A new spectral Jacobi–Gauss–Lobatto collocation (J–GL–C) method is developed and analyzed to solve numerically parabolic partial differential equations (PPDEs) subject to initial and nonlocal boundary conditions. The method depends basically on the fact that an expansion in a series of Jacobi polynomials Jn(θ,ϑ)(x) is assumed, for the function and its space derivatives occurring in the partial differential equation (PDE), the expansion coefficients are then determined by reducing the PDE with its boundary conditions into a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit the Runge–Kutta (IRK) method of order four. The proposed method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the spatial discretizations. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.

References

References
1.
Canuto
,
C.
,
Hussaini
,
M. Y.
,
Quarteroni
,
A.
, and
Zang
,
T. A.
,
2006
,
Spectral Methods: Fundamentals in Single Domains
,
Springer-Verlag
,
New York
.
2.
Dehghan
,
M.
, and
Sabouri
,
M.
,
2013
, “
A Legendre Spectral Element Method on a Large Spatial Domain to Solve the Predator-Prey System Modeling Interacting Populations
,”
Appl. Math. Model.
,
37
, pp.
1028
1038
.10.1016/j.apm.2012.03.030
3.
Doha
,
E. H.
,
Bhrawy
,
A. H.
,
Baleanu
,
D.
, and
Hafez
,
R. M.
,
2014
, “
A New Jacobi Rational-Gauss Collocation Method for Numerical Solution of Generalized Pantograph Equations
,”
Appl. Numer. Math.
,
77
, pp.
43
54
.10.1016/j.apnum.2013.11.003
4.
Bhrawy
,
A. H.
,
2013
, “
A Jacobi-Gauss-Lobatto Collocation Method for Solving Generalized Fitzhugh-Nagumo Equation With Time-Dependent Coefficients
,”
Appl. Math. Comput.
,
222
, pp.
255
264
.10.1016/j.amc.2013.07.056
5.
Wan
,
Z.
,
Chen
,
Y.
, and
Huang
,
Y.
,
2009
, “
Legendre Spectral Galerkin Method for Second-Kind Volterra Integral Equations
,”
Front. Math. China
,
4
, pp.
181
193
.10.1007/s11464-009-0002-z
6.
Rahmoune
,
A.
,
2013
, “
Spectral Collocation Method for Solving Fredholm Integral Equations on the Half-Line
,”
Appl. Math. Comput.
,
219
, pp.
9254
9260
.10.1016/j.amc.2013.03.043
7.
Ahmadian
,
A.
,
Suleiman
,
M.
,
Salahshour
,
S.
, and
Baleanu
,
D.
,
2013
, “
A Jacobi Operational Matrix for Solving a Fuzzy Linear Fractional Differential Equation
,”
Adv. Difference Equations
,
2013
, p.
104
.10.1186/1687-1847-2013-104
8.
Bhrawy
,
A. H.
, and
Alghamdi
,
M.
,
2012
, “
A Shifted Jacobi-Gauss-Lobatto Collocation Method for Solving Nonlinear Fractional Langevin Equation Involving Two Fractional Orders in Different Intervals
,”
Boundary Value Problems
,
2012
, p.
62
.10.1186/1687-2770-2012-62
9.
Doha
,
E. H.
,
Bhrawy
,
A. H.
,
Hafez
,
R. M.
, and
Abdelkawy
,
M. A.
,
2014
, “
A Chebyshev-Gauss-Radau Scheme for Nonlinear Hyperbolic System of First Order
,”
Appl. Math. Info. Sci.
,
8
(
2
), pp.
535
544
.10.12785/amis/080211
10.
Bhrawy
,
A. H.
, and
Alofi
,
A. S.
,
2012
, “
A Jacobi-Gauss Collocation Method for Solving Nonlinear Lane-Emden Type Equations
,”
Commun. Nonlin. Sci. Numer. Simul.
,
17
, pp.
62
70
.10.1016/j.cnsns.2011.04.025
11.
Bialecki
,
B.
, and
Karageorghis
,
A.
,
2008
, “
Spectral Chebyshev-Fourier Collocation for the Helmholtz and Variable Coefficient Equations in a Disk
,”
J. Comput. Phys.
,
227
(
19
), pp.
8588
8603
.10.1016/j.jcp.2008.06.009
12.
Yuzbasi
,
S.
,
Sezer
,
M.
, and
Kemanci
,
B.
,
2013
, “
Numerical Solutions of Integro-differential Equations and Application of a Population Model With an Improved Legendre Method
,”
Appl. Math. Model.
,
37
, pp.
2086
2101
.10.1016/j.apm.2012.05.012
13.
Yuzbasi
,
S.
, and
Sahin
,
N.
,
2013
, “
Numerical Solutions of Singularly Perturbed One-Dimensional Parabolic Convection. Diffusion Problems by the Bessel Collocation Method
,”
Appl. Math. Comput.
,
220
, pp.
305
315
.10.1016/j.amc.2013.06.027
14.
Esmaeili
,
S.
,
Shamsi
,
M.
, and
Luchko
,
Y.
,
2011
, “
Numerical Solution of Fractional Differential Equations With a Collocation Method Based on Mntz Polynomials
,”
Comput. Math. Appl.
,
62
(
3
), pp.
918
929
.10.1016/j.camwa.2011.04.023
15.
Doha
,
E. H.
,
Bhrawy
,
A. H.
,
Abdelkawy
,
M. A.
, and
Hafez
,
R. M.
,
2014
, “
A Jacobi Collocation Approximation for Nonlinear Coupled Viscous Burgers' Equation
,”
Cent. Eur. J. Phys.
,
12
, pp.
111
122
.10.2478/s11534-014-0429-z
16.
Eslahchi
,
M. R.
,
Dehghan
,
M.
, and
Parvizi
,
M.
,
2014
, “
Application of the Collocation Method for Solving Nonlinear Fractional Integro-differential Equations
,”
J. Comput. Appl. Math.
,
257
, pp.
105
128
.10.1016/j.cam.2013.07.044
17.
Capsso
,
V.
, and
Kunisch
,
K.
,
1988
, “
A Reaction-Diffusion System Arising in Modeling Man-Environment Diseases
,”
Q. Appl. Math.
,
48
, pp.
431
449
.
18.
Yurchuk
,
N. I.
,
1986
, “
Mixed Problem With an Integral Condition for Certain Parabolic Equations
,”
J. Differential Equations
,
22
, pp.
2117
2126
.
19.
Cannon
,
J. R.
,
Lin
,
Y.
, and
Matheson
,
A. L.
,
1993
, “
The Solution of the Diffusion Equation in Two-Space Variables Subject to the Specification of Mass
,”
Applicable Anal.
,
50
, pp.
1
19
.10.1080/00036819308840181
20.
Bouziani
,
A.
,
1999
, “
On a Class of Parabolic Equations With a Nonlocal Boundary Condition
,”
Acad. R. Belg., Bull. Classe Sci.
,
10
, pp.
61
77
.
21.
Diaz
,
J. I.
,
1997
, “
On a Nonlocal Elliptic Problem Arising in the Magnetic Confinement of a Plasma in a Stellarator
,”
Nonlin. Anal.
,
30
, pp.
3963
3974
.10.1016/S0362-546X(97)00269-1
22.
Farengo
,
R.
,
Lee
,
Y. C.
, and
Guzdar
,
P. N.
,
1983
, “
An Electromagnetic Integral Equation: Application to Microtearing Modes
,”
Phys. Fluids
,
26
, pp.
3515
3523
.10.1063/1.864112
23.
Ashyralyev
,
A.
, and
Aggez
,
N.
,
2004
, “
A Note on the Difference Schemes of the Nonlocal Boundary Value Problems for Hyperbolic Equations
,”
Numer. Funct. Anal. Opt.
,
25
, pp.
439
462
.10.1081/NFA-200041711
24.
Sajavičius
,
S.-U.
,
2012
, “
Stability of the Weighted Splitting Finite-Difference Scheme for a Two-Dimensional Parabolic Equation With Two Nonlocal Integral Conditions
,”
Comput. Math. Appl.
,
64
, pp.
3485
3499
.10.1016/j.camwa.2012.08.009
25.
Martín-Vaquero
,
J.
, and
Vigo-Aguiar
,
J.
,
2009
, “
On the Numerical Solution of the Heat Conduction Equations Subject to Nonlocal Conditions
,”
Appl. Numer. Math.
,
59
, pp.
2507
2514
.10.1016/j.apnum.2009.05.007
26.
Dehghan
,
M.
,
2004
, “
Numerical Schemes for One-Dimensional Parabolic Equations With Nonstandard Initial Condition
,”
Appl. Math. Comput.
,
147
, pp.
321
331
.10.1016/S0096-3003(02)00669-0
27.
Golbabai
,
A.
, and
Javidi
,
M.
,
2007
, “
Numerical Solution for Non-classical Parabolic Problem Based on Chebyshev Spectral Collocation Method
,”
Appl. Math. Comput.
,
190
, pp.
179
185
.10.1016/j.amc.2007.01.033
28.
Liu
,
Y.
,
1999
, “
Numerical Solution of the Heat Equation With Nonlocal Boundary Conditions
,”
J. Comput. Appl. Math.
,
110
, pp.
115
127
.10.1016/S0377-0427(99)00200-9
29.
Olmstead
,
W. E.
, and
Roberts
,
C. A.
,
1997
, “
The One-Dimensional Heat Equation With a Nonlocal Initial Condition
,”
Appl. Math. Lett.
,
10
, pp.
89
94
.10.1016/S0893-9659(97)00041-4
30.
Mu
,
L.
, and
Du
,
H.
,
2008
, “
The Solution of a Parabolic Differential Equation With Non-local Boundary Conditions in the Reproducing Kernel Space
,”
Appl. Math. Comput.
,
202
, pp.
708
714
.10.1016/j.amc.2008.03.008
31.
Ang
,
W.-T.
,
2006
, “
Numerical Solution of a Non-classical Parabolic Problem: An Integro-differential Approach
,”
Appl. Math. Comput.
,
175
, pp.
969
979
.10.1016/j.amc.2005.08.011
32.
Lardner
,
R. W.
,
1990
, “
Stability of the Numerical Solution of a Parabolic System With Integral Subsidiary Conditions
,”
Comput. Math. Appl.
,
19
, pp.
41
46
.10.1016/0898-1221(90)90249-J
33.
Slodička
,
M.
, and
Dehilis
,
S.
,
2009
, “
A Numerical Approach for a Semilinear Parabolic Equation With a Nonlocal Boundary Condition
,”
J. Comput. Appl. Math.
,
231
, pp.
715
724
.10.1016/j.cam.2009.04.016
34.
Doha
,
E. H.
,
Bhrawy
,
A. H.
, and
Hafez
,
R. M.
,
2011
, “
A Jacobi-Jacobi Dual-Petrov-Galerkin Method for Third- and Fifth-Order Differential Equations
,”
Math. Comput. Model.
,
53
, pp.
1820
1932
.10.1016/j.mcm.2011.01.002
35.
Tchiotsop
,
D.
,
Wolf
,
D.
,
Louis-Dorr
, V
.
, and
Husson
,
R.
,
2007
, “
ECG Data Compression Using Jacobi Polynomials
,”
Proceedings of the 29th Annual International Conference of the IEEE EMBS
, pp.
1863
1867
.
36.
Gottlieb
,
D.
, and
Shu
,
C.-W.
,
1997
, “
On the Gibbs Phenomenon and Its Resolution
,”
SIAM Rev.
,
29
, pp.
644
668
.10.1137/S0036144596301390
37.
Doha
,
E. H.
, and
Bhrawy
,
A. H.
,
2009
, “
A Jacobi Spectral Galerkin Method for the Integrated Forms of Fourth-Order Elliptic Differential Equations
,”
Numer. Methods Partial Differential Equations
,
25
(
3
), pp.
712
739
.10.1002/num.20369
38.
El-Kady
,
M.
,
2012
, “
Jacobi Discrete Approximation for Solving Optimal Control Problems
,”
J. Korean Math. Soc.
,
49
, pp.
99
112
.10.4134/JKMS.2012.49.1.099
39.
Szegö
,
G.
,
1939
,
Orthogonal Polynomials
, Colloquium Publications, XXIII, American Mathematical Society, MR 0372517G, Providence, RI.
40.
Doha
,
E. H.
,
Bhrawy
,
A. H.
,
Abdelkawy
,
M. A.
, and
Van Gorder
,
R. A.
,
2014
, “
Jacobi-Gauss-Lobatto Collocation Method for the Numerical Solution of 1+1 Nonlinear Schrodinger Equations
,”
J. Comput. Phys.
,
261
, pp.
244
255
.10.1016/j.jcp.2014.01.003
41.
Butcher
,
J. C.
,
1987
,
The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods
,
Wiley
,
New York
.
42.
Butcher
,
J. C.
,
1964
, “
Implicit Runge-Kutta Processes
,”
Math. Comput.
,
18
, pp.
50
64
.10.1090/S0025-5718-1964-0159424-9
43.
Doha
,
E. H.
,
1983
, “
On the Chebyshev Method to Linear Parabolic P. D. E. s With Inhomogeneous Mixed Boundary Conditions
,”
Arab J. Math.
,
4
, pp.
31
57
.
44.
Mittal
,
R. C.
, and
Jain
,
R. K.
,
2012
, “
Cubic B-Splines Collocation Method for Solving Nonlinear Parabolic Partial Differential Equations With Neumann Boundary Conditions
,”
Commun. Nonlin. Sci. Numer. Simul.
,
17
, pp.
4616
4625
.10.1016/j.cnsns.2012.05.007
45.
Dehghan
,
M.
,
2004
, “
Numerical Solution of a Parabolic Equation Subject to Specification of Energy
,”
Appl. Math. Comput.
,
149
, pp.
31
45
.10.1016/S0096-3003(02)00954-2
46.
Zhoua
,
Y.
,
Cui
,
M.
, and
Lin
,
Y.
,
2009
, “
Numerical Algorithm for Parabolic Problems With Non-classical Conditions
,”
J. Comput. Appl. Math.
,
230
, pp.
770
780
.10.1016/j.cam.2009.01.012
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