 NASRIN ， S. （2017）正交贝恩斯坦多项式的应用构建 了求解分数阶随机积分-微分方程的有效 方案。视力，132，第 262-273 页。 [15] PARISA ， R. 和 YADOLLAH ， O. （2020）使用分数替代勒让德函数的非线 性分数积分微分方程的近似解。计算与应 用数学杂志，365，112365。 [16] HENDI，F.A.，WAFA，S。和 AL-BADRANI，H。（2019）分数阶二次积分 -微分方程的存在结果和近似解。沙特国
王大学学报–科学，31（3），第 314-321 页。 [17] 李 华 ， XIAN-FANG ， L。， 玉 林， Z。，向祥，Y.D。（2011）用泰勒展开法 近似求解分数阶积分-微分方程。计算机 与数学与应用，62（3），第 1127-1134 页。 [18] KHADER ， M.M 。 和 SWEILAM ， N.H.（2013）使用切比雪夫伪谱方法求解 分数阶积分-微分方程组的近似解。应用 数学建模，37（2415），pp。9819-9828。 [19] LEI，Z. 和 BIN，L.（2019）积分-偏 微分方程的障碍问题及其在随机最优控制/ 停止问题中的应用。富兰克林学院学报， 356（3），第 1396-1423 页。 [20] KONSTANTINOS ，D.（2020）关于 征收型部分积分-微分方程的有限差分格 式。计算与应用数学杂志，368，112587。 [21] DANA ， Č 。 和 VÁCLAV ， F. （2020）的加勒金方法，带有新的二次样 条小波，用于积分和积分微分方程。计算 与应用数学杂志，363，第 426-443 页。 [22] SAHA ， R.S 。 和 BEHERA ， S. （2020）二维小波运算方法，用于求解沃 尔泰拉弱奇异部分积分微分方程。计算与 应用数学杂志，366，112411。 [23] SACHIN ， K. 和 PRASHANT ， P. （2020）使用阿坦加纳，巴利努时间分数 导数的非线性反应扩散和积分反应扩散方 程的拟小波数值方法。混沌，孤子和分形， 130，109456。 [24] NEMATI ， S. ， LIMA ， P.M. 和 SEDAGHAT，S.（2020）结合了高斯-雅 各比积分的勒让德小波配点方法来求解分 数延迟型积分微分方程。应用数学，149， 第 99-112 页。 [25] MAJID，E .和 AMIN，M.（2019）用 合理化的哈尔小波求解复杂平面中的非线 性积分-微分方程。数学与模拟计算机， 165，第 223-237 页。 [26] 新罕布什尔州贝扎德（2019）一种有 效的算法，基于阿尔珀特的多小波加勒金 方法求解沃尔泰拉积分微分方程。计算与 应用数学杂志，348，第 453-465 页。 [27] HARISH ， K.K. 和 安 东 尼 ·V.V 。 （2018）一类 2D 偏积分微分方程的基于 小波的迭代方法。计算机与数学与应用， 75（11），第 187-198 页。 [28] 焦 W，田田，X。，闫巧，W。，贾 全，X。（2018）非线性分数阶积分微分 方程耦合系统的小波数值模拟。应用数学 与计算，324，第 36-50 页。 [29] BABAAGHAIE ， A 。 和 MALEKNEJAD， K 。（ 2017）基于小波 的高阶积分微分方程的数值解，算法和收 敛性分析。计算与应用数学杂志 325，第 125-133 页。 [30] LI，Z. 和 QIBIN，F.（2012）通过第 二类切比雪夫小波求解分数阶非线性弗雷 德霍尔姆积分微分方程。非线性科学与数 值模拟中的通信，17（6），第 2333-2341 页。 [31] OUDA，E.H。（2018）使用布贝克 多项式的一些变分问题的近似解。巴格达 科学杂志，15（1），第 106-109 页。 [32] RASHEED ， M. 和 BARILLÉ ， R. （2017）直流电溅射衍生的 ITO，二氧化 钛和二氧化钛：铌薄膜的光学常数，其特 征在于用于光电子器件的分光光度法和椭 圆偏振光度法。非晶体固体杂志，476， 第 1-14 页。 [33] RASHEED ， M. 和 BARILLÉ ， R. （2017）通过直流溅射技术在玻璃和宠物 基板上室温沉积氧化锌和铝：氧化锌超薄 膜。光学与量子电子学，49（5），第 1-14 页。 [34] RASHEED ， M. 和 BARILLÉ ， R. （2017）比较通过透明溅射电子溅射技术 在不同基板上沉积的氧化铋和氧化镍超薄 膜的光学性能。合金与化合物学报，728， 第 1186-1198 页。 [35] AUKŠTUOLIS ， A 。 GIRTAN ， MOUSDIS，GA，MALLET，R.，SOCOL， M.，RASHEED，M。和 STANCULESCU， A。（2017）钙钛矿纳米线薄膜中载流子 迁移率的测量相片塞利夫方法。罗马尼亚 科学院院刊-一个系列：数学，物理学， 技术科学，信息科学，18（1），第 34-41 页。 [36] BOURAS ， D. ， MECIF ， A. ， BARILLÉ ， R. ， HARABI ， A. ，
RASHEED ， M. ， MAHDJOUB ， A 。 和 ZAABAT，M。（2018）铜：氧化锌通过 以下方法沉积在多孔陶瓷基板上用于光催 化 应 用 的 简 单 热 方 法 。 陶 瓷 国 际 ， 44 （17），第 21546-21555 页。 [37] KADRI ， E. ， DHAHRI ， K. ， ZAAFOURI ， A. ， KRICHEN ， M. ， RASHEED ， M. ， KHIROUNI ， K. 和 BARILLÉ，R.（2017）非晶硅的交流电导 率和介电行为通过分子束外延法合成的： H / c-硅 1-yGey / 硅薄膜。合金与化合物， 705，第 708-713 页。 [38] DKHILALLI，F.，MEGDICHE，S.， GUIDARA ， K. ， RASHEED ， M. ， BARILLÉ ， R. 和 MEGDICHE ， M. （2018）钨酸钠硫酸钠在体积和晶界响应 中的交流电导率演变。离子学，24（1）， 第 169-180 页。
[39] H.H. NASRAWY ， A.K.O 。 AL-JUBORY 和 K.A. HUSSAINA（2019）一 种解决积分微分方程的高效算法。西南交 通 大 学 学 报 ， 54 （ 6 ） 。 可 从 http://jsju.org/index.php/journal/article/view/ 453 获得。
