Basic Informations

C.V

                                                        

 

• Ph.D. Degree in Pure Mathematics (Numerical Analysis and Approximation Theory), Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt, Sept. 2014.
• M.Sc. Degree in Applied Mathematics, Department of Mathematics, Faculty of Science, Beni-Suef University, Egypt, July 2011.
• B.Sc. Degree, Department of Mathematics, Faculty of Science (Beni-Suef), Cairo University, Egypt, May 2003 Grade: “Excellent with Honor”.

• Lecturer: Department of Mathematics, Faculty of Science, Beni-Suef University, Egypt. October 2014.
• Assistant Lecturer: Department of Mathematics, Faculty of Science, Beni-Suef University, Egypt. July 2011- September 2014.
• Demonstrator: Department of Mathematics, Faculty of Science, Beni-Suef University, Egypt.  September 2005– July 2011.
• Demonstrator: Department of Mathematics, Faculty of Science, Beni-Suef   Branch, Cairo University, Egypt. April 2004 - September 2005.

• Numerical analysis and scientific computing.
• Spectral methods and their applications.
• Developing spectral methods for solving ordinary/partial functional differential equations.
• Nonlinear partial differential equations.
• Functional differential equations.
• Fractional Differential Equations.
• Variable order fractional differential equations.
• Integral equations.
• Fractional integral equations.
• Fractional integro-differential equations.
• Variable order fractional integro-differential equations.
• Distributed order fractional differential equations.
• Complex partial differential equations.
• Error and convergence analysis.
• Orthogonal polynomials.
• Exact solutions of nonlinear partial differential equations that describe nonlinear phenomena appear in many scientific and engineering fields.

 Papers published or to be published in refereed journals:

1. A.H. Bhrawy, M. A. Abdelkawy,  Fouad Mallawi, An Accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays, Boundary Value Problems, 2015 doi: 10.1186/s13661-015-0364-y (2015).                                                                      Citation index:   0                Impact Factor: 0.836        ISSN: 1687-2770
2. R.M. Hafez, M.A. Abdelkawy, E.H. Doha, A.H. Bhrawy, A new collocation scheme for solving hyperbolic equations of second order in a semi-infinite domain, Rom Rep Phys, Accepted                                                                      Citation index:                    Impact Factor: 1.137        ISSN: 1221-1451
3. A.H. Bhrawy, M.A. Zaky, D. Baleanu, M.A. Abdelkawy, A novel spectral approximation for the two-dimensional fractional sub-diffusion problems, Rom. Rep. Phys., 60  (2015)  344–359.                                                                      Citation index:  0                 Impact Factor: 1.137        ISSN: 1221-1451
4. M.A. Abdelkawy, M.A. Zaky, A.H. Bhrawy, D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys., (2015) In Press.                                                                      Citation index:      0              Impact Factor: 1.137        ISSN: 1221-1451

3. A.H. Bhrawy, E.H. Doha, S.S. Ezz-Eldien, M.A. Abdelkawy, A Jacobi spectral collocation scheme based on  operational matrix for  time-fractional modified Korteweg-de Vries equations, Computer Modeling in Engineering & Sciences, CMES201408093121.                                                                      Citation index:  0                  Impact Factor: 1.183         ISSN: 1526 - 1492
4. AH Bhrawy, M. A. Abdelkawy, A fully spectral collocation approximation for multi-dimensional fractional Schrodinger equations, J. Comput.  Phys.  294 (2015) 462–483.                                                                      Citation index: 5                   Impact Factor: 2.485         ISSN: 0021 - 9991
5. E. H Doha, A H Bhrawy, M A. Abdelkawy,  An accurate Jacobi  pseudo-spectral algorithm for parabolic partial differential equations with non-local boundary conditions, J.  Comput.  Nonlin. Dyn. 10 (2015)  021016-13.                                                                                                      Citation index:                    Impact Factor:   1.53          ISSN: 1555 - 1423
6. A.H. Bhrawy, T.M. Taha, M. A. Abdelkawy, R.M. Hafez, On numerical methods for fractional differential equation on a semi-infinite interval, A Book chapter.                                                                      Citation index:                    Impact Factor:         ISSN:

9. Solitons, cnoidal waves, snoidal waves Math. Model., 39 (18) (2015) 5616–5635.

 Citation index:   1                 Impact Factor: 2.158          ISSN: 0307-904X

10. M. A. Abdelkawy, Engy A. Ahmed  and P. Sanchez, A method based on Legendre pseudo-spectral approximations for solving inverse problems of parabolic types equations, Math. Sci. Lett., 4 (2015) 81-90.                                                                                                                        Citation index:                    Impact Factor:         ISSN:
11. A.H. Bhrawy, E.H. Doha, S.S. Ezz-Eldien, M.A. Abdelkawy, A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equation, Calcolo DOI:10.1007/s10092-014-0132-x.                                                                                                Citation index: 4                   Impact Factor: 0.71         ISSN: 0008 - 0624
12. A.H. Bhrawy, M.A. Abdelkawy, A.A. Alzahrani, D. Baleanu, E.O. Alzahrani, A Chebyshev-Laguerre Gauss-Radau collocation scheme for solving time fractional sub-diffusion equation on a semi-infinite domain, Proceedings of The Romanian Academy, Series A, (2015) Accepted.                                                                      Citation index:                    Impact Factor:  1.115       ISSN: 1454 - 9069
13. A.H. Bhrawy, E.H. Doha, D. Baleanu, S.S. Ezz-Eldien, M.A. Abdelkawy, An accurate numerical technique for solving fractional optimal control problems, Proceedings of The Romanian Academy, Series A, 16 (2015)  47–54.                                                                                      Citation index:   5               Impact Factor:  1.115       ISSN: 1454 - 9069
14. E. H Doha, A. H. Bhrawy, M. A. Abdelkawy, R. M. Hafez, Numerical solution of initialy-boundary system of nonlinear hyperbolic equations, Indian Journal of Pure and Applied Mathematics, (2015) Accepted.                                                                                                        Citation index:                    Impact Factor: 0.206          ISSN: 0019-5588
15. M. A. Abdelkawy, S. S. Ezz-Eldien, A. Z. M. Amin, A Jacobi Spectral Collocation Scheme for Solving Abel’s Integral Equations, Progress in Fractional Differentiation and Applications, 1(3) (2015) 187-200.                                                                                                                                Citation index:                    Impact Factor:         ISSN:
16. E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions,  Central European Journal of Physics,  12 (2014) 637-653.                                                                                                                                  Citation index:0               Impact Factor: 1.077          ISSN: 1895-1082
17. E. H Doha, A H Bhrawy, M A Abdelkawy, R. A. Van Gorder, Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations, J. Comput. Phys., 261 (2014) 244–255.                                                                                                                     Citation index:   21                 Impact Factor: 2.485         ISSN: 0021 - 9991
18. A. Biswas, A. H. Bhrawy, M. A. Abdelkawy, A. A. Alshaery, E. M. Hilal, Symbolic computation of some nonlinear fractional differential equations, Romanian Journal of Physics  59 (2013) 433-442.                                                                                                                                                 Citation index: 8                 Impact Factor: 0.745        ISSN: 1221-146X
19. E.H. Doha, A.H. Bhrawy D. Baleanu, M.A. Abdelkawy,  Numerical treatment of Coupled Nonlinear Hyperbolic Klein-Gordon Equations,  Romanian Journal of Physics, 59 (2014) 247–264                                                                      Citation index: 13                  Impact Factor: 0.745        ISSN: 1221-146X
20. A. H. Bhrawy, M. A. Abdelkawy, A. A. Alshaery, E. M. Hilal, Anjan Biswas,   Solitons, cnoidal waves, snoidal waves and other solutions to Whitham-Broer-Kaup system, Applied Mathematics & Information, 8 (2014) 2119-2128 .                                                                                                                          Citation index:    1             Impact Factor: 1.232        ISSN: 2325 - 0399
21. E.H. Doha, A.H. Bhrawy, M.A. Abdelkawy, R.M. Hafez, A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation, Central European Journal of Physics, 12 ( 2014) 111-122.                                                                                                                                                  Citation index:     6               Impact Factor: 1.077          ISSN: 1895-1082
22. E.H. Doha, A.H. Bhrawy, D. Baleanu, and M.A. Abdelkawy, An accurate Legendre collocation scheme for coupled hyperbolic equations with variable coefficients. Romanian Journal of Physics 59 (2014)  247-264.                                                                                                                                                     Citation index:    6               Impact Factor: 0.745        ISSN: 1221-146X
23. A. H. Bhrawy, M. A. Abdelkawy and Anjan Biswas, Optical solitons in (1+1) and (2+1) dimensions, Optik, 125  ( 2014) 1537–1549.                                                                       Citation index:     4               Impact Factor: 0.769          ISSN: 0030-4026
24. M. A. Abdelkawy and T.M. Taha, An operational matrix of fractional derivatives of Laguerre polynomials, Walailak J Sci & Tech  11(12) (2014)  1041-1055.                                                                      Citation index:  1                  Impact Factor:         ISSN: 1686-3933
25. E. H. Doha, A. H. Bhrawy, R.M. Hafez and  M. A. Abdelkawy, A Chebyshev-Gauss-Radau scheme for nonlinear hyperbolic system of first order, Applied Mathematics & Information Sciences, 8 (2014) 535-544.                                                                                                       Citation index:  16                  Impact Factor: 1.232        ISSN: 2325 - 0399
26. E. H. Doha, D. Baleanu, A. H. Bhrawy, and M. A. Abdelkawy, A Jacobi collocation method for solving nonlinear Burgers'-type equations , Abstract and applied analysis 2013, ID 760542, 12 pp. (2013).                                                                                                                                           Citation index:   4                 Impact Factor: 1.274        ISSN: 1085 - 3375
27. A. H. Bhrawy, M. A. Abdelkawy, Anjan Biswas, Topological solitons and cnoidal waves to a few nonlinear wave equations in theoretical physics, Indian Journal of Physics,   87 (2013) 1125-1131                                                                      Citation index:   5                 Impact Factor: 1.337         ISSN: 0973 - 1458
28. A. H. Bhrawy, M. A. Abdelkawy, Computational study of some nonlinear shallow water equations, Central European Journal of Physics, 11 (2013) 518-525.                                                                      Citation index:  6                 Impact Factor: 1.077          ISSN: 1895-1082
29. A.H. Bhrawy and M. A. Abdelkawy, Integrable system modeling shallow water waves: Kaup-Boussinesq shallow water system, Indian Journal of Physics, 87  (2013) 665-671.                                                                      Citation index: 8                   Impact Factor: 1.337         ISSN: 0973 - 1458
30. A.H. Khater, D.K. Callebaut, A.H. Bhrawy and M.A. Abdelkawy, Nonlinear periodic solutions for isothermal magnetostatic atmospheres, Journal of Computational and Applied Mathematics, 242 (2013) 28–40.                                                                                                                     Citation index:   5                 Impact Factor: 1.077          ISSN: 037 - 0427
31.  A. H. Bhrawy, M. A. Abdelkawy and A. Biswas, Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi's elliptic function method, Communications in Nonlinear Science and Numerical Simulation, 18 (2013) 915–925.                                                                      Citation index: 17                   Impact Factor: 2.866         ISSN: 1007 - 5704
32. M. A. Abdelkawy and A.H. Bhrawy, G'/G-expansion method for two-dimensional force-free magnetic fields described by some nonlinear equations, Indian Journal of Physics, 87 (2013) 555-565                                                                                                                                                Citation index:    8                Impact Factor: 1.337         ISSN: 0973 - 1458
33.  A. H. Bhrawy, M. A. Abdelkawy, Sachin Kumar, Stephen Johnson,  Anjan Biswas, Solitons and other solutions to quantum Zakharov-Kuznetsov equation in quantum magneto-plasmas, Indian Journal of Physics,  2013, 87, 455-463.                                                                                                  Citation index: 18                Impact Factor: 1.337         ISSN: 0973 - 1458
34. A.H. Bhrawy, K. Boubaker and M.A. Abdelkawy, Extended F-expansion method for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation, Physical Chemistry: An Indian Journal, 8 (2013) 8-16.                                                                                                                                    Citation index:                    Impact Factor:         ISSN: 0974 - 7524
35. A.H. Bhrawy, A. Yildirim, M. M. Tharwat and M. A. Abdelkawy, A Jacobi elliptic function method for nonlinear arrays of vortices,  Indian Journal of Physics, 86 (2012) 1107-1113.                                                                      Citation index: 13                 Impact Factor: 1.337         ISSN: 0973 - 1458
36. Ali H. Bhrawy, M. Sh. Alhuthali and Mohammed A. Abdelkawy, New solutions for (1+1)-dimensional and (2+1)-dimensional Ito equations, Mathematical Problems in Engineering, 2012, Article ID 537930, pp. 24 (2012).                                                                                                   Citation index:      4              Impact Factor: 1.082          ISSN: 1024-123X
37. A. S. Alofi and M.A. Abdelkawy, Jacobi elliptic function expansion method for Zakharov-Kuznetsov (ZK) equations and Kadomtsov-Petvtashtvtlli (KP) equations, ISST journal of applied physics, 3 (2012) 31-38.                                                                                                                     Citation index:                    Impact Factor:         ISSN: 0976 – 903X
38. A.H. Bhrawy, A.S. Alofi, M.A. Abdelkawy, Time-dependent two-dimensional Zakharov-Kuznetsov equation in the electron-positron-ion plasmas. Life Science Journal, 9 (2012) 1804-1813                                                                                                                                                Citation index:  1                  Impact Factor: 0.165        ISSN: 1097 - 8135
39. M. A. Abdelkawy, M. A. Alghamdi and A.H. Bhrawy, Jacobi doubly periodic wave solutions for three versions of Benjamin-Bona- Mahony equation, Scientific Research and Essays, 7 (2012) 2417-2423.                                                                                                                                      Citation index:                    Impact Factor:         ISSN: 1992-2248
40. A.H. Bhrawy, M. A. Abdelkawy, S. Kumar and A. Biswas, Solitons and other solutions to Kadomtsev-Petviashvili equation of  B-type, Romanian Journal of Physics, 58, (2013) 729-748                                                                      Citation index:  17                Impact Factor: 0.745        ISSN: 1221-146X
41. A.S. Alofi and M.A. Abdelkawy, New exact solutions of Boiti-Leon-Manna-Pempinelli equation using extended F-expansion method,  Life Science Journal, 9 (2012).                                                                      Citation index: 3                Impact Factor: 0.165        ISSN: 1097 - 8135
42. A. H. Khater, D. K. Callebaut and M. A. Abdelkawy, Two-dimensional force-free magnetic fields described by some nonlinear equations, Phys. of plasmas, 17 (2010) 122902.                                                                      Citation index:  9                 Impact Factor: 2.249          ISSN: 1070-664X

• Abstract and Applied Analysis.
• Journal of Optoelectronics and Advanced Materials.
• Applications and Applied Mathematics.
•  Advances in Difference Equations.

• International Association of Geomagnetism and Aeronomy (IAGA) 2nd Symposium Cairo, Egypt, 4th-8th December (2009).
• International Congress on Computational and Applied Mathematics, Leuven, Belgium 5th-9th July (2010).

• A member of the Egyptian Mathematical Society.

• Z. M. Amin, M.Sc. Jan. 2015, thesis entitled:  “Spectral methods for solving fractional integral equations”.
• T. M. Taha, Ph.D Jul. 2015, thesis entitled:  “”.
• According to the database of Scopus, the number of Citations is about 312 with 10 h-index.
• According to Google Search Scholar, the number of Citations is about 178 with 8 h-index.

• Project entitled “A new spectral method  for solving a two-dimensional fractional diffusion equation” funded by faculty of science, Beni-Suef University.

• The prize of the best Ph.D. thesis for 2014 awarded from faculty of science, Beni-Suef University.

 Teaching Experiences:

       1- Courses for undergraduates:

I have experience in teaching courses in numerical analysis, special functions and orthogonal polynomials, differential equations, precalculus, calculus, advanced calculus, analytical geometry, linear algebra, introduction to computer, mathematical statistics, theory of probability, mechanics (statics and dynamics), in Faculty of Science, Faculty of Education, Faculty of Engineering and Faculty of Pharmacy, Beni-Suef University.

 Computer science Experiences:

1. I have programming skills in MATHEMATICA.
2. I have experience in teaching course in numerical analysis using MATHEMATICA.

                                            _---- Dr. M. A. Abdelkawy _---

Master Title

"Nonlinear Models in Plasma Physics and its Applications"

Master Abstract

In this thesis we study some nonlinear models in plasma physics and its applications. We obtained exact solutions for some important equations which describe physical models and we obtained related physical quantities. This thesis consists of an introduction, five chapters, 54 figures and a list of references at the end of each chapter, together with english and arabic summary. This thesis is organized as follow: Introduction. In this introduction we give quick hint for the importance of Magnetohydrodynamics (MHD) and plasma and its applications. Chapter 1. Which considered as a background for the material used in this thesis it cover the fundamental concepts of known results concerning our objects to make this thesis somewhat self contained. Chapter 2. We have investigated isothermal magneto-static (MS) atmospheric Models. The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential A, known as the Grad-Shafranov equation. Specifying the arbitrary functions in the latter equation, one gets the nonlinear elliptic equation. Analytical nonlinear periodic solutions of the elliptic equation are obtained for the case of a nonlinear isothermal atmosphere in a uniform gravitational field. In this chapter we obtained several classes of exact traveling wave solutions of some nonlinear equations using the generalized tanh and Jacobi elliptic function methods, also Bäcklund transformations (BTs) are used to generate new classes of solutions. The final results are used to investigate some models in solar physics. Note: The results in this chapter obtained using generalized tanh method are published in International Association of Geomagnetism and Aeronomy1. The results in this chapter obtained using Bäcklund transformations are published in International Congress on Computational and Applied Mathematics2 and then submitted to Journal of Computational and Applied Mathematics3. Chapter 3. In this chapter we find exact solutions of two-dimensional force-free magnetic fields (FFMFs), the FFMF is a type of field which arises as a special case from the MS equation in plasmas. This special case arises when the plasma pressure is so small, relative to the magnetic pressure, that the plasma pressure may be ignored, and so only the magnetic pressure is considered. The name "force-free" comes from being able to neglect the force from the plasma. We find exact solution of two-dimensional FFMF described by Liouville, sine, double sine, sinh-poission and power force-free magnetic equations. By finding the exact solutions of these equations by using the generalized tanh, Jacobi elliptic function and G'/G-expansion methods. In all those cases the ratio of the current density and the magnetic field is not constant as happens e.g. in the solar atmosphere. Note: The results in this chapter obtained using generalized tanh method are published in Physics of Plasmas4. The results in this chapter obtained using Jacobi elliptic function method submitted to Plasma Physics and Controlled Fusion5. Chapter 4. We discussed the derivation of Sagdeev potential, as a results of which, could be analyzed to predict the existence of various features of localized solitons in various configurations of plasmas. The advantages of the method in finding the solitary waves or double layers stemming from the nonlinear waves was found useful in investigating the large and small amplitude wave propagation. The study advances to describe the spiky and explosive solitary waves along with the possible existence of double layers causeway from the interaction of trapped electrons which are to be expected as common features in space plasmas. Moreover we find exact wave solution of the Korteweg-de Vries (KdV) equation, the KdV equation derived with mixed nonlinearity, the equation in general form and generalized Schamel equation by using generalized tanh and Jacobi elliptic function methods. Chapter 5. we discussed the nonlinear development of ion-acoustic waves in a magnetized plasma under the restrictions of small wave amplitude, weak dispersion, and strong magnetic fields is described by the Zakharov–Kuznetsov (ZK) equation. Kadomtsev-Petviashvili (KP) equation is derived for unmagnetized hot dust plasmas. It suggests that the nonlinear dust acoustic solitary waves in a hot dusty plasma are stable even there are some higher order transverse perturbations. Moreover we find exact wave solution of the ZK, generalized ZK, generalized form of modified ZK, KP, potential KP and Gardner Kadomtsov –Petviashivilli (GKP) equations by using generalized tanh and Jacobi elliptic function methods.

PHD Title

Spectral collocation methods for solving time-dependent partial dierential equations

PHD Abstract

The aim of the present thesis is to investigate the features of Jacobi collocation method for numerical solutions of different types of partial differential equations (PDEs) subject to various kinds of non-local conditions. The speed of convergence is one of the great advantages of spectral collocation method. Moreover, it has exponential rates of convergence; it also has high level of accuracy. In Chapter 1, we present a general introduction to the spectral methods and their advantages over the standard numerical methods. We also clarify the differences between the three most commonly used spectral methods, namely, the Galerkin, collocation and tau methods. A brief discussion is presented for classifying PDEs and non-local conditions. The orthogonal polynomials, their properties and expansion of functions in terms of them are introduced. In Chapter 2, we propose two efficient algorithms for solving parabolic PDEs and hyperbolic PDEs in bounded and semi-infinite domains, respectively. A Jacobi Gauss-Lobatto collocation (J-GL-C) method in conjunction with the two stage implicit Runge-Kutta (IRK) scheme are developed for numerical treatment of parabolic PDEs in bounded domain. A new collocation approach is presented to solve hyperbolic PDEs in semi-infinite domain. In this approach, the Jacobi rational Gauss-Radau collocation (JR-GR-C) method is proposed for spatial discretization, with a special choice of the parameters of Jacobi rational functions, and the JR-GR-C method is adopted for temporal discretization with another choice of these parameters. Several illustrative examples are implemented to reveal that the present methods are very effective and convenient for parabolic PDEs and hyperbolic PDEs in bounded and semi-infinite domains. In Chapter 3, two efficient numerical algorithms are proposed to obtain high accurate numerical solutions for different types of systems of PDEs. In the first one, a Chebyshev-Gauss-Radau collocation (C-GR-C) method in combination with the two stage IRK scheme are employed to obtain highly accurate approximations to the system of nonlinear hyperbolic equations of first order. In the second algorithm, the J-GL-C method is extended to reduce the nonlinear coupled hyperbolic equations of second order with variable coefficients to a system of algebraic equations, which solved by diagonally-implicit Runge-Kutta-Nystr\"{o}m (DIRKN) method. Finally, the second algorithm is implemented to solve the nonlinear coupled viscous Burgers' equation. Special attention is given to the comparison of the numerical results obtained by the new algorithms with those found by other known methods. In Chapter 4, we are concerned with the use of Legendre pseudo-spectral approximation in spatial direction to solve numerically parabolic partial differential equations with time-delay. A scalar delay parabolic partial differential equation is then converted into a system of delay differential equations (DDEs) in time direction that can be solved by continuous Runge-Kutta (CRK) scheme. We adapt the algorithm to solve singularly perturbed and coupled time delay parabolic equations. We extend this algorithm to two-dimensional time delay parabolic equations. Some numerical examples are considered to show the effectiveness and accuracy of the present algorithm for solving stiff partial differential equations. In Chapter 5, we propose an efficient spectral collocation algorithm to solve numerically parabolic and wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation are investigated for the discretization of the spatial variable of such equations. It possesses the spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions is reduced to a system of ordinary differential equations (ODEs) in temporal variable. This system is solved by two-stage forth-order A-stable IRK scheme. Several numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach. Finally in Chapter 6, %\ref{Q6}, a J-GL-C method, used in combination with the two stage IRK method, is proposed as a numerical algorithm for the approximation of solutions to nonlinear Schr\"{o}dinger equations (NLSE) with initial-boundary data in $(1+1)$ dimensions. Our procedure is implemented in two successive steps. In the first one, the J-GL-C method is employed for approximating the functional dependence on the spatial variable, using $(N-1)$ nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon two general Jacobi parameters. The resulting equations together with the two-point boundary conditions induce a system of $2(N-1)$ first-order ODEs in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve this temporal system. The proposed J-GL-C method, used in combination with the implicit Runge-Kutta method of fourth order, is employed to obtain highly accurate numerical approximations to four types of NLSE, including the attractive and repulsive NLSE and a Gross-Pitaevskii equation with space-periodic potential. The proposed techinque is extended and developed to solve the complex Schr\"{o}dinger equations in $(2+1)$ dimensions. The numerical results obtained by these algorithms are compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed methods. Indeed, for relatively few nodes used, the absolute error in our numerical solutions is sufficiently small. The obtained numerical results are tabulated and displayed graphically whenever possible. These results show that our proposed algorithms of solutions are reliable and accurate. Comparisons with previously obtained results by other researchers or exact known solutions are made throughout the context whenever available. To the best of our knowledge, the formulae and algorithms stated and proved in Chapters 2 up to 6 are completely new. The Programs used in this thesis are performed using the PC machine, with ICPU Intel(R) Core(TM) i3-2350M 2 Duo CPU 2.30 GHz, 6.00 GB of RAM.