In this paper, we examine the space discretization of time fractional telegraph equation (TFTE) with Mamadu-Njoseh orthogonal basis functions. For ease and convenience, we deal with the fractional derivative by first converting from Caputo’s type to Riemann-Liouville’s type. The proposed method was constrained to precise error analysis to establish the accuracy of the method. Numerical experimentation was implemented with the aid of MAPLE 18 to show convergence of the method as compared with the analytic solution.
The popularity of fractional partial differential equations (FPDEs) gained momentum in science and engineering due to its involvement in many areas of applications ( [
A time fractional telegraph equation (TFTE) has the form ( [
∂ 0 C β u ( x , t ) ∂ t β + ∂ u ( x , t ) ∂ t − ∂ 2 u ( x , t ) ∂ x 2 = g ( x , t ) , 0 ≤ x ≤ T , t > 0 , (1.1)
with the initial conditions
u ( x , 0 ) = u 0 ( x ) , ∂ u ( x , 0 ) ∂ t = u 1 ( x ) , 0 ≤ x ≤ T , t > 0 , (1.2)
and boundary conditions
u ( 0 , t ) = u ( 0 , T ) = 0 , 0 ≤ x ≤ T , t > 0 , (1.3)
where 1 < β < 2 , g ( x , t ) is the source term and ∂ 0 C β u ( x , t ) ∂ t β is Caputo fractional derivative of u ( x , t ) .
The TFTE is a hyperbolic partial differential equation responsible for modeling many physical phenomena, such as wave propagation, signal processing, random walk theory and so on. Consequently, TFTE has been studied by many authors. Riemann-Liouville’s method was adopted by Cascaval et al. ( [
In this paper, we solve (1.1)-(1.3) with Mamadu-Njoseh orthogonal basis functions in a space discretization approach. Here, the process of discretization is quite different from the classical numerical method—finite difference method. In FEM, the given differential equation has to be reformulated as a variational problem leading to the solution via the following steps:
1) Finite dimensional space construction, U h . This is the discretization process;
2) Seeking solution to the resultant discrete problem; and
3) Implementation through a computer programming.
This paper is organized as follows. Section 2 constitutes preliminaries. Finite element method for time fractional telegraph equation is given in Section 3. Error analysis is given in Section 4. Numerical illustrations, tables of results and graphical simulations are given in Section 5 and Section 6. Discussion of results and conclusions are presented in Sections 7 and Section 8, respectively.
Let’s use the notation
α ≤ τ A and α ≤ Q A ,
where τ and Q are constants free of α and A, and are discretization parameters.
Let R and γ be two given Hilbert spaces, ‖ . ‖ R → γ is defined as
‖ G ‖ R → γ = sup θ ∈ R , θ ≠ 0 ‖ G ( θ ) ‖ y ‖ θ ‖ x .
Suppose β = ( β 1 , β 2 , ⋯ , β n ) represent a multi-index and | β | = ∑ i = 1 n β j . For a well defined smooth function U ∈ Ω , Dβ, being the differential operator is given by ( [
D β U = ∂ | β | U .
Now, an integrable function V is said to possess a weak derivative U, if U satisfies
∫ Ω U ϕ d x = ( − 1 ) | β | ∫ Ω D β ϕ d x , ∀ ϕ ∈ C 0 ∞ ( Ω ) ,
where, C 0 ∞ ( Ω ) denotes the space of infinity differentiable functions supported compactly in Ω. We assume Dβ to be weak derivative throughout this research.
Let U ∈ Ω be a lebesque measurable function and q ≥ 1 . The norm ‖ . ‖ L P ( Ω ) be defined by ( [
‖ U ‖ L P ( Ω ) = ( ∫ Ω | u ( x ) | q d x ) 1 / q ,
where L P ( Ω ) denotes the set of all U such that ‖ U ‖ L P ( Ω ) is finite. Given an integer k ≥ 0 , we have the Sobolev space W k , p ( Ω ) given as
W k , p ( Ω ) = { U ∈ L P ( Ω ) : D β U ∈ L P ( Ω ) , ∀ | β | ≤ k } .
Also,
‖ U ‖ W k , p ( Ω ) = ( ∑ ‖ β ‖ ≤ k ‖ D β U ‖ L P ( Ω ) ) 1 / 2 ,
are the corresponding Sobolev and Semi norms of W k , p ( Ω ) respectively.
Now for 0 < k < 1 , | ⋅ | W k , p ( Ω ) is defined by
| U | W k , p ( Ω ) = ∫ Ω ∫ Ω | u ( x ) − y ( v ) | p | x ∀ v | d + k p ( d x d p ) 1 / p ,
called the fractional Sobolev Semi norm with
W k , p ( Ω ) = { U ∈ L P ( Ω ) : | U | W k , p ( Ω ) < ∞ } .
For q ≥ 0 , we write q = n + k , n ≥ q , k ∈ ( 0 , 1 ) .
Thus, the Sobolev space becomes
W q , p ( Ω ) = { u ∈ W n , p : D β U ∈ W k , p ( Ω ) , ∀ | β | = n } ,
and
‖ U ‖ W q , p ( Ω ) = ( U p W n , p ( Ω ) + ∑ | β | = n | D β U | p W k , p ( Ω ) ) 1 2 ,
is the full norm. For q ≥ 0 , Sobolev space W q , p ( Ω ) is a Banach space ( [
Similarly,
when p = 2 , the sobolev space W q , 2 ( Ω ) is a Hilbert space, that is,
H 2 ( Ω ) = W q , 2 ( Ω ) .
In particular, to solve our model equation we define the Sobolev space as
H 0 n ( Ω ) = { U ∈ H n ( Ω ) : ∂ β | ∂ Ω = 0 , ∀ | β | ≤ n − 1 } .
To establish the equivalences of certain norms in the subspaces of H 0 n ( Ω ) , we shall rely in the famous Poincaré inequalities.
Lemma 2.1. ( [
‖ V ‖ L P ( Ω ) ≤ C | V | H ′ ( Ω ) , ∀ V ∈ H ′ 0 ( Ω ) .
Lemma 2.2. ( [
‖ V − ∫ Ω v d x m e a s ( Ω ) ‖ L P ( Ω ) ≤ C ‖ V ‖ H ′ ( Ω ) , ∀ V ∈ H ′ ( Ω ) .
Lemma 2.3( [
‖ V ‖ H S ( Ω ) ≤ C | V | H n ( Ω ) , ∀ s ≤ n , ∀ V ∈ H n ( Ω ) ,
which is generalized poincare inequality.
Thus, | ⋅ | n , Ω over the space H 0 n ( Ω ) is equivalent to ‖ ⋅ ‖ n , Ω .
Let [ a , b ] ∈ ℝ , D a + β [ U ( t ) ] ( x ) ≡ ( D a + β U ) ( x ) , and D b − β [ U ( t ) ] ( x ) ≡ ( D − b β U ) ( x ) be the Reimann-Liouville (R-L) fractional derivatives of order β. The fractional derivatives of order ( D c a + β U ) ( x ) and ( D c b − β U ) ( x ) of order β on [ a , b ] ∈ ℝ > 0 , are as ( [
( D c a + β U ) ( x ) = ( D a + β [ u ( t ) − ∑ i = 0 m − 1 u ( k ) ( a ) i ! ( t − a ) i ] ) ( x ) (2.1)
( D c b − β U ) ( x ) = ( D b − β [ u ( t ) − ∑ i = 0 m − 1 u ( k ) ( b ) i ! ( b − a ) i ] ) ( x ) (2.2)
respectively, where m = [ ℝ ( β ) ] + 1 for β ∉ ℕ 0 , m = β for β ∈ ℕ 0 .
The above Equations (2.1) and (2.2) are called left- and right-sided Caputo fractional derivatives of order β.
Lemma 2.4 ( [
Let r ( x ) ∈ C − 1 n , n ∈ ℕ ∪ { 0 } . Then the caputo fractional derivative of r ( x ) is given as D β U ( x ) = I λ − γ D n U ( x ) , satisfying the following properties:
(a) D β ( I β U ( x ) ) = U ( x )
(b) I β ( D β U ( x ) ) = γ ( x ) − ∑ i = 1 m − 1 U k ( 0 + ) ( x i i ! )
(c) D β x γ = { 0 , γ ∈ ℕ a , γ < β a Γ ( γ + 1 ) − Γ ( γ − β + 1 ) x γ − β , γ ∈ ℕ a , γ ≥ β a , (2.3)
where β a ≥ a and ℕ a = { 0 , 1 , 2 , 3 , ⋯ } .
These are orthoponal polynomials generated with reference to the properties ( [
φ m ( x ) = ∑ j = 0 m a j x j , x ∈ [ − 1 , 1 ] , (2.4)
B [ j + ] = ∫ a b ( 1 + x 2 ) φ j − 1 ( x ) ( ∑ j = 0 m a j x j ) d x = 0 , j = 1 ( 2 ) n , (2.5)
subject to the initial conditions
φ 0 ( x ) = 1 and φ n ( 1 ) = 1 , (2.6)
where j + denotes a unit step increment, w ( x ) is weight function.
Lemma 2.5: For any Z + ∪ { 0 } value of j, ∃ a partition j 0 < j 1 < j 2 < ⋯ < j n − 1 < j n with a unit step size.
Theorem 2.1. For m = j, there exists n system of linear algebraic Equations generated from using (2.4)-(2.6) at the ( j 0 , m ) , ( j 1 , m ) , ⋯ , ( j n − 1 , m ) , ( j n , m ) , respectively.
Proof:Let B [ j + ] be given by lemma (2.5), we have j 0 < j 1 < ⋯ < j r − 1 < j r . Thus, for m = j = r , the grid points of the partition by refinement would ( j 0 , m ) , ( j 1 , m ) , ⋯ , ( j r − 1 , m ) , ( j r , m ) . Hence, we have
B [ j n + 1 ] = ∫ a b w ( x ) φ n ( x ) φ m ( x ) d x = 0 at ( j n , m ) .
The first Mamadu-Njoseh polynomials are general via MAPLE 18 via theorem 2.1, and are presented in
| n | Mamadu-Njoseh polynomials, φ n ( x ) |
|---|---|
| 0 1 2 3 4 5 6 | 1 x 1 3 ( 5 x 2 − 2 ) 1 5 ( 14 x 3 − 9 x ) 1 648 ( 333 − 2898 x 2 + 3213 x 4 ) 1 136 ( 325 x − 1410 x 3 + 1221 x 5 ) 1 1064 ( − 460 + 8685 x 2 − 24750 x 4 + 17589 x 6 ) |
We consider the space discretization time functional telegraph Equations (1.1)-(1.3) with Mamadu-Njoseh basis function using the finite element method.
Let a piecewise finite element space that is linear and continuous be given as V h . Let [0, 1] be partitioned as
0 = x 0 < x 1 < x 2 < ⋯ < x n = 1 ,
called the space partitioning of [ a , b ] .
Let V h = { S h ( x ) : S h ( x ) iscontinuousandlinearin [ 0 , 1 ] } .
The variational formulation for the time – fractional telegraph Equation (1.1) is to compute u ( t ) ∈ H 0 1 ( a , b ) such that
( D 0 R t β [ u ( x , t ) − U 0 ] , S ( x ) ) + ( U t , S ( x ) ) − ( U x , S ( x ) ) = ( g ( x , t ) , S ( x ) ) , S ( x ) ∈ H 0 1 . (3.1)
The essence of FEM is to compute U h ( t ) ∈ V h , such that
( D 0 R t β [ u ( x , t ) − U 0 ] , γ ) + ( ∂ u ∂ t , γ ) − ( ∂ u ∂ x , ∂ γ ∂ x ) = ( γ , ∂ γ ∂ x ) , γ ∈ V h (3.2)
Let B h = − Δ h : V h → V h satisfies
( B h U h , γ ) = ( ∂ u ∂ t , γ ) − ( ∂ u ∂ x , ∂ γ ∂ x ) , γ ∈ V h . (3.3)
Suppose G h : G → V h defined a L 2 operator given by
( G h s , γ ) = ( s , γ ) , ∀ γ ∈ V h , s ∈ L 2 .
Thus, Equation (3.2) can be written in the abstract sense as
( D 0 R t β [ u ( x , t ) − U 0 ] , S ( x ) ) + B h U h = G h g , t > 0 , (3.4)
where
D 0 R t β ( u ( x , t ) ) = 1 Γ ( 1 − β ) ∂ ∂ t ∫ 0 t u ( s − x ) ( t − s ) − β d s , β t ( 0 , 1 ) , (3.5)
called the Riemann-Liouville fractional derivative, and Γ is the Gamma function.
Using quadrative formula (( [
D 0 R t β u ( t j ) = ∑ r = 0 j w r j [ u ( t j − t r ) − U 0 ] + t j − β Δ t β Γ ( − β ) G j ( g ) , (3.6)
where w r j
w r j = 1 Γ ( 2 − β ) { 1 , r = 0 − 2 r 1 − β + ( r − 1 ) 1 − β + ( r + 1 ) 1 − β , r = 1 , 2 , ⋯ , j , (3.6a)
and G j ( g ) satisfies
‖ G j ( g ) ‖ ≤ K j β − 2 sup 0 ≤ t ≤ T ‖ U ″ ( t J − t j w ) ‖ , w ∈ [ 0 , 1 ] .
Now, let u ( x , t ) = U j ≈ U h ( t J ) = ∑ J = 1 N − 1 α j φ j ( x , t j ) , be an approximation of U h ( t j ) , where φ j ( x ) , j = 0 ( 1 ) ( N − 1 ) , are Mamadu-Njoseh Basis function of V h .
Also, let g j = g ( t j ) defines the time discretization such that
Δ t − β ∑ r = 0 j w r j ( U j − r − U 0 , γ ) + ( ∂ U j ∂ t , γ ) − ( ∂ U j ∂ x , ∂ γ ∂ x ) = ( g j , ∂ γ ∂ x ) , j = 0 ( 1 ) n , ∀ γ ∈ V h . (3.7)
Now, we consider the following steps for j = 0 ( 1 ) n .
Step 1: Suppose j = 0 , then U j = 0 .
Step 2: Set j = 1 , we get,
Δ t − β W 0 , 1 ( U 1 , γ ) + ( ∂ U 1 ∂ t , γ ) − ( ∂ U 1 ∂ x , ∂ γ ∂ x ) + Δ t − β ( ∑ r = 1 g W r 1 ( ( U j − U 0 ) , γ ) − W 01 ( U 0 , γ ) ) , γ ∈ V h . (3.8)
Since U j ≈ u ( t j ) = ∑ r = j N − 1 a r φ r ( x , t j ) , we have that,
Δ t − β ( ∑ r = j N − 1 a r ( φ r ( x , t j ) , γ ) ) + ∑ r = j N − 1 a r ( ∂ φ r ( x , t j ) ∂ t , γ ) − ∑ r = j N − 1 a r ( ∂ φ r ( x , t j ) ∂ x , ∂ γ ∂ x ) = ( g 1 , ∂ γ ∂ x ) − Δ t − β ( W 11 ( U 0 − u 0 , γ ) + W 01 ( U 0 , γ ) ) , ∀ γ ∈ V h
⇒ Δ t − β ( ∑ r = j N − 1 a r ( φ r ( x , t j ) , γ ) ) + ∑ r = j N − 1 a r ( ( ∂ φ r ( x , t j ) ∂ t , γ ) − ( ∂ φ r ( x , t j ) ∂ x , ∂ γ ∂ x ) ) = ( g 1 , ∂ γ ∂ x ) − Δ t − β ( W 11 ( U 0 − u 0 , γ ) + W 01 ( U 0 , γ ) ) , ∀ γ ∈ V h (3.9)
Let γ = φ k ( x , t j ) , k = 1 ( 2 ) ( N − 1 ) , we have,
Δ t − β ( ∑ r = j N − 1 a r ( φ r ( x ) , φ k ( x , t j ) ) ) + ∑ r = j N − 1 a r ( ( ∂ φ r ( x , t j ) ∂ t , φ k ( x , t j ) ) − ( ∂ φ r ( x , t j ) ∂ x , φ k ( x , t j ) ∂ x ) ) = ( g 1 , ∂ γ ∂ x ) − Δ t − β ( W 11 ( U 0 − u 0 , γ ) + W 01 ( U 0 , φ k ( x , t j ) ) ) , ∀ γ ∈ V h (3.10)
Thus,
Δ t − β W 0 , 1 ( M ∗ R 1 ) + Q ∗ R 1 = G 1 − Δ t − β W 11 R 0 + Δ t − β ∑ r = 0 1 W r 1 U 0 (3.11)
where,
M = ( ( φ 1 ( x , t j ) , φ 1 ( x , t j ) ) ( ( φ 2 ( x , t j ) , φ 1 ( x , t j ) ) ⋯ ( φ N − 1 ( x , t j ) , φ 1 ( x , t j ) ) ( φ 1 ( x , t j ) , φ 2 ( x , t j ) ) ( φ 2 ( x , t j ) , φ 2 ( x , t j ) ) ⋯ ( φ N − 1 ( x , t j ) , φ 2 ( x , t j ) ) ⋮ ⋮ ⋮ ( φ 1 ( x , t j ) , φ ( N − 1 ) ( x , t j ) ) ( φ 2 ( x , t j ) , φ ( N − 1 ) ( x , t j ) ) ⋯ ( φ 1 ( x , t j ) , φ ( N − 1 ) ( x , t j ) ) ) ,
R 1 = ( ( ∂ φ 1 ( x , t j ) ∂ x , φ 1 ( x , t j ) ) − ( ∂ φ 1 ( x , t j ) ∂ x , ∂ φ 1 ( x , t j ) ∂ x ) ⋯ ( ∂ φ N − 1 ( x , t j ) ∂ t , φ 1 ( x , t j ) ) − ( ∂ φ N − 1 ( x , t j ) ∂ x , ∂ φ N − 1 ( x , t j ) ∂ x ) ⋮ ⋮ ( ∂ φ 1 ( x , t j ) ∂ x , φ N − 1 ( x , t j ) ) − ( ∂ φ 1 ( x , t j ) ∂ x , ∂ φ N − 1 ( x , t j ) ∂ x ) ⋯ ( ∂ φ N − 1 ( x , t j ) ∂ t , φ N − 1 ( x , t j ) ) − ( φ N − 1 ( x , t j ) ∂ x , φ N − 1 ( x , t j ) ∂ x ) )
G 1 = [ ( g 1 , φ 1 ( x , t j ) ) ( g 1 , φ 2 ( x , t j ) ) ( g 1 , φ 3 ( x , t j ) ) ⋯ ( g 1 , φ N − 1 ( x , t j ) ) ] T ,
R 0 = [ ( U 0 , φ 1 ( x , t j ) ) ( U 0 , φ 2 ( x , t j ) ) ( U 0 , φ 3 ( x , t j ) ) ⋯ ( U 0 , φ N − 1 ( x , t j ) ) ] T ,
U 0 = [ ( u 0 , φ 1 ( x , t j ) ) ( u 0 , φ 2 ( x , t j ) ) ( u 0 , φ 3 ( x , t j ) ) ⋯ ( u 0 , φ N − 1 ( x , t j ) ) ] T ,
R 1 = [ a 0 a 1 a 2 ⋯ a N − 1 ] T .
Step 3: To compute U n ≈ U h ( t j ) we repeat the above steps as 1 and 2. Thus, with the above idea, the finite element method can be formulated and solve the resulting system via MAPLE 18 Software.
We consider the lemma below
Lemma 4.1: Let u j be the approximate solution of
D 0 R t β u ( t j ) = ∑ r = 0 j w r j [ u ( t j − t r ) − u 0 ] + t j − β Δ t β Γ ( − β ) G j ( g ) (4.1)
Then we have
‖ u j ‖ ≤ 2 u j + sin π β π | Γ ( − β ) | t j β ‖ g ‖ L ∞ .
Theorem 4.1: Let u ( t j ) and U j be the solutions (3.4) and (4.1), then we have
‖ U j − u ( t j ) ‖ ≤ 2 ‖ U 0 − Q h u 0 ‖ + O ( Δ t 2 − β + h 2 ) ,
where h is the space step size. Let Q h : H 0 1 → V h defines an elliptic or Ritz propectim given by
( ∇ Q h s , ∇ γ ) = ( ∇ s , ∇ γ ) , ∀ γ ∈ V h .
Let e j = U j − u ( t j ) = U j − Q h u ( t j ) + Q h u ( t j ) − u ( t j ) = α j + q j , j = 1 , 2 , 3 , ⋯
where, α j = U j − Q h u ( t j ) , q j = Q h u ( t j ) − u ( t j ) .
Now, the error equation obtained from (4.1),
t j − β Γ ( − β ) ∑ r = 0 i w r j ( α j − r − α 0 ) + B h α j = t j − β Γ ( − β ) ∑ r = 0 j w r j [ ( U j − r − u 0 ) + B h U j − Q h ( u ( t j − r ) − u 0 ) + B h Q h u ( t j ) ] = G h g j + G h B h u ( t j ) − Q h [ t j − β Γ ( − β ) ∑ r = 0 j w r j ( u ( t j − r ) − u 0 ) ] = − G h y j
where,
Y j = − D 0 R t β [ u ( t j ) − u 0 ] + Q h t j − β Γ ( − β ) ∑ r = 0 j w r j ( u ( t j − r − u 0 ) − u 0 ) = P j + K j ,
where,
P j = ( Q h − t ) t j − β Γ ( − β ) ∑ r = 0 j w r j ( u ( t j − r − u 0 ) ) ,
K j = t j − β Γ ( − β ) ∑ r = 0 j w r j ( u ( t j − r − u 0 ) ) − D 0 R t β [ u ( t j ) − u 0 ] .
Thus, we have,
t j − β Γ ( − β ) ∑ r = 0 i w r j u ( α j − r − α 0 ) + B h α j = G h ( P j + K j ) .
By Lemma 4.1, we have
‖ α j ‖ + 2 ‖ α 0 ‖ + sin π β π | Γ ( − β ) | t j − β ‖ G h ( P j + K j ) ‖ .
Here
‖ α j ‖ ≤ K t − β ‖ U ″ ( t j − t j t ) ‖ = K Δ t 2 − β ‖ U ″ ( t j − t j t ) ‖ ,
and
‖ K j ‖ ≤ K h 2 ‖ ∑ r = 0 j w r j ( t j − r ) ‖ G 2 + ‖ ∑ r = 0 j w r j u 0 ‖ G 2 ,
where ‖ . ‖ G 2 is Sobolev norm.
Let denote f ( t ) = u ( t j − t j t ) , then,
∑ r = 0 j w r j ( t j − r ) = ∫ 0 1 u ( t j − t j t ) t − 1 − β d t + G j = ∫ 0 1 f ( t ) t − 1 − β d t + G j ,
obtained via Hamamard d integral formulation ( [
| G j | ≤ j β − 2 ‖ f t ‖ G 2 ≤ Δ t 2 ‖ u ‖ G 2 ≤ Δ t j 2 − β ‖ u t ‖ G 2 .
Let q = t j − t j t into ∫ 0 1 f ( t ) t − 1 − β d t , to obtain,
∫ 0 1 f ( t ) t − 1 − β d t = t j β D 0 R t β u ( t j ) Γ ( − β ) .
Thus,
‖ ∑ r = 0 j w r j u ( t j − r ) ‖ ≤ t j β | Γ ( − α ) | ‖ D 0 R t β u ( t j ) ‖ G 2 + Δ t 2 − β ‖ U t t ‖ G 2 ,
t j β ‖ α j ‖ ≤ K h 2 t j β ( | Γ ( − β ) | ‖ D 0 R t β u ( t j ) ‖ G 2 + Δ t 2 − β ‖ U t t ‖ G 2 ) .
Thus, we have that,
‖ α j ‖ ≤ 2 ‖ α 0 ‖ + sin π β π | Γ ( − β ) | t j − β ‖ G h ( α j + q j ) ‖ ≤ 2 ‖ α 0 ‖ + K t j β Δ t 2 − β ‖ U t t ‖ G 2 + K h 2 t j β ( ‖ D 0 R t β u ( t j ) ‖ G 2 + Δ t 2 − β ‖ U t t ‖ G 2 ) ≤ 2 ‖ α 0 ‖ + O ( Δ t 2 − β + h 2 )
Hence,
‖ e j ‖ ≤ ‖ α j ‖ + ‖ G j ‖ ≤ 2 ‖ α 0 ‖ + O ( Δ t 2 − β + h 2 ) + ‖ G j ‖ .
Therefore,
‖ G j ‖ = ‖ Q h u ( t j ) − u ( t j ) ‖ = K h 2 ‖ u ( t j ) ‖ G 2 .
Obtained via elliptic projection of error estimation. Thus, we finally obtain
‖ e j ‖ ≤ 2 ‖ α j ‖ + B ( Δ t 2 − β + h 2 ) .
In this section, we carry out numerical simulations to verify the accuracy of the proposed method.
Let in (1.1) be given
g ( x , t ) = 2 ( x 2 − x ) t ( Γ ( 3 − β ) + t 1 − β Γ ( 3 − β ) ) − 2 t 2 , 0 ≤ x ≤ 1 , t ∈ ( 0 , 1 ] (5.1)
with initial conditions
u ( x , 0 ) = ∂ u ( x , 0 ) ∂ t = 0 , 0 ≤ x ≤ 1 (5.2)
and boundary conditions
u ( 0 , t ) = u ( 1 , t ) = 0 , 0 ≤ x ≤ 1. (5.3)
The exact solution is given as u ( x , t ) = ( x 2 − x ) t 2 .
Using (3.11) on (5.1) at N = 3 with w r 3 , r = 0 ( 1 ) 3 , estimated using (3.6a),
and Δ t = 1 / 1000 at t = 1 , results are presented below with the aid of MAPLE 18.
The proposed method has been successively implemented for the time fractional telegraph equation. Maximum errors in L 2 and L ∞ were obtained as shown in
| N | L2 Error (Proposed method) | L∞ Error | β |
|---|---|---|---|
| 20 40 80 160 | 3.8141E−006 1.0594E−005 3.0311E−005 4.0142E−005 | 3.6141E−006 1.2242E−005 4.0311E−005 3.0142E−004 | 1.5 |
| 20 40 80 160 | 3.7337E−003 5.2802E−003 1.6125E−003 2.2804E−003 | 3.3327E−003 4.2982E−003 2.0125E−003 3.2907E−003 | 1.8 |
The space discretization scheme was developed and implemented with the aid of Mamadu-Njoseh orthogonal basis functions. Satisfactory numerical evidence was obtained as the order of convergence of the proposed method is in total agreement with the theoretical analysis. Also, The L 2 and L ∞ errors and the numerical order are in agreement in space for β = 1.5 and 1.8.
The authors declare no conflicts of interest regarding the publication of this paper.
Mamadu, E.J., Njoseh, I.N. and Ojarikre, H.I. (2022) Space Discretization of Time-Fractional Telegraph Equation with Mamadu-Njoseh Basis Functions. Applied Mathematics, 13, 760-773. https://doi.org/10.4236/am.2022.139048