In this work, a highly efficient algorithm is developed for solving the parabolic partial differential equation (PDE) with the nonlocal condition. For this purpose, we employ orthogonal Chelyshkov polynomials as the basis. The convergence analysis of the proposed scheme is derived. Numerical experiments are carried out to explain the efficiency and precision of the proposed scheme. Furthermore, the reliability of the scheme is verified by comparisons with assured existing methods.
In the else decades, nonlocal boundary value problems have become a rapidly increasing field of research. The study of this type of problem is driven not only by a theoretical interest, but also by the fact that several phenomena in engineering, physics and life sciences can be modelled in this way. For example, problems with feedback controls such as the steady-states of a thermostat, transfer reactive and passive pollutants in underground water [1] [2], heat transfer, radioactive nuclear decay in fluid streams [3], viscoelastic material malformation in polymers [3], semiconductor modelling [4] and bioengineering.
The variety of physical phenomena developed on a (PDEs) concerning non-local integral terms is constantly increasing. The authors of [5] have given an example from metrology. This example is a prototype for the evolution of the system temperature distribution of air above the ground during calm nights. Specific problems occur in thermodynamics in thermoelasticity [6] [7] [8], heat transfer [9] [10] [11] and plasma physics [12]. The above-mentioned articles focus on the problems described in terms of parabolic equations. However, there are some problems dealing with the dynamics of the ground waters which are described in terms of hyperbolic equations [13].
The numerical research for PDEs with distinct types of non-local conditions is of great interest due to their broad range of applications. Several methods for solving nonlocal boundary problems have been developed such as finite-difference schemes [14], finite volume element method [15] [16], implicit finite difference scheme [17], Galerkin procedure [18] [19] [20], spectral collocation with preconditioning method [21], Chebyshev spectral collocation techniques [22] [23], Tau scheme [24], sinc method [25] [26] [27], sinc-Galerkin method [28], finite difference methods [14], spectral collocation method with preconditioning [21], Gaussian radial basis functions method [29], Legendre-Gauss-Lobatto pseudo-spectral method [30] [31]. Recently, El-Gamel and Abd El-Hady applied sinc collocation approach for solving a parabolic PDE with nonlocal boundary conditions [32]. Existence, uniqueness and some characteristics of the solution to these issues have been developed in [33].
In this paper, we attempt to introduce a new method, based on Chelyshkov polynomials for solving
∂ u ( η , t ) ∂ t = ∂ 2 u ( η , t ) ∂ η 2 + q ( η , t ) , 0 < η < 1 , 0 < t ≤ 1 (1)
with initial condition
u ( η , 0 ) = f ( η ) (2)
subject to boundary conditions
∑ k = 0 1 α k ( t ) ∂ k ∂ η k u ( 0 , t ) + ∫ 0 1 k 1 ( η , t ) u ( η , t ) d η = g 1 ( t ) (3)
∑ k = 0 1 β k ( t ) ∂ k ∂ η k u ( 1 , t ) + ∫ 0 1 k 2 ( η , t ) u ( η , t ) d η = g 2 ( t ) (4)
where q , f , k i , g i , α i and β i , i = 0 , 1 , are known functions.
The area of orthogonal polynomials is a very strong research area in mathematics as well as in purposes in mathematical physics, engineering and computer science. One of orthogonal polynomials is the set of the Chelyshkov polynomials. These polynomials have formed by Chelyshkov [34] [35], which are orthogonal over the interval [ 0,1 ] with respect to the weight function w ( x ) = 1 . Chelyshkov orthogonal basis has been used for solving several kinds of integral and differential equations. For example, nonlinear weakly singular integral equations In [36], a class of mixed functional integro-differential equations In [37] [38], Volterra-Hammerstein delay integral equations In [39], the two-dimensional Fredholm-Volterra integral equations In [40], a systems of linear functional differential equations In [41] and distributed-order fractional differential equations In [42]. Moradi et al. In [43] applied Chelyshkov wavelets of time-delay fractional optimal control problems and El-Gamel et al. In [44] used Chelyshkov-Tau approach for solving Bagley-Torvik equation.
The layout of this paper is as follows. Section 2, below briefly references, in which the reader can find an excellent summary of Chelyshkov polynomials which are required for establishing our results. Section 3 is dedicated to the formulation of Chelyshkov collocation scheme. In Section 4, the convergence analysis of the proposed method is investigated and error estimation for the fully discrete problem. Section 5 includes test examples to illustrate the accuracy and the performance of our scheme. Conclusion is made in Section 6.
2. An Overview and Relations of Chelyshkov Polynomials
An appropriate solution is expressed in the following form
u N ( η ) ≅ ∑ j = 0 N a j ψ N , j ( η )
so that a j and ψ N , j ( η ) , j = 0 , 1 , 2 , ⋯ , N , respectively, are the unknown Chelyshkov coefficients and Chelyshkov orthogonal polynomials of the degree N
ψ N , j ( η ) = ∑ i = 0 N − j ( − 1 ) i ( N − j i ) ( N + j + i + 1 N − j ) η i + j , j = 0 , 1 , ⋯ , N (5)
The Chelyshkov polynomials can be connected to Rodrigues’ type expansion by
ψ N , j ( η ) = 1 ( N − j ) ! 1 η j − 1 d N − j d η N − j [ η N + j + 1 ( 1 − η ) N − j ] , j = 0 , 1 , ⋯ , N
with orthogonality relation
∫ 0 1 ψ N , i ( η ) ψ N , j ( η ) d η = { 1 i + j + 1 , for i = j , i , j = 0 , 1 , ⋯ , N , N + 1 0 for i ≠ j . (6)
Another relation to Chelyshkov polynomials from the previous one is
∫ 0 1 ψ N , j ( η ) d η = ∫ 0 1 η j d η = 1 j + 1
The Chelyshkov polynomials could be represented in terms of the Jacobi polynomials P k ( α , β ) by the following relation
ψ N , j ( η ) = ( − 1 ) N − j η N − j P N − j ( 0,2 j + 1 ) ( 2 η − 1 ) , j = 0,1, ⋯ , N .
We can write u ( η ) and its derivative in matrix forms as follows:
u ( η ) = ψ ( η ) A = χ ( η ) H A u ( 1 ) ( η ) = ψ ( 1 ) ( η ) A = χ ( η ) M H A , u ( 2 ) ( η ) = ψ ( 2 ) ( η ) A = χ ( η ) M 2 H A , (7)
where
ψ = [ ψ N , 0 , ψ N , 1 , ⋯ , ψ N , N ]
A = [ a 0 , ⋯ , a N ] τ , and χ ( η ) = [ 1 η η 2 ⋯ η N ]
for N is odd,
H = [ ( N 0 ) ( N + 1 N ) 0 ⋯ 0 0 − ( N 1 ) ( N + 2 N ) ( N − 1 0 ) ( N + 2 N − 1 ) ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ ( N N − 1 ) ( 2 N N ) − ( N − 1 N − 2 ) ( 2 N N − 1 ) ⋯ ( 1 0 ) ( 2 N 1 ) 0 − ( N N ) ( 2 N + 1 N ) ( N − 1 N − 1 ) ( 2 N + 1 N − 1 ) ⋯ − ( 1 1 ) ( 2 N + 1 1 ) 1 ] ( N + 1 ) × ( N + 1 )
for N is even,
H = [ ( N 0 ) ( N + 1 N ) 0 ⋯ 0 0 − ( N 1 ) ( N + 2 N ) ( N − 1 0 ) ( N + 2 N − 1 ) ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ − ( N N − 1 ) ( 2 N N ) ( N − 1 N − 2 ) ( 2 N N − 1 ) ⋯ ( 1 0 ) ( 2 N 1 ) 0 ( N N ) ( 2 N + 1 N ) − ( N − 1 N − 1 ) ( 2 N + 1 N − 1 ) ⋯ − ( 1 1 ) ( 2 N + 1 1 ) 1 ] ( N + 1 ) × ( N + 1 )
and
M = [ 0 1 0 ⋯ 0 0 0 2 ⋯ 0 0 0 0 ⋱ 0 0 0 0 ⋯ N 0 0 0 ⋯ 0 ] ( N + 1 ) × ( N + 1 )
3. Direct Chelyshkov Collocation Method
In this part, we will approximate the solution of the Equation (1) as follows
u ( η , t ) ≈ u N ( η , t ) = ∑ i = 0 N ∑ j = 0 N c i j ψ N , i ( η ) ψ N , j ( t ) = ψ ( η , t ) C , (8)
in which
ψ ( η , t ) = ψ ( η ) ⊗ ψ ( t ) = [ ψ N , 0 ( η ) ψ N , 0 ( t ) ⋯ ψ N , 0 ( η ) ψ N , N ( t ) ⋯ ψ N , N ( η ) ψ N ,0 ( t ) ⋯ ψ N , N ( η ) ψ N , N ( t ) ]
where ⊗ represent for the kronecker product and also C = [ c 00 ⋯ c 0 N ⋯ c N 0 ⋯ c N N ] τ which be an unknown vector and will be obtained by our scheme.
Clearly,
∂ ∂ t u N ( η , t ) = [ ψ ( η ) ⊗ ψ ( t ) ] t C = [ ψ ( η ) ⊗ [ χ ( t ) M H ] ] C (9)
Likewise, we should deduce that
∂ 2 ∂ η 2 u N ( x , t ) = [ ψ ( η ) ⊗ ψ ( t ) ] η η C = [ [ χ ( η ) M 2 H ] ⊗ χ ( t ) ] C (10)
consider that u ( η , t ) is approximated by u N ( η , t ) and we discretize the Equation (1) in the following form
∂ u N ( η i , t j ) ∂ t = ∂ 2 u N ( η i , t j ) ∂ η 2 + q ( η i , t j ) , i = 1 , 2 , ⋯ , N − 1 ; j = 1 , 2 , ⋯ , N (11)
where
η i = t i = i N , i = 0 , 1 , 2 , ⋯ , N
by substituting (9) and (10) into Equation (11) we obtain
[ ψ ( η i ) ⊗ [ χ ( t j ) M H ] − [ χ ( η i ) M 2 H ] ⊗ ψ ( t j ) ] C = q ( η i , t j ) , i = 1 , 2 , ⋯ , N − 1 ; j = 1 , 2 , ⋯ , N (12)
The initial conditions (2) can be discretized in the form
ψ ( η i , 0 ) C = f ( η i ) , i = 0 , 1 , ⋯ , N (13)
The boundary condition (3) can be discretized in the form
V 1 ( t j ) C = g 1 ( t j ) , j = 1 , 2 , ⋯ , N (14)
where
V 1 ( t j ) = ∑ k = 0 1 α k ( t j ) [ χ ( 0 ) M k H ] ⊗ ψ ( t j ) + [ ∫ 0 1 k 1 ( η , t j ) ψ ( η ) d η ] ⊗ ψ ( t j ) , j = 1,2, ⋯ , N
The boundary condition (4) is discretized in the form below
V 2 ( t j ) C = g 2 ( t j ) , j = 1 , 2 , ⋯ , N (15)
where
V 2 ( t j ) = ∑ k = 0 1 β k ( t j ) [ χ ( 1 ) M k H ] ⊗ ψ ( t j ) + [ ∫ 0 1 k 2 ( η , t j ) ψ ( η ) d η ] ⊗ ψ ( t j ) , j = 1 , 2 , ⋯ , N
Equations (12)-(15) can be rewritten in the form
Λ C = Φ (16)
where
Λ = [ ψ ( η 1 ) ⊗ [ χ ( t 1 ) M H ] − [ χ ( η 1 ) M 2 H ] ⊗ ψ ( t 1 ) ⋮ ψ ( η 1 ) ⊗ [ χ ( t N ) M H ] − [ χ ( η 1 ) M 2 H ] ⊗ ψ ( t N ) ⋮ ψ ( η N − 1 ) ⊗ [ χ ( t 1 ) M H ] − [ χ ( η N − 1 ) M 2 H ] ⊗ ψ ( t 1 ) ⋮ ψ ( η N − 1 ) ⊗ [ χ ( t N ) M H ] − [ χ ( η N − 1 ) M 2 H ] ⊗ ψ ( t N ) ψ ( η 0 ,0 ) ⋮ ψ ( η N ,0 ) V 1 ( t 1 ) ⋮ V 1 ( t N ) V 2 ( t 1 ) ⋮ V 2 ( t N ) ] ( N + 1 ) 2 × ( N + 1 ) 2
Φ = [ q ( η 1 , t 1 ) ⋮ q ( η 1 , t N ) ⋮ q ( η N − 1 , t 1 ) ⋮ q ( η N − 1 , t N ) f ( η 0 ) ⋮ f ( η N ) g 1 ( t 1 ) ⋮ g 1 ( t N ) g 2 ( t 1 ) ⋮ g 2 ( t N ) ] ( N + 1 ) 2 × 1
Then we solve the generated linear system of ( N + 1 ) 2 × ( N + 1 ) 2 equations by using Q-R method.
4. Convergence and Error Estimation4.1. Convergence of Chelyshkov Polynomial
Now, we will introduce the convergence and error bound to Chelyshkov polynomials.
Theorem 4.1. Suppose that the function u ( η , t ) : [ 0,1 ] × [ 0,1 ] → ℝ 2 is k + 1 times continuously differentiable, u ( η , t ) ∈ C k + 1 [ 0,1 ] × [ 0,1 ] and u ^ ( η , t ) is the best approximation of u ( η , t ) in the space X × T where
X = S p a n { ψ N ,0 ( η ) , ψ N ,1 ( η ) , ψ N ,2 ( η ) , ⋯ , ψ N , N ( η ) }
and
T = S p a n { ψ N ,0 ( t ) , ψ N ,1 ( t ) , ψ N ,2 ( t ) , ⋯ , ψ N , N ( t ) }
then the error bound is
‖ u ( η , t ) − u ^ ( η , t ) ‖ 2 ≤ M ( k + 1 ) ! 2 2 k + 4 − 2 ( 2 k + 3 ) ( 2 k + 4 ) (17)
where, ‖ . ‖ 2 refers to the norm defined as
‖ u ( η , t ) ‖ 2 2 = ∫ 0 1 ∫ 0 1 | u ( η , t ) | 2 d η d t ,
and M = max | u ( ϑ ) ( η , t ) | , ϑ = 0 , 1 , ⋯ , k + 1
Proof. Due to u ( η , t ) ∈ C k + 1 [ 0,1 ] × [ 0,1 ] , then
∃ M > 0 : ∀ ( η , t ) ∈ [ 0 , 1 ] × [ 0 , 1 ] , | u ( ϑ ) ( η , t ) | ≤ M , ϑ = 0 , 1 , ⋯ , k + 1 .
We use the Maclaurin expansion of u ( η , t ) as follows:
u ( η , t ) = ∑ ϑ = 0 k 1 ϑ ! ( η ∂ ∂ η + t ∂ ∂ t ) ϑ u ( η , t ) | ( η , t ) = ( 0 , 0 ) + 1 ( k + 1 ) ! ( η ∂ ∂ η + t ∂ ∂ t ) k + 1 u ( η , t ) | ( η , t ) = ( η ^ , t ^ )
with η ^ , t ^ ∈ [ 0,1 ] and let
u ˜ ( η , t ) = ∑ ϑ = 0 k 1 ϑ ! ( η ∂ ∂ η + t ∂ ∂ t ) ϑ u ( η , t ) | ( η , t ) = ( 0 , 0 )
then the error bound
‖ u ( η , t ) − u ^ ( η , t ) ‖ 2 ≤ ‖ u ( η , t ) − u ˜ ( η , t ) ‖ 2 = ‖ 1 ( k + 1 ) ! ( η ∂ ∂ η + t ∂ ∂ t ) k + 1 u ( η ^ , t ^ ) ‖ 2 = ( ∫ 0 1 ∫ 0 1 ( 1 ( k + 1 ) ! ( η ∂ ∂ η + t ∂ ∂ t ) k + 1 u ( η ^ , t ^ ) ) 2 d η d t ) 1 2 (18)
but
( η ∂ ∂ η + t ∂ ∂ t ) k + 1 u ( η ^ , t ^ ) = ∑ i = 0 k + 1 ( k + 1 i ) η k − i + 1 t i ∂ k + 1 ∂ η k − i + 1 ∂ t i u ( η , t ^ ) ≤ M ( η + t ) k + 1 (19)
by using two Equations (18), (19) we obtain
( ∫ 0 1 ∫ 0 1 ( 1 ( k + 1 ) ! ( η ∂ ∂ η + t ∂ ∂ t ) k + 1 u ( η ^ , t ^ ) ) 2 d η d t ) 1 2 ≤ ( ∫ 0 1 ∫ 0 1 ( M ( k + 1 ) ! ( η + t ) k + 1 ) 2 d η d t ) 1 2 = ( ∫ 0 1 ∫ 0 1 ( ( η + t ) k + 1 ) 2 d η d t ) 1 2 = M ( k + 1 ) ! ( ∫ 0 1 ∫ 0 1 ( η + t ) 2 k + 2 d η d t ) 1 2 = M ( k + 1 ) ! 2 2 k + 4 − 2 ( 2 k + 3 ) ( 2 k + 4 ) (20)
4.2. Error Estimation of Chelyshkov-Collocation Method
In this section, the error estimation for the Chelyshkov-collocation method has been employed with the residual error function [45] [46] [47] [48]. One can obtain the residual function first, we can display the residual function R N ( η , t ) as
R N ( η , t ) = u ^ t ( η , t ) − u ^ η η ( η , t ) − q ( η , t ) . (21)
where u ^ ( η , t ) is approximate solution given by (8) of Equation (1). Thus, u ^ ( η , t ) fulfills the equation
u ^ t ( η , t ) − u ^ η η ( η , t ) = q ( η , t ) + R N ( η , t ) ,
u ^ ( η , 0 ) = f ( η )
∑ k = 0 1 α k ( t ) ∂ k ∂ η k u ^ ( 0 , t ) + ∫ 0 1 k 1 ( η , t ) u ^ ( η , t ) d η = g 1 ( t )
∑ k = 0 1 β k ( t ) ∂ k ∂ η k u ^ ( 1 , t ) + ∫ 0 1 k 2 ( η , t ) u ^ ( η , t ) d η = g 2 ( t )
so, we can obtain the error function
ε ( η ) = u ( η , t ) − u ^ ( η , t ) , (22)
such that u ( η , t ) is the exact solution of Equation (1).
Accordingly, the error differential equation is
ε t ( η , t ) − ε η η ( η , t ) = − R N ( η , t ) , ∑ k = 0 1 α k ( t ) ∂ k ∂ η k ε ( 0, t ) + ∫ 0 1 k 1 ( η , t ) ε ( η , t ) d η = 0 ∑ k = 0 1 β k ( t ) ∂ k ∂ η k ε ( 1, t ) + ∫ 0 1 k 2 ( η , t ) ε ( η , t ) d η = 0 (23)
The solution of Equation (23) is
ε ( η , t ) = ∑ i = 0 N ∑ j = 0 N c ˜ i j ψ N , i ( η ) ψ N , j ( t )
In the same manner as Section 3, we obtain unknown coefficients c ˜ i j , i , j = 0 , 1 , 2 , ⋯ , N . so, the maximum absolute error can be determined by
E max = max | ε ( η , t ) | , 0 < η < 1 , 0 < t ≤ 1
Using maximum error estimation, we can test the reliability of the results especially if the exact solution is unknown.
5. Numerical Results
In this section, we experimentally illustrate the performance of the proposed scheme. Numerically, we verify that our proposed scheme can deal with the parabolic PDE equation with the nonlocal condition. We consider the following five examples, namely the nonlocal problems from [14] [21] [25] [29]. The formula of the maximum absolute error is
‖ E J ‖ = max { | u ( η i , t j ) − u N ( η i , t j ) | } , i = 0 , 1 , ⋯ , N ; j = 0 , 1 , ⋯ , N
Example 1: [14] consider the following parabolic PDE
∂ u ∂ t = ∂ 2 u ∂ η 2 − e − t [ η ( η − 1 ) + δ 2 6 ( 1 + δ 2 ) + 2 ]
subject to the boundary conditions
u ( 0 , t ) + δ 2 ∫ 0 1 u ( η , t ) d η = 0 u ( 1 , t ) + δ 2 ∫ 0 1 u ( η , t ) d η = 0
and the initial condition
u ( η , 0 ) = η ( η − 1 ) + δ 2 6 ( 1 + δ 2 )
whose the exact solution is
u ( η , t ) = e − t [ η ( η − 1 ) + δ 2 6 ( 1 + δ 2 ) ] .
where δ = 0.12 The maximum absolute error, ‖ E N ‖ is reported in Table 1 as N increases from N = 3 to N = 9 . Maximum absolute error is tabulated in Table 2 for Chelyshkov collocation with the analogous results of Ekolin [14] who used finite difference methods (Forward Euler, backward Euler and Crank-Nicolson methods) to obtain his numerical solution. The exact and approximating solutions and errors are shown in Figure 1.
Example 2: [25] consider the following
∂ u ∂ t = ∂ 2 u ∂ η 2
subject to the boundary conditions
u ( 0 , t ) − ∫ 0 1 ( η + t ) u ( η , t ) d η = g 1 ( t ) u ( 1 , t ) − ∫ 0 1 t e − η u ( η , t ) d η = g 2 ( t )
where
g 1 ( t ) = 1 2 − t − e − t [ cos 1 + sin 1 + t sin 1 − 2 ]
Maximum absolute error ‖ E N ‖ for example 1
N
‖ E N ‖
3
3.4117E-05
5
4.6525E-08
7
5.2898E-11
9
4.5264E-14
Comparison of the numerical results for example 1
‖ E N ‖ , N = 9
4.5264E-14
Forward Euler N = 130 [14]
1.50E-07
backward Euler, N = 130 [14]
6.70E-06
Crank-Nicolson, N = 130 [14]
1.30E-06
g 2 ( t ) = 1 + e − t cos 1 − t 2 e [ 2 ( e − 1 ) + e − t ( e − cos 1 + sin 1 ) ]
and the initial condition
u ( η , 0 ) = 1 + cos ( η )
whose the exact solution is
u ( η , t ) = 1 + e − t cos ( η ) .
The maximum absolute error, ‖ E N ‖ is reported in Table 3 as N increases from N = 3 to N = 9 . Maximum absolute error is tabulated in Table 4 for Chelyshkov collocation method with the analogous results of Shidfar [25] who used sinc-collocation method to obtain this numerical solution. The exact and approximating solutions and error are shown in Figure 2.
Example 3: [21] [29] consider the following parabolic PDE
Maximum absolute error ‖ E N ‖ for example 2
N
‖ E N ‖
3
3.89250E-03
5
2.42266E-06
7
4.23364E-08
9
1.87322E-11
Comparison of the numerical results for example 2
‖ E N ‖ , N = 9
Method [25] , N = 30
1.87E-11
1.83E-05
∂ u ∂ t = ∂ 2 u ∂ η 2 , 0 < η < 1 , 0 < t < 1
subject to the boundary conditions
u ( 0 , t ) = e − π 2 4 t ∫ 0 1 u ( η , t ) d η = 2 π e − π 2 4 t
whose the exact solution is
u ( η , t ) = exp ( − π 2 t 4 ) cos ( π η 2 ) .
In Table 5 we display the comparison of absolute errors between our scheme and the absolute errors result from Gaussian radial method [29] at t = 0.1 . The exact and approximating solutions and errors are shown in Figure 3.
Example 4: [21] consider the following parabolic PDE
∂ u ∂ t = ∂ 2 u ∂ η 2 + e t ( η 2 − 2 ) , 0 < η < 1 , 0 < t < 1
subject to the boundary conditions
u ( 0 , t ) = η 2 ∫ 0 1 u ( η , t ) d η = e t 3
and the initial condition
u ( η , 0 ) = η 2
whose the exact solution is
u ( η ,0 ) = e t η 2 .
In Table 6, we display the comparison of absolute errors between our scheme and the absolute errors in [21] at different values of x and t. The exact and approximating solutions and error are shown in Figure 4.
Example 5: [29] consider the following parabolic PDE
∂ u ∂ t = ∂ 2 u ∂ η 2 + ( π 2 − 1 ) e − t ( sin ( π η ) + cos ( π η ) ) , 0 < η < 1 , 0 < t < 1
subject to the boundary conditions
u ( 0 , t ) − 2 ∫ 0 1 sin ( π η ) u ( η , t ) d η = 0 u ( 0 , t ) + 2 ∫ 0 1 cos ( π η ) u ( η , t ) d η = 0
and the initial condition
Absolute error with t = 0.1 and N = 9 for example 3
x
our method
Method [29]
0.1
7.6734E-09
2.6837E-08
0.2
1.2885E-08
3.0350E-08
0.3
1.4747E-08
2.3109E-08
0.4
1.3092E-08
1.1866E-08
0.5
8.2383E-09
1.0594E-09
0.6
1.0779E-09
3.7093E-09
0.7
7.0853E-09
7.5678E-09
0.8
1.4621E-08
5.5621E-08
0.9
1.9894E-08
1.7974E-07
1.0
2.0407E-08
4.5256E-07
Comparison the absolute error of the our scheme with N = 9 and method in [<xref ref-type="bibr" rid="scirp.115097-ref21">21</xref>] for example 4
( x , t )
our method
Method [21]
( 0.1,0.1 )
9.74824E-13
1.19E-08
( 0.2,0.2 )
6.14349E-14
2.81E-11
( 0.4,0.4 )
1.03907E-14
3.98E-11
( 0.6,0.6 )
2.53647E-14
2.52E-11
( 0.8,0.8 )
2.19862E-13
1.38E-13
u ( η , 0 ) = sin ( π η ) + cos ( π η )
whose the exact solution is
u ( η , t ) = e − t [ sin ( π η ) + cos ( π η ) ]
In Table 7 we display the comparison of absolute errors between our scheme and the absolute errors result from Crank-Nicolson scheme [49] and Gaussian radial method [29] at x = 0.25 . The exact and approximating solutions and errors are shown in Figure 5.
Absolute values of error with x = 0.25 and N = 9 for example 5
t
present method
Method [49]
Method [30]
0.1
1.5887E-06
5.17E-05
1.09E-06
0.2
2.1365E-06
6.19E-05
3.04E-06
0.3
2.2131E-06
6.49E-05
9.01E-06
0.4
2.1091E-06
6.45E-05
1.56E-06
0.5
1.9485E-06
6.21E-05
2.02E-06
0.6
1.6145E-06
5.64E-05
2.19E-06
0.7
1.6145E-06
4.99E-05
2.08E-06
0.8
1.4628E-06
4.49E-05
1.68E-06
0.9
1.3246E-06
4.08E-05
1.05E-06
1.0
1.1984E-06
3.64E-05
3.32E-06
6. Conclusion
This paper solved partial differential equations with nonlocal boundary conditions by applying Chelyshkov matrix collocation method. The numerical experiments demonstrated the efficiency of Chelyshkov matrix collocation method. In addition, the accuracy of the scheme was tested on five examples. The study found that the computational by Chelyshkov matrix collocation method can be an efficient numerical method to solve nonlocal problems.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
El-Gamel, M., El-Baghdady, G.I. and El-Hady, M.A. (2022) Highly Efficient Method for Solving Parabolic PDE with Nonlocal Boundary Conditions. Applied Mathematics, 13, 101-119. https://doi.org/10.4236/am.2022.132009
ReferencesCushman, J. and Ginn, T. (1993) Nonlocal Dispersion in Porous Media with Continuously Evolving Scales of Heterogeneity. Transport in Porous Media, 13, 123-138. https://doi.org/10.1007/BF00613273Dagan, G. (1994) The Significance of Heterogeneity of Evolving Scales to Transport in Porous Formations. Water Resources Research, 13, 3327-3336. https://doi.org/10.1029/94WR01798Renardy, M., Hrusa, W. and Nohel, J. (1987) Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, No. 35, Longman Scientific Technical, Harlow.Allegretto, W., Lin, Y. and Zhou, A. (1999) A Box Scheme for Coupled Systems Resulting from Microsensor Thermistor Problems. DCDIS: Dynamics of Continuous, Discrete and Impulsive Systems, 5, 209-223.Murthy, A. and Verwer, J. (1992) Solving Parabolic Integro-Differential Equations by an Explicit Integration Method. Journal of Computational and Applied Mathematics, 39, 121-132. https://doi.org/10.1016/0377-0427(92)90229-QDay, W. (1982) A Decreasing Property of Solutions of Parabolic Equations with Applications to Thermoelasticity. Quarterly of Applied Mathematics, 40, 468-475. https://doi.org/10.1090/qam/693879Day, W. (1983) Extensions of a Property of the Heat Equation to Linear Thermoelasticity and Other Theories. Quarterly of Applied Mathematics, 41, 319-330. https://doi.org/10.1090/qam/678203Fairweather, G. and Saylor, R.D. (1991) The Reformulation and Numerical Solution of Certain Nonclassical Initial-Boundary Value Problems. SIAM Journal on Scientific and Statistical Computing, 12, 127-144. https://doi.org/10.1137/0912007Bouziani, A. (1996) Mixed Problem with Boundary Integral Conditions for a Certain Parabolic Equation. Journal of Applied Mathematics and Stochastic Analysis, 9, 323-330. https://doi.org/10.1155/S1048953396000305Bouziani, A. (1999) On a Class of Parabolic Equations with a Nonlocal Boundary Condition. Academie Royale de Belgique. Bulletin de la Classe des Sciences, 10, 61-77. https://doi.org/10.3406/barb.1999.27977Bouziani, A. (1997) Strong Solution for a Mixed Problem with Nonlocal Condition for a Certain Pluriparabolic Equations. Hiroshima Mathematical Journal, 27, 373-390. https://doi.org/10.32917/hmj/1206126957Samarskii A. ,et al. (1980)Some Problems in Differential Equations Theory Differential Equations 16, 1221-1228.Nakhushev A. ,et al. (1982)On Certain Approximate Method for Boundary-Value Problems for Differential Equations and Its Applications in Ground Waters Dynamics Differentsialnye Uravneniya 18, 72-81.Ekolin, G. (1991) Finite Difference Methods for a Nonlocal Boundary Value Problem for the Heat Equation. BIT, 31, 245-261. https://doi.org/10.1007/BF01931285Ewing, R., Lazarov, R. and Lin, Y. (2000) Finite Volume Element Approximations of Nonlocal Reactive Flows in Porous Media. Numerical Methods for Partial Differential Equations, 16, 285-311. https://doi.org/10.1002/(SICI)1098-2426(200005)16:3<285::AID-NUM2>3.0.CO;2-3Pani, A. (1993) A Finite Element Method for a Diffusion Equation with Constrained Energy and Nonlinear Boundary Conditions. Journal of the Australian Mathematical Society Series B, 35, 87-102. https://doi.org/10.1017/S0334270000007281Cannon, J., Lin, Y. and Wang, S. (1990) An Implicit Finite Difference Scheme for the Diffusion Equation Subject to Mass Specification. International Journal of Engineering Science, 28, 573-578. https://doi.org/10.1016/0020-7225(90)90086-XCannon, J. and Matheson, A. (1993) A Numerical Procedure for Diffusion Subject to the Specification of Mass. International Journal of Engineering Science, 31, 347-355. https://doi.org/10.1016/0020-7225(93)90010-RCannon, J., Esteva, S. and Van der Hoek, J. (1987) A Galerkin Procedure for the Diffusion Equation Subject to the Specification of Mass. SIAM Journal on Numerical Analysis, 24, 499-515. https://doi.org/10.1137/0724036Bouziani, A., Merazga, N. and Benamira, S. (2008) Galerkin Method Applied to a Parabolic Evolution Problem with Nonlocal Boundary Conditions. Nonlinear Analysis, Theory, Methods and Applications, 69, 1515-1524. https://doi.org/10.1016/j.na.2007.07.008Bastani, M. and Salkuyeh, D. (2013) Numerical Studies of a Non-Local Parabolic Partial Differential Equations by Spectral Collocation Method with Preconditioning. Computational Mathematics and Modeling, 24, 81-89. https://doi.org/10.1007/s10598-013-9161-6Golbabai, A. and Javidi, M. (2007) A Numerical Solution for Nonclassical Parabolic Problem Based on Chebyshev Spectral Collocation Method. Applied Mathematics and Computation, 190, 179-185. https://doi.org/10.1016/j.amc.2007.01.033El-Gamel, M. and Sameeh, M. (2013) A Chebychev Collocation Method for Solving Troesch’s Problem. International Journal of Mathematics and Computer Applications Research (IJMCAR), 3, 23-32.Yousefi, S., Behroozifar, M. and Dehghan, M. (2011) The Operational Matrices of Bernstein Polynomials for Solving the Parabolic Equation Subject to Specification of the Mass. Journal of Computational and Applied Mathematics, 235, 5272-5283. https://doi.org/10.1016/j.cam.2011.05.038Zolfaghari, R. and Shidfar, A. (2013) Solving a Parabolic PDE with Nonlocal Boundary Conditions Using the Sinc Method. Numerical Algorithms, 62, 411-427. https://doi.org/10.1007/s11075-012-9595-5El-Gamel, M. and Abd El-Hady, M. (2020) Novel Efficient Collocation Method for Sturm-Liouville Problems with Nonlocal Integral Boundary Conditions. SeMA Journal, 77, 375-388. https://doi.org/10.1007/s40324-020-00220-3El-Gamel, M. (2013) Numerical Solution of Troesch’s Problem by Sinc-Collocation Method. Applied Mathematics, 4, 707-712. https://doi.org/10.4236/am.2013.44098El-Gamel, M. (2006) A Numerical Scheme for Solving Nonhomogeneous Time-Dependent Problems. Zeitschrift für angewandte Mathematik und Physik (ZAMP), 57, 369-383. https://doi.org/10.1007/s00033-005-0022-9Khaksarfard, M., Ordokhani, A. and Babolian, A. (2019) An Approximate Method for Solution of Nonlocal Boundary Value Problems via Gaussian Radial Basis Functions. SeMA Journal, 76, 123-142. https://doi.org/10.1007/s40324-018-0165-1El-Baghdady, G., El-Azab, M. and El-Beshbeshy, W. (2021) Determination of Legendre-Gauss-Lobatto Pseudo-Spectral Method for One-Dimensional Advection-Diffusion Equation. Current Topics on Mathematics and Computer Science, 4, 27-40. https://doi.org/10.9734/bpi/ctmcs/v4/10245DEl-Baghdady, G. and El-Azab, M. (2015) Numerical Solution of One-Dimensional Advection-Diffusion Equationwith Variable Coefficients via Legendre-Gauss-Lobatto Time-Space Pseudo-Spectracl Method. Electronic Journal of Mathematical Analysis and Applications, 3, 1-14. https://doi.org/10.18576/sjm/030102El-Gamel, M. and Abd El-Hady, M. (2021) On Using Sinc Collocation Approach for Solving a Parabolic PDE with Nonlocal Boundary Conditions. Journal of Nonlinear Sciences and Applications, 14, 29-38. https://doi.org/10.22436/jnsa.014.01.04Hokkanen, V. and Morosanu, G. (2002) Functional Methods in Differential Equations. Chapman and Hall/CRC, London. https://doi.org/10.1201/9781420035360Chelyshkov V. ,et al. (2006)Alternative Orthogonal Polynomials and Quadratures Electronic Transactions on Numerical Analysis 25, 17-26.Chelyshkov V. ,et al. (1994)A Variant of Spectral Method in the Theory of Hydrodynamic Stability Hydromech 68, 105-109.Rasty, M. and Hadizadeh, M. (2010) A Product Integration Approach on New Orthogonal Polynomials for Nonlinear Weakly Singular Integral Equations. Acta Applicandae Mathematicae, 109, 861-873. https://doi.org/10.1007/s10440-008-9351-yOguza, C. and Sezer, M. (2015) Chelyshkov Collocation Method for a Class of Mixed Functional Integro-Differential Equations. Applied Mathematics and Computation, 259, 943-954. https://doi.org/10.1016/j.amc.2015.03.024Oguza, C. and Sezer, M. (2017) A Novel Chelyshkov Approach Technique for Solving Functional Integro-Differential Equations with Mixed Delays. Journal of Science and Arts, 3, 477-490.Rahimkhani, P. and Ordokhani, Y. (2020) Numerical Solution of Volterra-Hammerstein Delay Integral Equations. Iranian Journal of Science and Technology. Transaction A, Science, 44, 445-457. https://doi.org/10.1007/s40995-020-00846-yArdabili, S. and Talaei, Y. (2018) Chelyshkov Collocation Method for Solving the Two-Dimensional Fredholm-Volterra Integral Equations. International Journal of Applied and Computational Mathematics, 4, 1-13. https://doi.org/10.1007/s40819-017-0433-2Oguz, C., Sezer, M. and Oguza, A. (2015) Chelyshkov Collocation Approach to Solve the Systems of Linear Functional Differential Equations. New Trends in Mathematical Sciences, 3, 83-97.Rahimkhani, P., Ordokhani, Y. and Lima, P. (2019) An Improved Composite Collocation Method for Distributed-Order Fractional Differential Equations Based on Fractional Chelyshkov Wavelets. Applied Numerical Mathematics, 145, 1-27. https://doi.org/10.1016/j.apnum.2019.05.023Moradi, L., Mohammadi, F. and Baleanu, D. (2019) A Direct Numerical Solution of Time-Delay Fractional Optimal Control Problems by Using Chelyshkov Wavelets. Journal of Vibration and Control, 25, 310-324. https://doi.org/10.1177/1077546318777338El-Gamel, M., Abd-El-Hady, M. and El-Azab, M. (2017) Chelyshkov-Tau Approach for Solving Bagley-Torvik Equation. Applied Mathematics, 8, 1795-1807. https://doi.org/10.4236/am.2017.812128Oliveira, F. (1980) Collocation and Residual Correction. Numerische Mathematik, 36, 27-31. https://doi.org/10.1007/BF01395986Çelik, I. (2005) Approximate Calculation of Eigenvalues with the Method of Weighted Residuals-Collocation Method. Applied Mathematics and Computation, 160, 401-410. https://doi.org/10.1016/j.amc.2003.11.011Shahmorad, S. (2005) Numerical Solution of the General Form Linear Fredholm-Volterra Integro-Differential Equations by the Tau Method with an Error Estimation. Applied Mathematics and Computation, 167, 1418-1429. https://doi.org/10.1016/j.amc.2004.08.045Çelik, I. (2006) Collocation Method and Residual Correction Using Chebyshev Series. Applied Mathematics and Computation, 174, 910-920. https://doi.org/10.1016/j.amc.2005.05.019Wang, S.M. and Lin, Y.P. (1990) A Numerical Method for the Diffusion Equation with Nonlocal Boundary Specifications. International Journal of Engineering Science, 28, 543-546. https://doi.org/10.1016/0020-7225(90)90056-O