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Study of time-fractional delayed differential equations via new integral transform-based variation iteration technique

  • Brajesh K. Singh , Mukesh Kumar Awasthi , Saloni Agrawal , Mukesh Gupta and Ravi Tomar EMAIL logo
Published/Copyright: May 5, 2023
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

The present article proposes a new-integral transform-based variational iteration technique (NTVIT) to study the behavior of higher-order nonlinear time-fractional delayed differential equations. The NTVIT is a hybrid technique that is developed via the concept of variational theory with the use of the properties of a new integral transform. The stability and convergence of NTVIT are analyzed via Banach’s fixed point theory. The effectiveness and validity of NTVIT solutions are demonstrated via the evaluation of error norms: relative/absolute errors for some test suitable delayed problems of different fractional order. The numerical experiments confirms that NTVIT is capable of producing highly accurate behaviors as compared to some existing techniques.

Keywords: Caputo derivative; nonlinear time-fractional delayed partial differential equations; proportional delay; new integral transform

1 Introduction

Fractional calculus is a very meaningful concept in the field of applied mathematics motivated toward the study of integral (and differential) operators of arbitrary positive fraction [1,2]. The fractional partial differential equations, generalization of classical partial differential equations (PDEs), are the popular selection for modeling of complicated physical phenomena marked by memory/history times, which facilitates the analysis of realistic situations, e.g., the nonlinear fractional differential equations are comparatively more suitable for modeling numerous realistic situations: earthquake propagation, population growth, volcanic eruption, signal processing, reaction/diffusion processes, electrical networks, nonlinear control theory, nonlinear biological systems, and astrophysics [3–9]. Because of so wide range of applications of fractional differential equations, the study of the behaviors of the fractional differential equations is essential. But the study of solution behavior of the differential equations is very complicated task. Many researchers are continuously working on developing numerical/analytical techniques to study these kinds of fractional differential equations, see [10–16].

The class of fractional delay differential equations is suitable for modeling certain physical phenomena involving present not only time but also memory/history time and is mainly adopted for modeling numerous popular real word situations in environmental sciences, economics, chemistry, physics, and life sciences. The delayed differential equations and their fractional models appear in the modeling of various kinds of physical phenomena that occurred in many branches of engineering/science: biological models [17,18], the control theory [19,20], the oscillation theory [21,22], and theory of transport [23]. Delay differential equation in pantograph equations (pantograph: z -shaped mechanical linkage attached with the roof of an electric train/bus to receive power supply via an overhead electric wire) is the one of the most popular application in mechanical/electro-dynamic system [15,24,25]. In addition, the proportional delayed time-fractional nonlinear PDE (PDTF-NPDE) of the form

D C α τ ϕ ( σ , τ ) = b ϕ ( σ , τ ) ∂ ∂ σ ϕ ( p 0 σ , q 0 τ ) + ∂ 3 ∂ σ 3 ϕ ( p 1 σ , q 1 τ ) , σ , τ ≥ 0 , 0 < α ≤ 1 , b ∈ R

arises in numerous study based on shallow water waves and is the well-known time fractional model of Korteweg-de Vries equation with proportional delay.

The PDTF-NPDE usually adopted for the description of nonlinear wave interaction arises in the quantum field theory, and is the time fractional model of the Klein–Gordon equation with proportional delay as follows:

D C α τ ϕ ( σ , τ ) = ∂ 2 ∂ σ 2 ϕ ( p 0 σ , q 0 τ ) − b ϕ ( p 1 σ , q 1 τ ) − F ( ϕ ( p 2 σ , q 2 τ ) ) + h ( σ , τ ) , σ , τ ≥ 0 , 1 < α ≤ 2 ,

where F is the nonlinear function of ϕ ( σ , τ ) and h ( σ , τ ) be a given smooth function.

The PDTF-NPDEs are very challenging, and the transcendental nature of these delay terms makes them more complicated to evaluate numerical solutions behavior accurately. Even if numerous vigorous techniques has been developed recently for solving physical models occurred in the form of PDTF-NDEs or PDTF-NPDEs, among them, vigorous techniques for the study of delay differential equations are Chebyshev pseudospectral method [26], waveform relaxation methods [27], iterated pseudo-spectral method [28], two-dimensional differential transform method [29], and the variational iteration method (VIM) for solution of a neutral functional PDEs with proportional delays in ref. [30], and for PDTF-NPDEs: homotopy perturbation method (HPM) [31], homotopy perturbation transform method (HPTM) [32], alternative variational iteration method [33], fractional reduced differential transform method [25,34,35], residual power series method, homotopy analysis transform method [36], and optimal homotopy analysis method [37], which provided approximate numerical solutions, are in a good agreement with analytical solutions. To the best of my knowledge, the proposed NTVIT is not proposed for the study of behavior of the referred delayed model equation.

In the present article, a new technique called variational iteration new integral transform (NIT) technique is developed by utilizing interesting properties of the NIT along with the variational calculus. An interesting feature of this technique is that the valuation procedure of the Lagrange multipliers, a crucial step in standard variational iteration, needs comparatively very less computational effort and handled easily. Especially, the proposed technique is utilized in the evaluation of time-fractional delayed nonlinear DEs/PDEs:

(i) PDTF-NDEs: proportional delayed time-fractional nonlinear differential equation:

(1) D C α τ ϕ ( τ ) = T τ , ϕ ( p 0 τ ) , d d τ ϕ ( p 1 τ ) , … , d m d τ m ϕ ( p m τ ) , κ − 1 < α ≤ κ , τ ≥ 0 , d k d τ k ϕ ( τ ) τ = 0 = C k , k ∈ { 0 , 1 , … , κ − 1 } .

(ii) PDTF-NPDEs: proportional delayed time-fractional nonlinear PDE:

(2) D C α τ ϕ ( σ , τ ) = T σ , τ , ϕ ( p 0 σ , q 0 τ ) , ∂ ∂ τ ϕ ( p 1 σ , q 1 τ ) , … , ∂ m ∂ τ m ϕ ( p m σ , q m τ ) , σ , τ ≥ 0 , κ − 1 < α ≤ κ , ∂ k ∂ τ k ϕ ( σ , τ ) τ = 0 = ψ k ( σ ) , k ∈ { 0 , 1 , … , κ − 1 } ,

where C k and ψ k ( σ ) are specified initial values/functions of systems (1) and (2), respectively, for k ∈ { 0 , 1 , … , κ − 1 } , κ ∈ N ; p i , q i ∈ ( 0 , 1 ) for i ∈ { 0 , 1 , … , m } ; D C α τ ( ⋅ ) Caputo-fractional differential operator (FDO), and T be the partial differential operator.

2 Basic literature

This section revisits basic concepts of Banach’s fixed point theorem, fractional derivatives, and NIT for a crucial understanding of the rest part of the work.

Let Π = ( Π , d ) be a metric space. A point z ∈ Π with property T z = z refers to the fixed point of the map T : Π → Π , and the map T : Π → Π recalls as contraction on Π whenever for each z , z 1 ∈ Π the property d ( T z , T z 1 ) ≤ γ d ( z , z 1 ) holds for some γ , 0 < γ < 1 , see [38].

Theorem 1

(Banach’s fixed point theorem) [38] Banach’s fixed point theorem states that each contraction defined over a Banach space has a unique fixed point.

Let { x λ } 1 ∞ be a Picard iterative sequence obtained via an iterative procedure x λ + 1 = T x λ with arbitrary x 0 ∈ Π converges to the unique fixed point x of T . Prior estimate of error for the iterative procedure d ( x λ , x ) ≤ γ λ 1 − γ d ( x 0 , x 1 ) , and the posterior estimate of the error via d ( x λ , x ) ≤ x 1 − γ d ( x λ − 1 , x λ ) . It is worth mentioning that a function T : Π → Π , where Π is Banach space is termed as Picard T -stable whenever the following property hold d ( T σ , T σ 1 ) ≤ K d ( σ , T σ ) + γ d ( z , z 1 ) , ∀ z , z 1 ∈ Π , for some K ≥ 0 and 0 ≤ γ < 1 .

Let ϕ be a function taken from the space C μ [1,2]. Riemann-Liouville fractional integral-operator [1,2] D C − α τ ϕ ( τ ) of fractional order α of ϕ ∈ C μ with μ ≥ − 1 is defined by

D C − α τ ϕ ( τ ) = 1 Γ ( α ) ∫ 0 τ ( τ − y ) α − 1 ϕ ( y ) d y , α > 0 , τ > 0 ,

and Caputo-FDO D C α τ ϕ ( τ ) FDO of order κ − 1 < α ≤ κ of ϕ ∈ C μ , μ ≥ − 1 is defined by D C κ τ ϕ ( τ ) ≔ ∂ κ ϕ ( τ ) ∂ τ κ and

D C α τ ϕ ( τ ) = D C − ( κ − α ) D C κ ϕ ( τ ) = 1 Γ ( κ − α ) ∫ 0 τ ( κ − y ) κ − ( α + 1 ) ∂ κ ϕ ( y ) ∂ y κ d y , κ − 1 < α < κ .

Definition 1

(NIT, [39–41]) The NIT of ϕ ( τ ) ∈ F is defined by

(3) Φ ( μ ) = K { ϕ ( τ ) } ≔ 1 μ ∫ 0 ∞ exp − τ μ 2 ϕ ( τ ) d τ ,

where F be family of real-valued exponential-order functions, expressed as follows:

F = ϕ ( τ ) : ∃ r 1 , r 2 > 0 , 0 < K < ∞ with property ∣ ϕ ( τ ) ∣ ≤ K exp ∣ τ ∣ r ℓ 2 , if τ ∈ ( − 1 ) ℓ × [ 0 , ∞ ) ,

Some properties of NIT that requires to complete understanding of the present work are reported as follows:

Theorem 2

(Properties of NIT [39–41])

  1. The NIT of ∂ κ g ( σ , τ ) ∂ τ κ ∈ F is defined by

    K ∂ κ g ( σ , τ ) ∂ τ κ = G ( σ , μ ) μ 2 κ − ∑ ℓ = 0 κ − 1 1 μ 2 ( κ − ℓ ) − 1 ∂ ℓ g ( σ , 0 + ) ∂ τ ℓ , κ ≥ 1 .

  2. The NIT of Caputo fractional derivative D C α τ g ( σ , τ ) and Riemann-Liouville fractional integral D C − α τ g ( σ , τ ) , read as follows:

    1. K { D C − α τ g ( σ , τ ) } = K { g ( σ , τ ) } = μ 2 α G ( σ , μ ) ,

    2. K { D C α τ g ( σ , τ ) } = G ( σ , μ ) μ 2 α − ∑ ℓ = 0 κ − 1 1 μ 2 ( α − ℓ ) − 1 ∂ ℓ g ( σ , 0 ) ∂ τ ℓ , κ − 1 < α ≤ κ ∈ N ,

  3. K τ κ α Γ ( 1 + κ α ) = μ 2 κ α + 1 , κ = 0 , 1 , 2 , … , and K { 1 } = μ .

3 Variation iteration technique (VIT)

The VIT, introduced by He [42], is one of the very efficient numerical techniques. After that many researchers have adopted VIT or its modified form the so-called mVIT for the study of different types of nonlinear differential equations or its systems that occurred in various realistic physical phenomena, see [43–47] the references therein. For the last one decades, VIM is also utilized in the study of nonlinear fractional PDEs [33,48–50].

Suppose a time-fractional nonlinear PDE of the following form:

(4) D C α τ ϕ ( σ , τ ) + T ϕ ( σ , τ ) − h ( σ , τ ) = 0 , κ − 1 < α ≤ κ ,

where D C α τ ( ⋅ ) is Caputo-FDO [11,13,14], T ϕ ( σ , τ ) is nonlinear differential operator, h ( σ , τ ) is a smooth function, and κ ∈ N .

In mVIT, the correction functional of Eq. (4) [47] can be evaluated from the following:

(5) ϕ λ + 1 ( σ , τ ) = ϕ λ ( σ , τ ) + ∫ 0 τ θ ( τ , ε ) [ D C α τ ϕ ( σ , τ ) + T ϕ ˜ λ ( σ , ε ) − h ( σ , ε ) ] d ε ,

where θ ( τ , ε ) is termed as a Lagrange multiplier. The evaluation of the multiplier θ ( τ , ε ) is a very tough task, which is a significant part of the iteration scheme. In the iteration procedure, terms ϕ ˜ λ refers the restricted variation [51]. The stationary property of the functional (5) leads to

(6) δ ϕ λ + 1 ( σ , τ ) = δ ϕ λ ( σ , τ ) + δ ∫ 0 t θ ( τ , ε ) D C α τ ϕ λ ( σ , ε ) d ε = 0 .

Our main goal is to evaluate the multiplier θ from Eq. (6). In the case when α = κ , the optimal θ is obtained via integration by parts and variational theory. But it is very tough for fractional case ( α ≠ κ ) [50]. In the following section, it is seen that how easily this evaluation is processed after imposing the properties of NIT with variational theory.

3.1 NIT-based VIT for PDTF-NDEs

In the main section, a new hybrid technique, based variational iteration technique (NTVIT) (NIT-based VIT) is developed for the evaluation of the behavior of nonlinear models of time fractional delayed PDEs of higher orders.

Case A: Imposing NIT operator to PDTF-NDEs (1) and utilize the property of NIT of D C α τ ϕ ( σ , τ ) from Theorem 2(b)(ii), we obtain

K { D C α τ ϕ ( τ ) } = Φ ( μ ) μ 2 α − ∑ ℓ = 0 κ − 1 1 μ 2 ( α − ℓ ) − 1 d ℓ ϕ ( τ ) d τ ℓ τ = 0 ,

and we also obtain

(7) 1 μ 2 α Φ ( μ ) − ∑ ℓ = 0 κ − 1 1 μ 2 ( α − ℓ ) − 1 C ℓ − K T τ , ϕ ( p 0 τ ) , d d τ ϕ ( p 1 τ ) , … , d m d τ m ϕ ( p m τ ) = 0 .

In sequel to mVIT, a correction functional for (7) can be constructed as follows:

(8) Φ λ + 1 ( μ ) = Φ λ ( μ ) + θ ( μ ) × 1 μ 2 α Φ λ ( μ ) − ∑ ℓ = 0 κ − 1 1 μ 2 ( α − ℓ ) − 1 C ℓ − θ ( μ ) K T ˜ τ , ϕ λ ( p 0 τ ) , d d τ ϕ λ ( p 1 τ ) , … , d m d τ m ϕ λ ( p m τ ) ,

where ϕ ˜ λ and T ˜ restricted variations, and so, δ ϕ ˜ λ = 0 and δ T ˜ = 0 .

By utilizing variational operator δ on (8) with the aforementioned property, we obtain

(9) δ Φ λ + 1 ( μ ) = δ Φ λ ( μ ) + δ θ ( μ ) 1 μ 2 α Φ λ ( μ ) .

By utilizing the stationary condition: δ Φ λ + 1 ( μ ) = 0 for (8) in (9), we obtain

(10) 1 + θ ( μ ) μ 2 α = 0 → θ ( μ ) = − μ 2 α .

By utilizing the optimal value of the multiplier θ ( μ ) = − μ 2 α from (10) in (8), we obtain

(11) Φ λ + 1 ( μ ) = ∑ ℓ = 0 κ − 1 μ 2 ℓ + 1 C ℓ + μ 2 α K { T ( τ , ϕ λ ( p 0 τ ) , × d d τ ϕ λ ( p 1 τ ) , … , d m d τ m ϕ λ ( p m τ ) .

By imposing inverse NIT operator in (11), we obtain

(12) ϕ λ + 1 ( τ ) = T ϕ λ ( τ ) ,

where

T ϕ λ ( τ ) = ϕ λ 0 ( τ ) + K − 1 μ 2 α K T τ , ϕ λ ( p 0 τ ) , d d τ ϕ λ ( p 1 τ ) , … , d m d τ m ϕ λ ( p m τ )

and

ϕ λ 0 ( τ ) = ∑ ℓ = 0 κ − 1 τ ℓ Γ ( ℓ + 1 ) ∂ ℓ ∂ τ ℓ ϕ λ ( 0 + ) .

which is the required iteration formula derived from NTVIT for solving PDTF-NDEs (1).

Case B: Analogous to case A iteration formula derived from NTVIT for solving PDTF-NPDEs (2) is obtained as follows:

(13) ϕ λ + 1 ( σ , τ ) = T ϕ λ ( σ , τ ) ,

where

T ϕ λ ( σ , τ ) = ϕ λ 0 ( σ , τ ) + K − 1 μ 2 α K T σ , τ , ϕ λ ( p 0 σ , q 0 τ ) , ∂ ∂ τ ϕ λ × ( p 1 σ , q 1 τ ) , … , ∂ m ∂ τ m ϕ λ ( p m σ , q m τ ) ,

and

ϕ λ 0 ( σ , τ ) = ∑ ℓ = 0 κ − 1 τ ℓ Γ ( ℓ + 1 ) ∂ ℓ ∂ τ ℓ ϕ λ ( σ , 0 + ) .

4 Stability analysis and convergence of NTVIT

The stability analysis of the aforesaid method is provided in the following theorem.

Theorem 3

(Stability analysis of NTVIT) Let T : Π → Π be a self-map on Banach space ( Π , ‖ ⋅ ‖ ) . The solution behavior from the iteration procedure: ϕ λ + 1 ( τ ) = T ϕ λ ( τ ) as defined in (12) is Picard T -stable whenever the following conditions hold true for each τ .

  1. ‖ ϕ p 0 ( τ ) − ϕ n 0 ( τ ) ‖ ≤ δ 0 ‖ ϕ p ( τ ) − ϕ n ( τ ) ‖ for some δ 0 > 0 ,

  2. T τ , ϕ p ( p 0 τ ) , d d τ ϕ p ( p 1 τ ) , … , d m d τ m ϕ p ( p m τ ) − T τ , ϕ n ( p 0 τ ) , d d τ ϕ n ( p 1 τ ) , … , d m d τ m ϕ n ( p m τ ) ≤ T ϕ p ( p 0 τ ) − ϕ n ( p 0 τ ) , … , d m d τ m ϕ p ( p m τ ) − d m d τ m ϕ n ( p m τ ) ≤ δ 1 ‖ ϕ p ( τ ) − ϕ n ( τ ) ‖ , for s o m e δ 1 > 0 ,

  3. θ = δ 0 + δ 1 t α Γ ( α + 1 ) < 1 .

Proof

Let n , p ∈ N . Then

(14) T ϕ p − T ϕ n = ϕ p 0 ( τ ) − ϕ n 0 ( τ ) + K − 1 μ 2 α K T τ , ϕ p ( p 0 τ ) , d d τ ϕ p × ( p 1 τ ) , … , d m d τ m ϕ p ( p m τ ) − K − 1 μ 2 α K T τ , ϕ n ( p 0 τ ) , d d τ ϕ n × ( p 1 τ ) , … , d m d τ m ϕ n ( p m τ ) , = ϕ p 0 ( τ ) − ϕ n 0 ( τ ) + K − 1 μ 2 α K T ϕ p ( p 0 τ ) − ϕ n ( p 0 τ ) , … , d m d τ m ϕ p ( p m τ ) − d m d τ m ϕ n ( p m τ ) .

and after imposing norm on both sides to (14):

∥ T ϕ p − T ϕ n ∥ ≤ ∥ ϕ p 0 ( τ ) − ϕ n 0 ( τ ) ∥ + K − 1 μ 2 α K T ϕ p ( p 0 τ ) − ϕ n ( p 0 τ ) , … , d m d τ m ϕ p ( p m τ ) − d m d τ m ϕ n ( p m τ ) .

By utilizing condition ( a ) and ( b ) , we obtain

(15) ∥ T ϕ p − T ϕ n ∥ ≤ δ 0 ∥ ϕ p ( τ ) − ϕ n ( τ ) ∥ + δ 1 ∥ ϕ p ( τ ) − ϕ n ( τ ) ∥ K − 1 { μ 2 α K { 1 } } , = δ 0 + δ 1 t α Γ ( α + 1 ) ∥ ϕ p ( τ ) − ϕ n ( τ ) ∥ ≤ θ ∥ ϕ p ( τ ) − ϕ n ( τ ) ∥ .

Thus, we obtain

(16) ∥ T ϕ p − T ϕ n ∥ ≤ θ ∥ ϕ p ( τ ) − ϕ n ( τ ) ∥ ≤ β ‖ ϕ p − T ϕ p ‖ + θ ‖ ϕ p − ϕ n ‖ , for β ≥ 0

Hence, NTVIT is Picard T -stable provided θ < 1 .

Convergence of NTVIT and its error estimates are stated and proved in the following. For the sake of convenience, we read ϕ n in place of ϕ n ( τ ) or ϕ n ( σ , τ ) .

Theorem 4

(Convergence analysis) Let { ϕ n } 1 ∞ be a sequence of a Banach space Π = ( C [ Ω × ( 0 , T ) ] , ‖ ⋅ ‖ ) generated by iteration formula of NTVIT derived in (12) or (13), and T : Π → Π be its associated self-map. Then

  1. T has a unique fixed point in Π , and

  2. The sequence { ϕ n } 1 ∞ with initial value ϕ 0 ∈ Π converges to the fixed point.

Proof

  1. Eq. (15) confirms that T is a contraction on Banach space Π , and so, T has a fixed point in Π , see Theorem 1.

  2. By assumption, the sequence { ϕ n } 1 ∞ with initial value ϕ 0 ∈ Π is generated by iteration formula of NTVIT as derived in (12) (or (13)), that is,

    (17) ϕ λ + 1 = T ϕ λ .

Iteration formula (17) can be re-written in the following form:

(18) ϕ 1 = T ϕ 0 , ϕ 2 = T ϕ 1 = T 2 ϕ 0 , … , ϕ k = T k ϕ 0 , … ,

To check the convergence of the iteration procedure { ϕ n } 1 ∞ , it is sufficient to show that it is a Cauchy sequence.

Write λ , κ ∈ N such that λ > κ . From Eq. (15) and iteration formula (17), we obtain

(19) ‖ ϕ 2 − ϕ 1 ‖ = ‖ T ϕ 1 − T ϕ 0 ‖ ≤ θ ‖ ϕ 1 − ϕ 0 ‖ , … , ‖ ϕ λ + 1 − ϕ λ ‖ ≤ θ λ ‖ ϕ 1 − ϕ 0 ‖ .

Cauchy-Schwarz inequality and (19),

(20) ‖ ϕ λ − ϕ κ ‖ = ∑ ℓ = 0 λ − κ − 1 ( ϕ κ + ℓ + 1 − ϕ κ + ℓ ) ≤ ∑ ℓ = 0 λ − κ − 1 θ κ + ℓ ‖ ϕ 1 − ϕ 0 ‖ = θ κ ‖ ϕ 1 − ϕ 0 ‖ ∑ ℓ = 0 λ − κ − 1 θ ℓ = θ κ ‖ ϕ 1 − ϕ 0 ‖ 1 − θ λ − κ 1 − θ ≤ ‖ ϕ 1 − ϕ 0 ‖ θ κ 1 − θ ,

as 0 < 1 − θ λ − κ < 1 whenever 0 < θ < 1 .

Moreover, ‖ ϕ 1 − ϕ 0 ‖ is finite and fixed, so, for any given quantity ε > 0 , there exists a sufficiently large value κ 0 for κ for which

(21) ‖ ϕ λ − ϕ κ ‖ < ε , λ , κ ≥ κ 0 ,

confirming that the sequence { ϕ λ } λ = 1 ∞ is Cauchy. In consequence, the sequence converges to ϕ ∈ Π , that is,

lim κ → ∞ ϕ κ = ϕ .

Next, we show that ϕ ∈ Π is the fixed point of T .

By utilize triangle inequality and Eq. (15), we obtain

(22) ‖ ϕ − T ϕ ‖ ≤ ‖ ϕ − ϕ κ ‖ + ‖ T ϕ κ − 1 − T ϕ ‖ ≤ ‖ ϕ − ϕ κ ‖ + θ ‖ ϕ κ − 1 − ϕ ‖ = 0 as κ → ∞ ,

confirming that ϕ = T ϕ , and so, ϕ ∈ Π is the unique fixed point of T . T is contraction of Banach space Π .

5 Test examples

In the main section of this article, the effectiveness and validity of the proposed technique are demonstrated in terms of relative/absolute error norm for six test problems of PDTF PDEs of various fractional order. The relative error in κ th iterative results is computed by

RE ( ϕ κ ) = ϕ κ ( τ ) − ϕ κ − 1 ( τ ) ϕ κ ( τ ) ,

where ϕ κ ( τ ) is the approximate results at κ th iteration.

Example 1

The nonlinear model of pantograph delay differential equation with suitable initial conditions as in ref. [15] is expressed as follows:

(23) D C α τ ϕ ( τ ) = 2 ϕ 2 τ 2 − 1 ϕ ( 0 ) = 0 , ϕ ′ ( 0 ) = 1 , ϕ ″ ( 0 ) = 0 τ ≥ 0 , 2 < α ≤ 3 .

In special case when α = 3 , the exact solution of this problem is ϕ ( τ ) = sin τ .

On implementing iteration formula of NTVIT as mentioned in (12)–(23), we obtain

(24) ϕ λ + 1 ( τ ) = ϕ λ 0 ( τ ) + K − 1 μ 2 α K 2 ϕ λ 2 τ 2 − 1 , λ ≥ 0 ,

where ϕ λ 0 ( τ ) = τ ϕ ′ ( 0 ) = τ , for λ = 0 .

The evaluation of recurrence (24) yields the following the first three iterations as follows:

(25) ϕ 1 ( τ ) = τ − τ α Γ ( α + 1 ) , ϕ 2 ( τ ) = 2 1 − 2 α τ 3 α Γ ( 2 α + 1 ) Γ ( α + 1 ) 2 Γ ( 3 α + 1 ) + τ α + 2 Γ ( α + 3 ) − 2 1 − α τ 2 α + 1 Γ ( α + 2 ) + τ α Γ ( 2 α + 2 ) Γ ( α + 1 ) Γ ( 2 α + 2 ) + τ , ϕ 3 ( τ ) = 2 1 − 2 α Γ ( 2 α + 1 ) τ 3 α Γ ( α + 1 ) 2 Γ ( 3 α + 1 ) − 2 3 − 6 α Γ ( 2 α + 1 ) Γ ( 4 α + 1 ) τ 5 α Γ ( α + 1 ) 3 Γ ( 3 α + 1 ) Γ ( 5 α + 1 ) + 2 3 − 10 α Γ ( 2 α + 1 ) 2 Γ ( 6 α + 1 ) τ 7 α Γ ( α + 1 ) 4 Γ ( 3 α + 1 ) 2 Γ ( 7 α + 1 ) + τ α + 2 Γ ( α + 3 ) − 2 1 − α Γ ( α + 2 ) τ 2 α + 1 Γ ( α + 1 ) Γ ( 2 α + 2 ) + 2 − α − 1 Γ ( α + 4 ) τ 2 α + 3 Γ ( α + 3 ) Γ ( 2 α + 4 ) − 2 − 2 α Γ ( 2 α + 3 ) τ 3 α + 2 Γ ( α + 1 ) Γ ( α + 3 ) Γ ( 3 α + 3 ) − 2 1 − 3 α Γ ( α + 2 ) Γ ( 2 α + 3 ) τ 3 α + 2 Γ ( α + 1 ) Γ ( 2 α + 2 ) Γ ( 3 α + 3 ) + 2 − 2 α − 3 Γ ( 2 α + 5 ) τ 3 α + 4 Γ ( α + 3 ) 2 Γ ( 3 α + 5 ) + 2 2 − 4 α Γ ( α + 2 ) Γ ( 3 α + 2 ) τ 4 α + 1 Γ ( α + 1 ) 2 Γ ( 2 α + 2 ) Γ ( 4 α + 2 ) + 2 2 − 5 α Γ ( 2 α + 1 ) Γ ( 3 α + 2 ) τ 4 α + 1 Γ ( α + 1 ) 2 Γ ( 3 α + 1 ) Γ ( 4 α + 2 ) − 2 − 4 α Γ ( α + 2 ) Γ ( 3 α + 4 ) τ 4 α + 3 Γ ( α + 1 ) Γ ( α + 3 ) Γ ( 2 α + 2 ) Γ ( 4 α + 4 ) + 2 1 − 6 α Γ ( α + 2 ) 2 Γ ( 4 α + 3 ) τ 5 α + 2 Γ ( α + 1 ) 2 Γ ( 2 α + 2 ) 2 Γ ( 5 α + 3 ) + 2 1 − 6 α Γ ( 2 α + 1 ) Γ ( 4 α + 3 ) τ 5 α + 2 Γ ( α + 1 ) 2 Γ ( α + 3 ) Γ ( 3 α + 1 ) Γ ( 5 α + 3 ) − 2 3 − 8 α Γ ( α + 2 ) Γ ( 2 α + 1 ) Γ ( 5 α + 2 ) τ 6 α + 1 Γ ( α + 1 ) 3 Γ ( 2 α + 2 ) Γ ( 3 α + 1 ) Γ ( 6 α + 2 ) − τ α Γ ( α + 1 ) + τ .

In consequence, λ th iterative solutions ϕ λ ( τ ) for λ ≥ 4 can be computed from (24). Table 1 reports the comparison between the relative errors in λ th-order ( λ = 4 , 5 ) evaluated results for α = 2.8 , 2.9 , 3 and the absolute errors in 4th iterative results for α = 3 is compared with the error obtained by Muntz-Legendre wavelet operational matrix method [15], which concludes that computed results are highly improved the results in ref. [15]. Figure 1(a) depicts the comparison of evaluated results at second/third iteration with exact results; Figure 1(b) depicts the comparison of evaluated results at fifth-iteration with exact results; Figure 1(c) and (d) depicts logarithmic plots of relative errors in λ th iterative results ( λ = 3 , 4 , 5 ) for α = 2.8 and α = 3 , respectively. Figure 1(e) depicts logarithmic plots of absolute errors in λ th iterative results ( λ = 3 , 4 , 5 ) for α = 3 . The numerical testing from Table 1 and Figure 1 shows that NTVIT is capable of providing the approximations with high accuracy, and converging very fast to the exact behavior.

Table 1

Relative errors in λ th-order results ( λ = 4 , 5 ) of Example 1 for α = 2.8 , 2.9 , 3 , and comparison of absolute error in fourth-order results in ref. [15] at different time levels 0 < τ ≤ 1

τ RE ϕ 4 RE ϕ 5 ϕ 4 ( τ ) M = 4 , γ = 2 [15]
α = 2.8 α = 2.9 α = 3 α = 2.8 α = 2.9 α = 3 Absolute error
0.2 1.6895 × 1 0 − 19 4.0602 × 1 0 − 20 9.6883 × 1 0 − 21 6.6701 × 1 0 − 29 7.9051 × 1 0 − 30 9.2741 × 1 0 − 31 2.7756 × 1 0 − 17 1.56 × 1 0 − 12
0.4 4.6719 × 1 0 − 16 1.3775 × 1 0 − 16 4.0345 × 1 0 − 17 2.5667 × 1 0 − 24 4.0010 × 1 0 − 25 6.1758 × 1 0 − 26 5.5511 × 1 0 − 17 4.84 × 1 0 − 11
0.6 4.9243 × 1 0 − 14 1.6334 × 1 0 − 14 5.3828 × 1 0 − 15 1.2612 × 1 0 − 21 2.3037 × 1 0 − 22 4.1676 × 1 0 − 23 1.1102 × 1 0 − 16 9.59 × 1 0 − 11
0.8 1.3724 × 1 0 − 12 4.9424 × 1 0 − 13 1.7684 × 1 0 − 13 1.0469 × 1 0 − 19 2.1373 × 1 0 − 20 4.3218 × 1 0 − 21 0.0000 × 1 0 00 4.03 × 1 0 − 10
1 1.8568 × 1 0 − 11 7.1202 × 1 0 − 12 2.7126 × 1 0 − 12 3.2996 × 1 0 − 18 7.3374 × 1 0 − 19 1.6158 × 1 0 − 19 0.0000 × 1 0 00
Figure 1 (a) Comparison of iterative results with the exact results; (b) comparison of 5th-order results with exact results; logarithmic plots of relative errors in λ \lambda th iterative results ( λ = 3 , 4 , 5 ) \left(\lambda =3,4,5) for (c) α = 2.8 \alpha =2.8 , (d) α = 3 \alpha =3 ; and (e) logarithmic plots of absolute errors in λ \lambda th iterative results ( λ = 3 , 4 , 5 ) \left(\lambda =3,4,5) for α = 3 \alpha =3 in Example 1. (a) Convergence plot of the evaluated results for τ ∈ ( 0 , 0.5 ) \tau \in \left(0,0.5) . (b) Comparison of fifth-order results with exact results in τ ∈ ( 0 , 0.5 ) \tau \in \left(0,0.5) . (c) Logarithmic plots of relative errors for α = 2.8 \alpha =2.8 . (d) Logarithmic plots of relative error for α = 3 \alpha =3 . (e) Logarithmic plots of absolute error for α = 3 \alpha =3 .
Figure 1

(a) Comparison of iterative results with the exact results; (b) comparison of 5th-order results with exact results; logarithmic plots of relative errors in λ th iterative results ( λ = 3 , 4 , 5 ) for (c) α = 2.8 , (d) α = 3 ; and (e) logarithmic plots of absolute errors in λ th iterative results ( λ = 3 , 4 , 5 ) for α = 3 in Example 1. (a) Convergence plot of the evaluated results for τ ∈ ( 0 , 0.5 ) . (b) Comparison of fifth-order results with exact results in τ ∈ ( 0 , 0.5 ) . (c) Logarithmic plots of relative errors for α = 2.8 . (d) Logarithmic plots of relative error for α = 3 . (e) Logarithmic plots of absolute error for α = 3 .

Example 2

A PDTF-NPDES with suitable initial conditions as in ref. [37] is expressed as follows:

(26) D C α τ ϕ ( τ ) − 2 ϕ τ 2 + ϕ ( τ ) + 1 + τ 2 = 0 τ > 0 , 2 < α ≤ 3 , ϕ ( 0 ) = 1 ϕ ′ ( 0 ) = 0 ϕ ″ ( 0 ) = − 4 .

In special case when α = 3 , the exact solution of this problem is ϕ ( τ ) = 1 − 2 τ 2 .

On implementing the iteration formula of NTVIT as mentioned in (12) on Eq. (26), we obtain

(27) ϕ λ + 1 ( τ ) = ϕ λ 0 ( τ ) + K − 1 μ 2 α K 2 ϕ τ 2 − ϕ λ ( τ ) − τ 2 − 1 , λ ≥ 0 ,

where ϕ λ 0 ( τ ) = 1 − 2 τ 2 for λ = 0 .

On solving recurrence (27),

(28) ϕ 1 ( τ ) = − 2 τ α + 2 Γ ( α + 3 ) − 2 τ 2 + 1 , ϕ 2 ( τ ) = − 2 − α τ 2 α + 2 Γ ( 2 α + 3 ) + 2 τ 2 α + 2 Γ ( 2 α + 3 ) − 2 τ 2 + 1 , ϕ 3 ( τ ) = 2 − 2 α τ 3 α + 2 Γ ( 3 α + 3 ) + 2 − α τ 3 α + 2 Γ ( 3 α + 3 ) − 2 − 3 α − 1 t 3 α + 2 Γ ( 3 α + 3 ) − 2 τ 3 α + 2 Γ ( 3 α + 3 ) − 2 τ 2 + 1 , …

In consequence, the λ th iterative solutions ϕ λ ( τ ) for λ ≥ 4 can be computed from (27). Table 2 reports the comparison between the relative errors in λ th-order solutions ϕ λ ( λ = 5 , 6 ) for α = 2.8 , 2.9 while absolute errors in ϕ λ ( τ ) ( λ = 4 , 6 ) solutions. Third-order approximate solutions are compared with the recently developed optimal homotopy analysis method (OHAM) [37] in Table 3, which shows that NTVIT results are highly improved upon OHAM results.

Table 2

Relative errors in fifth/sixth-order results of Example 2 for different values α = 2.8 , 2.9 , 3 at different time levels 0 < τ ≤ 1

τ RE ϕ 5 RE ϕ 6 Absolute error ( α = 3 )
α = 2.8 α = 2.9 α = 3 α = 2.8 α = 2.9 α = 3 In ϕ 4 ( τ ) In ϕ 6 ( τ )
0.2 1.810 × 1 0 − 16 6.034 × 1 0 − 17 6.034 × 1 0 − 17 1.810 × 1 0 − 16 6.034 × 1 0 − 17 6.034 × 1 0 − 17 0.000 × 1 0 00 1.110 × 1 0 − 16
0.4 1.469 × 1 0 − 15 3.265 × 1 0 − 16 0.000 × 1 0 00 1.633 × 1 0 − 16 3.265 × 1 0 − 16 0.000 × 1 0 00 0.000 × 1 0 00 0.000 × 1 0 00
0.6 7.359 × 1 0 − 13 2.101 × 1 0 − 13 5.948 × 1 0 − 14 0.000 × 1 0 00 7.930 × 1 0 − 16 0.000 × 1 0 00 1.665 × 1 0 − 14 0.000 × 1 0 00
0.8 3.283 × 1 0 − 11 1.055 × 1 0 − 11 3.350 × 1 0 − 12 8.723 × 1 0 − 15 7.930 × 1 0 − 16 7.930 × 1 0 − 16 9.377 × 1 0 − 13 1.665 × 1 0 − 16
1 1.75 × 1 0 − 10 6.14 × 1 0 − 11 2.132 × 1 0 − 11 8.79 × 1 0 − 14 2.13 × 1 0 − 14 5.33 × 1 0 − 15 2.13 × 1 0 − 11 5.33 × 1 0 − 15
Table 3

Comparison of NTVIT with OHAM [37] third-order solutions of Example 2 for different α = 2.8 , 2.9 , 3 at 0 < τ ≤ 1

ϕ 3 ( τ ) ( α = 2.8 ) ϕ 3 ( τ ) ( α = 3 ) Absolute error ( α = 3 )
τ OHAM [37] NTVIT OHAM [37] NTVIT Exact OHAM [37] NTVIT
0.2 0.9199 0.9200 0.9200 0.9200 0.9200 3.957 × 1 0 − 5 7.772 × 1 0 − 16
0.4 0.6795 0.6800 0.6797 0.6800 0.6800 3.168 × 1 0 − 4 1.955 × 1 0 − 12
0.6 0.2785 0.2800 0.2789 0.2800 0.2800 1.057 × 1 0 − 3 1.691 × 1 0 − 10
0.8 0.2830 − 0.2800 0.2824 − 0.2800 − 0.2800 2.416 × 1 0 − 3 4.003 × 1 0 − 9
1 1.0049 − 1.0000 1.0044 − 1.0000 − 1.0000 4.373 × 1 0 − 3 4.661 × 1 0 − 8

Figure 2(a) depicts the comparison of computed results at second/third iteration with exact results. Figure 2(b) depicts the comparison of computed results at fifth iteration with exact results; Figure 2(c) and (d) depicts plots of relative errors in λ th iterative results ( λ = 3 , 4 , 5 ) for α = 2.8 and α = 3 , respectively, while the absolute errors for α = 3 are depicted in Figure 2(e). From Tables 2 and 3 and Figure 2, it is seen that NTVIT is capable of producing highly accurate results, which converge very fast to the exact results.

Figure 2 (a) Comparison of computed results at second/third iteration with exact results; (b) comparison of computed results at fifth iteration with exact results; plots of relative errors in λ \lambda th iterative results ( λ = 3 , 4 , 5 ) \left(\lambda =3,4,5) for (c) α = 2.8 \alpha =2.8 , and (d) α = 3 \alpha =3 ; and (e) plots of absolute errors in λ \lambda th iterative results ( λ = 3 , 4 , 5 ) \left(\lambda =3,4,5) for α = 3 \alpha =3 in Example (2). (a) Convergence plot of the computed results for τ ∈ ( 0 , 0.5 ) \tau \in \left(0,0.5) . (b) Comparison of fifth-order results with exact results in τ ∈ ( 0 , 0.5 ) \tau \in \left(0,0.5) . (c) Plots of relative errors for α = 2.8 \alpha =2.8 . (d) Plots of relative error for α = 3 \alpha =3 . (e) Plots of absolute error for α = 3 \alpha =3 .
Figure 2

(a) Comparison of computed results at second/third iteration with exact results; (b) comparison of computed results at fifth iteration with exact results; plots of relative errors in λ th iterative results ( λ = 3 , 4 , 5 ) for (c) α = 2.8 , and (d) α = 3 ; and (e) plots of absolute errors in λ th iterative results ( λ = 3 , 4 , 5 ) for α = 3 in Example (2). (a) Convergence plot of the computed results for τ ∈ ( 0 , 0.5 ) . (b) Comparison of fifth-order results with exact results in τ ∈ ( 0 , 0.5 ) . (c) Plots of relative errors for α = 2.8 . (d) Plots of relative error for α = 3 . (e) Plots of absolute error for α = 3 .

Example 3

(System of delay differential equation) Let us consider the system of delay differential equation with suitable initial conditions [15,25]:

(29) D C α τ ϕ ( τ ) = ϕ ( τ ) − ψ ( τ ) + ϕ τ 2 − exp τ 2 + exp ( − τ ) D C α τ ψ ( τ ) = − ϕ ( τ ) − ψ ( τ ) − ψ τ 2 + exp − τ 2 + exp ( τ ) ϕ ( 0 ) = 1 , ψ ( 0 ) = 1 , τ > 0 , 0 < α ≤ 1 .

In special case when α = 1 the exact solution is ϕ ( τ ) = exp τ , ϕ ( τ ) = exp ( − τ ) (Table 4).

Table 4

Comparison of seventh-order results with exact results and absolute errors in λ th-order ( λ = 5 , 7 ) results in example 3 with λ = 1 at different time levels 0 < τ ≤ 1

τ ϕ 7 Exact ∣ E ϕ 5 ∣ ∣ E ϕ 7 ∣ ψ 7 Exact ∣ E ψ 5 ∣ ∣ E ψ 7 ∣
0.2 1.2214 1.2214 1.1372 × 1 0 − 6 8.2179 × 1 0 − 8 8.1873 × 1 0 − 1 8.1873 × 1 0 − 1 1.2325 × 1 0 − 7 4.7335 × 1 0 − 8
0.3 1.3499 1.3499 1.2973 × 1 0 − 5 3.7545 × 1 0 − 8 7.4082 × 1 0 − 1 7.4082 × 1 0 − 1 1.2571 × 1 0 − 6 9.6883 × 1 0 − 8
0.4 1.4918 1.4918 7.3017 × 1 0 − 5 4.1740 × 1 0 − 7 6.7032 × 1 0 − 1 6.7032 × 1 0 − 1 6.2374 × 1 0 − 6 1.3871 × 1 0 − 8
0.5 1.6487 1.6487 2.7909 × 1 0 − 4 2.6012 × 1 0 − 6 6.0653 × 1 0 − 1 6.0653 × 1 0 − 1 2.0637 × 1 0 − 5 2.5134 × 1 0 − 7
0.6 1.8221 1.8221 8.3521 × 1 0 − 4 1.0868 × 1 0 − 5 5.4881 × 1 0 − 1 5.4881 × 1 0 − 1 5.2171 × 1 0 − 5 7.7130 × 1 0 − 7
0.7 2.0137 2.0138 2.1113 × 1 0 − 3 3.7369 × 1 0 − 5 4.9658 × 1 0 − 1 4.9659 × 1 0 − 1 1.0764 × 1 0 − 4 2.4157 × 1 0 − 6
0.8 2.2254 2.2255 4.7173 × 1 0 − 3 1.0858 × 1 0 − 4 4.4932 × 1 0 − 1 4.4933 × 1 0 − 1 1.8636 × 1 0 − 4 6.0531 × 1 0 − 6
0.9 2.4593 2.4596 9.5918 × 1 0 − 3 2.7923 × 1 0 − 4 4.0656 × 1 0 − 1 4.0657 × 1 0 − 1 2.6885 × 1 0 − 4 1.3312 × 1 0 − 5
1.0 2.7176 2.7183 1.8107 × 1 0 − 2 6.5018 × 1 0 − 4 3.6785 × 1 0 − 1 3.6788 × 1 0 − 1 2.9976 × 1 0 − 4 2.5053 × 1 0 − 5

On implementing the iteration formula of NTVIT as mentioned in (12) on Eq. (29), we obtain

(30) ϕ λ + 1 ( τ ) = ϕ λ ( 0 ) + K − 1 μ 2 α K ϕ λ ( τ ) − ψ λ ( τ ) + ϕ λ τ 2 − exp τ 2 + exp ( − τ ) ψ λ + 1 ( τ ) = ψ λ ( 0 ) + K − 1 μ 2 α K − ϕ λ ( τ ) − ψ λ ( τ ) − ψ λ τ 2 + exp − τ 2 + exp ( τ ) .

On solving recurrence (30),

(31) ϕ 1 ( τ ) = exp ( − τ ) τ α ( − τ ) − α + τ α Γ ( α + 1 ) − exp ( − τ ) τ α ( − τ ) − α Γ ( α , − τ ) Γ ( α ) − 2 α exp ( τ ∕ 2 ) + 2 α exp ( τ ∕ 2 ) Γ α , τ 2 Γ ( α ) + 1 , ψ 1 ( τ ) = 2 α exp − τ 2 τ α ( − τ ) − α − 3 τ α Γ ( α + 1 ) − 2 α exp − τ 2 τ α ( − τ ) − α Γ α , − τ 2 Γ ( α ) − exp ( τ ) Γ ( α , τ ) Γ ( α ) + exp ( τ ) + 1 .

In consequence, the λ th iterative solutions ϕ λ ( τ ) for λ ≥ 2 can be evaluated from (30). Table 5 reports evaluated solution ϕ 7 ( τ ) , ψ 7 ( τ ) and the comparison between the absolute errors in λ th-order ( λ = 5 , 7 ) evaluated results for α = 1 , which shows that computed results converges to the exact results.

Table 5

Comparison of the absolute errors in λ th-order results ( λ = 4 , 6 ) with various methods: HPM [31], HPTM [32], VIM [33], and FRDTM [34] for different values of σ , τ in Example 4

σ τ [31–34] in ϕ 4 In ϕ 6 [33] In ϕ 6 [34] NTVIT in ϕ 4 NTVIT in ϕ 6
0.25 0.25 2.1230 × 1 0 − 6 8.7895 × 1 0 − 8 3.9529 × 1 0 − 9 5.7849 × 1 0 − 7 2.3288 × 1 0 − 10
0.5 7.0943 × 1 0 − 5 5.8385 × 1 0 − 6 4.1348 × 1 0 − 7 2.0000 × 1 0 − 5 3.1798 × 1 0 − 8
0.75 5.6348 × 1 0 − 4 6.9095 × 1 0 − 5 7.2987 × 1 0 − 6 1.6439 × 1 0 − 4 5.8049 × 1 0 − 7
1 2.4871 × 1 0 − 3 4.0379 × 1 0 − 4 5.6569 × 1 0 − 5 7.5129 × 1 0 − 4 4.6544 × 1 0 − 6
0.5 0.25 4.2450 × 1 0 − 6 1.7579 × 1 0 − 7 7.9058 × 1 0 − 9 1.1570 × 1 0 − 6 4.6576 × 1 0 − 10
0.5 1.4189 × 1 0 − 4 1.1677 × 1 0 − 5 8.2597 × 1 0 − 7 4.0000 × 1 0 − 5 6.3595 × 1 0 − 8
0.75 1.1270 × 1 0 − 3 1.3819 × 1 0 − 4 1.4597 × 1 0 − 5 3.2879 × 1 0 − 4 1.1610 × 1 0 − 6
1 4.9743 × 1 0 − 3 8.0758 × 1 0 − 4 1.1314 × 1 0 − 4 1.5026 × 1 0 − 3 9.3087 × 1 0 − 6
0.75 0.25 6.3670 × 1 0 − 6 2.6368 × 1 0 − 7 1.1858 × 1 0 − 8 1.7355 × 1 0 − 6 6.9865 × 1 0 − 10
0.5 2.1283 × 1 0 − 4 1.7515 × 1 0 − 5 1.2364 × 1 0 − 6 6.0000 × 1 0 − 5 9.5393 × 1 0 − 8
0.75 1.6905 × 1 0 − 3 2.0728 × 1 0 − 4 2.1901 × 1 0 − 5 4.9318 × 1 0 − 4 1.7415 × 1 0 − 6
1 7.4614 × 1 0 − 3 1.2113 × 1 0 − 3 1.6970 × 1 0 − 4 2.2539 × 1 0 − 3 1.3963 × 1 0 − 5

Figure 3(a) and (b) depicts the comparison of evaluated results at λ th iteration ( λ = 2 , 3 , 4 ) with exact results for ϕ and ψ , respectively, whereas the absolute errors in λ th iterative ( λ = 4 , 5 , 6 ) results are depicted in Figure 3(c) and (d). These findings demonstrate that NTVIT is capable of producing the results with high accuracy, which converges very fast to the exact results.

Figure 3 Comparison of computed results at λ \lambda th iteration ( λ = 2 , 3 , 4 ) \left(\lambda =2,3,4) with exact results for (a) ϕ \phi and (b) ψ \psi , and logarithmic plots of absolute errors in λ \lambda th iterative ( λ = 3 , 5 , 7 ) \left(\lambda =3,5,7) results for (c) ϕ \phi , (d) ψ \psi in example (3) with α = 1 \alpha =1 . (a) Convergence plot of the computed results of ϕ \phi for τ ∈ ( 0 , 0.5 ) \tau \in \left(0,0.5) . (b) Convergence plot of the computed results of ψ \psi for τ ∈ ( 0 , 0.5 ) \tau \in \left(0,0.5) . (c) Logarithmic plots of absolute errors in ϕ λ {\phi }_{\lambda } ( λ = 3 , 5 , 7 \lambda =3,5,7 ) α = 1 \alpha =1 . (d) Logarithmic plots of absolute errors in ψ λ {\psi }_{\lambda } ( λ = 3 , 5 , 7 \lambda =3,5,7 ) α = 1 \alpha =1 .
Figure 3

Comparison of computed results at λ th iteration ( λ = 2 , 3 , 4 ) with exact results for (a) ϕ and (b) ψ , and logarithmic plots of absolute errors in λ th iterative ( λ = 3 , 5 , 7 ) results for (c) ϕ , (d) ψ in example (3) with α = 1 . (a) Convergence plot of the computed results of ϕ for τ ∈ ( 0 , 0.5 ) . (b) Convergence plot of the computed results of ψ for τ ∈ ( 0 , 0.5 ) . (c) Logarithmic plots of absolute errors in ϕ λ ( λ = 3 , 5 , 7 ) α = 1 . (d) Logarithmic plots of absolute errors in ψ λ ( λ = 3 , 5 , 7 ) α = 1 .

Example 4

Consider time fractional generalized Burgers equation with proportional delay as in refs [31,32]:

(32) D C α τ ϕ ( σ , τ ) = ∂ 2 ∂ σ 2 [ ϕ ( σ , τ ) ] + ∂ ∂ σ ϕ σ , τ 2 × ϕ σ 2 , τ 2 + 1 2 ϕ ( σ , τ ) , ϕ ( σ , 0 ) = σ , σ , τ > 0 , 0 < α ≤ 1 .

In special case when α = 1 , the exact solution of (32) is ϕ ( σ , τ ) = σ exp τ .

On implementing the iteration formula of NTVIT as mentioned in (13) on Eq. (32), we obtain

(33) ϕ λ + 1 ( σ , τ ) = ϕ λ ( σ , 0 ) + K − 1 μ 2 α K ∂ 2 ∂ σ 2 [ ϕ λ ( σ , τ ) ] + ∂ ∂ σ ϕ λ σ , τ 2 ϕ σ 2 , τ 2 + 1 2 ϕ λ ( σ , τ ) , λ ≥ 0 ,

with ϕ λ ( σ , 0 ) = σ for λ = 0 .

On solving recurrence (33), we obtain

(34) ϕ 1 ( σ , τ ) = σ τ α Γ ( α + 1 ) + σ ϕ 2 ( σ , τ ) = 2 − α σ t 2 α Γ ( 2 α + 1 ) + σ τ 2 α 2 Γ ( 2 α + 1 ) + 2 − 2 α − 1 σ τ 3 α Γ ( 2 α + 1 ) Γ ( α + 1 ) 2 Γ ( 3 α + 1 ) + σ τ α Γ ( α + 1 ) + σ ϕ 3 ( σ , τ ) = σ + σ τ α Γ ( α + 1 ) + 1 Γ ( 2 α + 1 ) ( 2 − α σ t 2 α + σ τ 2 α 2 ) + σ τ 3 α Γ ( 3 α + 1 ) 2 − α − 1 + 2 − 3 α + 2 − 2 α − 1 + 1 4 + 2 − 2 α − 1 Γ ( 2 α + 1 ) Γ ( α + 1 ) 2 + σ t 4 α Γ ( 4 α + 1 ) 2 − 2 α − 2 Γ ( 2 α + 1 ) Γ ( α + 1 ) 2 + 2 − 5 α − 1 Γ ( 2 α + 1 ) Γ ( 4 α + 1 ) + 2 − 4 a Γ ( 3 α + 1 ) Γ ( α + 1 ) Γ ( 2 α + 1 ) + 2 − 3 α − 1 Γ ( 3 α + 1 ) Γ ( α + 1 ) Γ ( 2 α + 1 ) + σ t 5 α Γ ( 5 α + 1 ) 2 − 4 α − 3 Γ ( 4 α + 1 ) Γ ( 2 α + 1 ) 2 + 2 − 5 α − 1 Γ ( 4 α + 1 ) Γ ( 2 α + 1 ) 2 + 2 − 6 α − 1 Γ ( 4 α + 1 ) Γ ( 2 α + 1 ) 2 + 2 − 6 α − 1 Γ ( 2 α + 1 ) Γ ( 4 α + 1 ) Γ ( α + 1 ) 3 Γ ( 3 α + 1 ) + σ t 6 α Γ ( 6 α + 1 ) 2 − 7 α − 2 Γ ( 5 α + 1 ) Γ ( α + 1 ) 2 Γ ( 3 α + 1 ) + 2 − 8 α − 1 Γ ( 5 α + 1 ) Γ ( α + 1 ) 2 Γ ( 3 α + 1 ) Γ ( 6 α + 1 ) + 2 − 10 α σ Γ ( 2 α + 1 ) 2 Γ ( 6 α + 1 ) τ 7 α 8 Γ ( α + 1 ) 4 Γ ( 3 α + 1 ) 2 Γ ( 7 α + 1 ) .

In consequence, the λ th iterative solutions ϕ λ ( τ ) for λ ≥ 4 can be computed from (33). Absolute errors in λ th-order computed results ( λ = 4 , 6 ) are compared in Table 5 are compared with the results via recent techniques [31–34] for α = 1 . The relative errors in (5,6)th-order results are reported in Table 6, which shows errors in the approximate results decreases rapidly with increasing iterations. The findings concludes that present results are more accurate as compared to published results in refs [31–34]. Figure 4(a) and (d)–(f) shows that the approximate results converges to exact results very fast on increasing iterations. Figure 4(b) and (c) depicts 2D and surface behavior of sixth iteration results for different α = 0.8 , 0.9 , 1 .

Table 6

Relative errors in λ th-order computed results ( λ = 5 , 6 ) in Example 4 for different values α = 0.8 , 0.9 , 1 , τ = 0.5

τ RE( ϕ 5 ) RE( ϕ 6 )
α = 0.8 α = 0.9 α = 1 α = 0.8 α = 0.9 α = 1
2.5 4.771 × 1 0 − 5 9.284 × 1 0 − 6 1.762 × 1 0 − 6 2.484 × 1 0 − 6 3.148 × 1 0 − 7 3.892 × 1 0 − 8
0.5 6.262 × 1 0 − 4 1.723 × 1 0 − 4 4.640 × 1 0 − 5 5.685 × 1 0 − 5 1.089 × 1 0 − 5 2.044 × 1 0 − 6
0.75 2.624 × 1 0 − 3 8.820 × 1 0 − 4 2.904 × 1 0 − 4 3.299 × 1 0 − 4 8.019 × 1 0 − 5 1.914 × 1 0 − 5
1 6.893 × 1 0 − 3 2.666 × 1 0 − 3 1.010 × 1 0 − 3 1.093 × 1 0 − 3 3.137 × 1 0 − 4 8.856 × 1 0 − 5
Figure 4 (a) Comparison of computed ϕ λ ( 0.5 , τ ) {\phi }_{\lambda }\left(0.5,\tau ) ( λ = 2 , 3 , 4 ) \left(\lambda =2,3,4) with exact results; (b) 2D plots of ϕ 6 ( 0.5 , τ ) {\phi }_{6}\left(0.5,\tau ) for α = 0.8 , 0.9 , 1 \alpha =0.8,0.9,1 and exact result for α = 1 \alpha =1 in τ ∈ ( 0 , 1 ) \tau \in \left(0,1) , (c) 3D plots of sixth-order results ϕ 6 ( σ , τ ) {\phi }_{6}\left(\sigma ,\tau ) for α = 0.8 , 0.9 , 1 \alpha =0.8,0.9,1 in τ ∈ ( 0 , 1 ) \tau \in \left(0,1) ; the relative errors in λ \lambda th-order results ( λ = 4 , 5 , 6 ) \left(\lambda =4,5,6) for (d) α = 0.8 \alpha =0.8 and (e) α = 1 \alpha =1 and (f) absolute error plots for α = 1 \alpha =1 in Example 4. (a) Convergence plot of the NTVIT solution for τ ∈ ( 0 , 1 ) \tau \in \left(0,1) . (b) 2D plots of ϕ 6 ( 0.5 , τ ) {\phi }_{6}\left(0.5,\tau ) in τ ∈ ( 0 , 1 ) \tau \in \left(0,1) for α = 0.8 , 0.9 , 1 \alpha =0.8,0.9,1 . (c) 3D plots of ϕ 6 ( σ , τ ) {\phi }_{6}\left(\sigma ,\tau ) in σ , τ ∈ ( 0 , 1 ) \sigma ,\tau \in \left(0,1) for α = 0.8 , 0.9 , 1 \alpha =0.8,0.9,1 . (d) Relative error in ϕ λ ( 0.5 , τ ) {\phi }_{\lambda }\left(0.5,\tau ) for α = 0.8 \alpha =0.8 . (e) Relative error in ϕ λ ( 0.5 , τ ) {\phi }_{\lambda }\left(0.5,\tau ) for α = 1 \alpha =1 . (f) Absolute error in ϕ λ ( 0.5 , τ ) {\phi }_{\lambda }\left(0.5,\tau ) for α = 1 \alpha =1 .
Figure 4

(a) Comparison of computed ϕ λ ( 0.5 , τ ) ( λ = 2 , 3 , 4 ) with exact results; (b) 2D plots of ϕ 6 ( 0.5 , τ ) for α = 0.8 , 0.9 , 1 and exact result for α = 1 in τ ∈ ( 0 , 1 ) , (c) 3D plots of sixth-order results ϕ 6 ( σ , τ ) for α = 0.8 , 0.9 , 1 in τ ∈ ( 0 , 1 ) ; the relative errors in λ th-order results ( λ = 4 , 5 , 6 ) for (d) α = 0.8 and (e) α = 1 and (f) absolute error plots for α = 1 in Example 4. (a) Convergence plot of the NTVIT solution for τ ∈ ( 0 , 1 ) . (b) 2D plots of ϕ 6 ( 0.5 , τ ) in τ ∈ ( 0 , 1 ) for α = 0.8 , 0.9 , 1 . (c) 3D plots of ϕ 6 ( σ , τ ) in σ , τ ∈ ( 0 , 1 ) for α = 0.8 , 0.9 , 1 . (d) Relative error in ϕ λ ( 0.5 , τ ) for α = 0.8 . (e) Relative error in ϕ λ ( 0.5 , τ ) for α = 1 . (f) Absolute error in ϕ λ ( 0.5 , τ ) for α = 1 .

Example 5

Consider time fractional nonlinear pantograph equation, as in ref. [37], of the following form

(35) D C α τ ϕ ( σ , τ ) = ∂ 2 ∂ σ 2 ϕ ( σ , τ ) + ϕ σ , τ 2 ∂ 2 ∂ σ 2 ϕ σ 2 , τ 2 + 2 ϕ σ 2 , τ , 1 < α ≤ 2 , ϕ ( σ , 0 ) = σ , ϕ τ ( σ , 0 ) = 0 , τ ≥ 0 , σ ∈ [ 0 , 1 ] ,

where ϕ τ ( σ , 0 ) is the derivative of ϕ ( σ , τ ) at initial time τ = 0 .

In special case when α = 2 , ϕ ( σ , τ ) = σ cosh τ is the exact solution of nonlinear pantograph Eq. (35).

Imposing NTVIT iteration formula (12) for Eq. (35), we obtain

(36) ϕ λ + 1 ( σ , τ ) = ϕ λ 0 ( σ , τ ) + K − 1 μ 2 α K ∂ 2 ∂ σ 2 [ ϕ λ ( σ , τ ) ] + ϕ λ σ , τ 2 ∂ 2 ∂ σ 2 ϕ λ σ 2 , τ 2 + 2 ϕ λ σ 2 , τ

with ϕ λ 0 ( σ , τ ) = ϕ λ ( σ , 0 ) + τ ∂ ϕ λ ∂ τ ( σ , 0 ) = σ for λ = 0 .

On solving recurrence (36), we obtain

(37) ϕ 1 ( σ , τ ) = σ τ α Γ ( α + 1 ) + σ , ϕ 2 ( σ , τ ) = σ τ 2 α Γ ( 2 α + 1 ) + σ τ α Γ ( α + 1 ) + σ , ϕ 3 ( σ , τ ) = σ τ 2 α Γ ( 2 α + 1 ) + σ τ 3 α Γ ( 3 α + 1 ) + σ τ α Γ ( α + 1 ) + σ , ϕ 4 ( σ , τ ) = σ τ 4 α Γ ( 4 α + 1 ) + σ τ 3 α Γ ( 3 α + 1 ) + σ τ 2 α Γ ( 2 α + 1 ) + σ τ α Γ ( α + 1 ) + σ , …

In consequence, the λ th iterative solutions ϕ λ ( τ ) for λ ≥ 5 can be computed from (36). Relative errors in λ th-order ( λ = 4 , 6 ) computed results for α = 1.8 , 1.9 and absolute errors for α = 2 is reported in Table 7. Computed results in third iteration are compared with OHAM [37] results for α = 1.8 , 2 . Figure 5(a) shows that λ th-order results for λ ≥ 3 approaches very fast toward exact results and conclusion is from the error plots 5 ( c − e ) of λ th iterative results ( λ = 4 , 5 , 6 ) for α = 1.8 , 2 . Figure 5(b) depicts 2D behavior of sixth iterative results for different α = 1.8 , 1.9 , 2 and exact result for α = 2 . The findings conclude that present results are more accurate as compared to [37], and errors in computed results decrease rapidly with increasing iterations (Table 8).

Table 7

Relative errors in λ th-order results ( λ = 4 , 6 ) with different α in Example 5 at different τ ≤ 1 , σ = 0.5

τ RE( ϕ 5 ) RE( ϕ 6 ) Absolute error
α = 1.8 α = 1.9 α = 2 α = 1.8 α = 1.9 α = 2 ϕ 5 ( σ , τ ) ϕ 6 ( σ , τ )
0.25 5.827 × 1 0 − 9 1.619 × 1 0 − 12 2.548 × 1 0 − 13 1.001 × 1 0 − 11 1.282 × 1 0 − 15 2.153 × 1 0 − 16 1.110 × 1 0 − 16 0.000 × 1 0 00
0.50 7.637 × 1 0 − 7 1.059 × 1 0 − 9 2.387 × 1 0 − 10 4.571 × 1 0 − 9 3.012 × 1 0 − 12 4.519 × 1 0 − 13 2.551 × 1 0 − 13 3.331 × 1 0 − 16
0.75 1.205 × 1 0 − 5 4.295 × 1 0 − 8 1.199 × 1 0 − 8 1.496 × 1 0 − 7 2.639 × 1 0 − 10 5.108 × 1 0 − 11 3.317 × 1 0 − 11 1.025 × 1 0 − 13
1.00 7.892 × 1 0 − 5 5.495 × 1 0 − 7 1.786 × 1 0 − 7 1.645 × 1 0 − 6 5.832 × 1 0 − 9 1.353 × 1 0 − 9 1.050 × 1 0 − 9 5.759 × 1 0 − 12
Figure 5 (a) Comparison of λ \lambda th iterative ( λ = 2 , 3 , 4 ) \left(\lambda =2,3,4) results with exact results; (b) 2D plots ϕ 6 ( 0.5 , τ ) {\phi }_{6}\left(0.5,\tau ) for α = 1.8 , 1.9 , 2 \alpha =1.8,1.9,2 and exact results for α = 2 \alpha =2 in τ ∈ ( 0 , 1 ) \tau \in \left(0,1) ; relative errors in λ \lambda th-order results ( λ = 4 , 5 , 6 ) \left(\lambda =4,5,6) for (c) α = 1.8 \alpha =1.8 , (e) α = 2 \alpha =2 ; (f) absolute error for α = 2 \alpha =2 in Example 5. (a) Convergence plot of the NTVIT results for τ ∈ ( 0 , 1 ) \tau \in \left(0,1) . (b) 2D behavior for α = 1.8 , 1.9 , 2 \alpha =1.8,1.9,2 and exact results for α = 2 \alpha =2 in τ ∈ ( 0 , 1 ) \tau \in \left(0,1) . (c) 2D plots of relative error for α = 2.8 \alpha =2.8 . (d) 2D plots of relative error plot for α = 2 \alpha =2 . (e) 2D plots of absolute error for α = 2 \alpha =2 .
Figure 5

(a) Comparison of λ th iterative ( λ = 2 , 3 , 4 ) results with exact results; (b) 2D plots ϕ 6 ( 0.5 , τ ) for α = 1.8 , 1.9 , 2 and exact results for α = 2 in τ ∈ ( 0 , 1 ) ; relative errors in λ th-order results ( λ = 4 , 5 , 6 ) for (c) α = 1.8 , (e) α = 2 ; (f) absolute error for α = 2 in Example 5. (a) Convergence plot of the NTVIT results for τ ∈ ( 0 , 1 ) . (b) 2D behavior for α = 1.8 , 1.9 , 2 and exact results for α = 2 in τ ∈ ( 0 , 1 ) . (c) 2D plots of relative error for α = 2.8 . (d) 2D plots of relative error plot for α = 2 . (e) 2D plots of absolute error for α = 2 .

Table 8

Comparison of third-order results with OHAM results for α = 1.8 , 2 , absolute error at different ( σ , τ ) in Example 5

σ τ α = 1.8 α = 2 Absolute error ( α = 2 )
OHAM [37] NTVIT OHAM [37] NTVIT Exact OHAM [37] NTVIT
0.1 0.5 1.199 × 1 0 − 1 1.178 × 1 0 − 1 1.145 × 1 0 − 1 1.128 × 1 0 − 1 1.128 × 1 0 − 1 3.044 × 1 0 − 5 9.715 × 1 0 − 9
0.2 0.4 2.267 × 1 0 − 1 2.235 × 1 0 − 1 2.186 × 1 0 − 1 2.162 × 1 0 − 1 2.162 × 1 0 − 1 4.600 × 1 0 − 5 3.257 × 1 0 − 9
0.3 0.3 3.238 × 1 0 − 1 3.208 × 1 0 − 1 3.157 × 1 0 − 1 3.136 × 1 0 − 1 3.136 × 1 0 − 1 4.386 × 1 0 − 5 4.887 × 1 0 − 10
0.4 0.2 4.153 × 1 0 − 1 5.166 × 1 0 − 1 4.093 × 1 0 − 1 4.080 × 1 0 − 1 4.080 × 1 0 − 1 2.826 × 1 0 − 5 2.541 × 1 0 − 11
0.5 0.1 5.055 × 1 0 − 1 5.888 × 1 0 − 1 5.029 × 1 0 − 1 5.025 × 1 0 − 1 5.025 × 1 0 − 1 9.272 × 1 0 − 7 1.240 × 1 0 − 13

Example 6

Consider the coupled system of PDTF-NPDEs, as in OHAM [37], of the form

(38) D C α τ ϕ ( σ , τ ) + ψ ( σ , τ ) − ∂ 2 ∂ σ 2 ϕ ( σ , τ ) + ϕ ( σ , τ ) ∂ ∂ σ ψ σ 2 , τ 2 = 1 2 ( τ + 3 σ − 2 ) D C α τ ψ ( σ , τ ) − ϕ σ 2 , τ 2 + ∂ 2 ∂ σ 2 ψ σ 3 , τ + ∂ ∂ σ ψ σ 2 , τ 2 = 1 2 ( τ − σ + 3 ) ϕ ( σ , 0 ) = σ , ψ ( σ , 0 ) = σ , τ ≥ 0 , σ ∈ [ 0 , 1 ] , 0 < α ≤ 1 .

In special case when α = 1 , the exact solution of this system is ϕ ( σ , τ ) = σ − τ , ψ ( σ , τ ) = σ + τ .

The following iteration formula for Eq. (38) is obtained on imposing NTVIT iteration formula (12):

(39) ϕ λ + 1 ( σ , τ ) = ϕ λ ( σ , 0 ) + K − 1 { μ 2 α K { − ψ λ ( σ , τ ) + ∂ 2 ∂ σ 2 [ ϕ λ ( σ , τ ) ] − ϕ λ ( σ , τ ) ∂ ∂ σ × ψ λ σ 2 , τ 2 + 1 2 ( τ + 3 σ − 2 )

ψ λ + 1 ( σ , τ ) = ψ λ ( σ , 0 ) + K − 1 μ 2 α K ϕ λ σ 2 , τ 2 − ∂ 2 ∂ σ 2 [ ψ λ ( σ 3 , τ ) ] − ∂ ∂ σ ψ λ σ 2 , τ 2 + 1 2 ( τ − σ + 3 ) .

On solving the above recurrence (39), we obtain

At first iteration:

(40) ϕ 1 ( σ , τ ) = τ α + 1 2 Γ ( α + 2 ) − τ α Γ ( α + 1 ) + σ , ψ 1 ( σ , τ ) = τ α + 1 2 Γ ( α + 2 ) + τ α Γ ( α + 1 ) + σ .

At second iteration:

(41) ϕ 2 ( σ , τ ) = − τ 2 α 2 Γ ( 2 α + 1 ) + τ α + 1 2 Γ ( α + 2 ) − 3 τ 2 α + 1 4 Γ ( 2 α + 2 ) − τ α Γ ( α + 1 ) + σ . ψ 2 ( σ , τ ) = − 2 − α τ 2 α Γ ( 2 α + 1 ) + τ α + 1 2 Γ ( α + 2 ) + 2 − α − 2 τ 2 α + 1 Γ ( 2 α + 2 ) + τ α Γ ( α + 1 ) + σ .

In sequel, the approximate solution at λ th iteration for λ ≥ 3 computed in the same fashion from (39). Further relative errors in λ th-order ( λ = 5 , 6 ) computed results for α = 0.8 , 0.9 , 1 and absolute errors for α = 1 for ϕ and ψ is reported in Tables 9 and 10, respectively. Also comparison of third-order iterative solution with exact results and absolute error ( λ = 3 ) with OHAM [37] is reported in Table 11, which shows that proposed results are adjacent to exact results and highly improved upon OHAM results. Figures 6 and 7(a) shows that computed results converges very fast toward exact results for small number of iterations, and the same observations are shown in Figures 6 and 7(c)–(e). Surface behavior of sixth iterative results of ϕ and ψ for α = 0.8 , 0.9 , 1 is depicted in Figures 6 and 7(b).

Table 9

Relative errors in λ th-order results ( λ = 5 , 6 ) for different values of α and absolute error in sixth-order NTVIT solution of ϕ ( σ , τ ) in Example 6 at different τ ≤ 1 and σ = 0.2

τ ∕ σ = 0.2 RE ϕ 5 RE ϕ 6 Abs error ϕ 6
α = 0.8 α = 0.9 α = 1 α = 0.8 α = 0.9 α = 1
0.25 3.3734 × 1 0 − 5 1.1601 × 1 0 − 6 5.0863 × 1 0 − 6 4.0279 × 1 0 − 6 4.3105 × 1 0 − 7 2.3653 × 1 0 − 10 1.1826 × 1 0 − 11
0.5 2.1360 × 1 0 − 4 6.1092 × 1 0 − 6 2.7127 × 1 0 − 5 4.3297 × 1 0 − 5 5.1930 × 1 0 − 6 5.0459 × 1 0 − 9 1.5138 × 1 0 − 9
0.75 7.3112 × 1 0 − 4 1.7920 × 1 0 − 5 1.1236 × 1 0 − 4 2.0014 × 1 0 − 4 2.8130 × 1 0 − 5 4.7026 × 1 0 − 8 2.5864 × 1 0 − 8
1 1.8273 × 1 0 − 3 3.4827 × 1 0 − 5 3.2552 × 1 0 − 4 6.1564 × 1 0 − 4 9.7650 × 1 0 − 5 2.4220 × 1 0 − 7 1.9376 × 1 0 − 7
Table 10

Relative errors in λ th-order results ( λ = 5 , 6 ) for different values of α and absolute error in sixth-order NTVIT solution of ψ ( σ , τ ) in Example 6 at different τ ≤ 1 and σ = 0.2

τ ∕ σ = 0.2 RE ψ 5 RE ψ 6 Abs error ψ 6
α = 0.8 α = 0.9 α = 1 α = 0.8 α = 0.9 α = 1
0.25 8.1897 × 1 0 − 6 1.5625 × 1 0 − 6 2.8331 × 1 0 − 7 5.8321 × 1 0 − 8 7.3238 × 1 0 − 10 7.3586 × 1 0 − 10 0.0000 × 1 0 00
0.5 9.1237 × 1 0 − 5 2.3591 × 1 0 − 5 5.8431 × 1 0 − 6 1.1512 × 1 0 − 6 1.9214 × 1 0 − 8 3.0275 × 1 0 − 8 0.0000 × 1 0 00
0.75 3.6443 × 1 0 − 4 1.1158 × 1 0 − 4 3.2779 × 1 0 − 5 6.4711 × 1 0 − 6 1.2124 × 1 0 − 7 2.5410 × 1 0 − 7 0.0000 × 1 0 00
1 9.6246 × 1 0 − 4 3.3175 × 1 0 − 4 1.0964 × 1 0 − 4 2.1882 × 1 0 − 5 4.2992 × 1 0 − 7 1.1303 × 1 0 − 6 0.0000 × 1 0 00
Table 11

Comparison of third-order results with exact and absolute error with OHAM in Example 6 at τ = 0.01 and different values of σ

Absolute error Absolute error
σ ∕ τ = 0.01 ϕ 3 Exact ϕ 3 OHAM ψ 3 Exact ψ 3 OHAM
0.25 2.4000 × 1 0 − 1 2.4000 × 1 0 − 1 1.0417 × 1 0 − 10 2.9262 × 1 0 − 3 2.6000 × 1 0 − 1 2.6000 × 1 0 − 1 3.9062 × 1 0 − 11 2.6819 × 1 0 − 3
0.5 4.9000 × 1 0 − 1 4.9000 × 1 0 − 1 1.0417 × 1 0 − 10 2.9251 × 1 0 − 3 5.1000 × 1 0 − 1 5.1000 × 1 0 − 1 3.9062 × 1 0 − 11 2.6784 × 1 0 − 3
0.75 7.4000 × 1 0 − 1 7.4000 × 1 0 − 1 1.0417 × 1 0 − 10 2.9240 × 1 0 − 3 7.6000 × 1 0 − 1 7.6000 × 1 0 − 1 3.9062 × 1 0 − 11 2.6750 × 1 0 − 3
1 9.9000 × 1 0 − 1 9.9000 × 1 0 − 1 1.0417 × 1 0 − 10 2.9229 × 1 0 − 3 1.0100 × 1 0 00 1.0100 × 1 0 00 3.9063 × 1 0 − 11 2.6716 × 1 0 − 3
Figure 6 (a) Comparison of computed ϕ λ ( λ = 2 , 3 , 4 ) {\phi }_{\lambda }\hspace{0.33em}\left(\lambda =2,3,4) with exact results; (b) 3D behavior of ϕ 5 ( σ , τ ) {\phi }_{5}\left(\sigma ,\tau ) for α = 0.8 , 0.9 , 1 \alpha =0.8,0.9,1 and exact result for α = 1 \alpha =1 in σ , τ ∈ ( 0 , 1 ) \sigma ,\tau \in \left(0,1) ; relative errors in λ \lambda th-order results ( κ = 4 , 5 , 6 ) \left(\kappa =4,5,6) for (c) α = 0.8 \alpha =0.8 , (d) α = 1 \alpha =1 ; (e) absolute error for α = 1 \alpha =1 in Example 6. (a) Convergence plot of the NTVIT solution ϕ \phi for τ ∈ ( 0 , 1 ) \tau \in \left(0,1) . (b) 3D behavior of ϕ 5 ( σ , τ ) {\phi }_{5}\left(\sigma ,\tau ) in σ , τ ∈ ( 0 , 1 ) \sigma ,\tau \in \left(0,1) . (c) 2D plots of relative error for α = 0.8 \alpha =0.8 . (d) 2D plots of relative error for α = 1 \alpha =1 . (e) 2D plots of absolute error for α = 1 \alpha =1 .
Figure 6

(a) Comparison of computed ϕ λ ( λ = 2 , 3 , 4 ) with exact results; (b) 3D behavior of ϕ 5 ( σ , τ ) for α = 0.8 , 0.9 , 1 and exact result for α = 1 in σ , τ ∈ ( 0 , 1 ) ; relative errors in λ th-order results ( κ = 4 , 5 , 6 ) for (c) α = 0.8 , (d) α = 1 ; (e) absolute error for α = 1 in Example 6. (a) Convergence plot of the NTVIT solution ϕ for τ ∈ ( 0 , 1 ) . (b) 3D behavior of ϕ 5 ( σ , τ ) in σ , τ ∈ ( 0 , 1 ) . (c) 2D plots of relative error for α = 0.8 . (d) 2D plots of relative error for α = 1 . (e) 2D plots of absolute error for α = 1 .

Figure 7 (a) Comparison of computed ψ λ {\psi }_{\lambda } ( λ = 2 , 3 , 4 ) \left(\lambda =2,3,4) with exact results; (b) 3D behavior of ψ 5 ( σ , τ ) {\psi }_{5}\left(\sigma ,\tau ) for α = 0.8 , 0.9 , 1 \alpha =0.8,0.9,1 and exact result for α = 1 \alpha =1 in σ , τ ∈ ( 0 , 1 ) \sigma ,\tau \in \left(0,1) ; relative errors in ψ λ {\psi }_{\lambda } : λ \lambda th-order results ( λ = 4 , 5 , 6 ) \left(\lambda =4,5,6) for (c) α = 0.8 \alpha =0.8 , (d) α = 1 \alpha =1 ; (e) absolute error for α = 1 \alpha =1 in Example 6. (a) Convergence plot of the NTVIT solution of ψ \psi for τ ∈ ( 0 , 1 ) \tau \in \left(0,1) . (b) 3D behavior of ψ 5 ( σ , τ ) {\psi }_{5}\left(\sigma ,\tau ) in σ , τ ∈ ( 0 , 1 ) \sigma ,\tau \in \left(0,1) . (c) 2D plots of relative error for α = 0.8 \alpha =0.8 . (d) 2D plots of relative error for α = 1 \alpha =1 . (e) 2D plots of absolute error for α = 1 \alpha =1 .
Figure 7

(a) Comparison of computed ψ λ ( λ = 2 , 3 , 4 ) with exact results; (b) 3D behavior of ψ 5 ( σ , τ ) for α = 0.8 , 0.9 , 1 and exact result for α = 1 in σ , τ ∈ ( 0 , 1 ) ; relative errors in ψ λ : λ th-order results ( λ = 4 , 5 , 6 ) for (c) α = 0.8 , (d) α = 1 ; (e) absolute error for α = 1 in Example 6. (a) Convergence plot of the NTVIT solution of ψ for τ ∈ ( 0 , 1 ) . (b) 3D behavior of ψ 5 ( σ , τ ) in σ , τ ∈ ( 0 , 1 ) . (c) 2D plots of relative error for α = 0.8 . (d) 2D plots of relative error for α = 1 . (e) 2D plots of absolute error for α = 1 .

6 Conclusion

The present article develops a novel technique: NTVIT for the numerical study of time-fractional delayed differential equations of higher order. Using Banach approach, some essential conditions are obtained for the stability and convergence of the proposed technique.

The effectiveness and validity of the NTVIT are tested via numerical study of some test examples of time fractional delayed differential equations of higher fractional order by computing the relative/absolute error norm, which demonstrate that NTVIT performs better than some recently developed techniques and enables to produces more accurate solutions.

Acknowledgments

The authors are grateful to the editor and anonymous reviewers for their comments and suggestions. Saloni Agrawal thanks the University Grant Commission, New Delhi, India, while Mukesh Gupta thanks the CSIR, Delhi, India, for financial assistant to carry out their research.

  1. Funding information: Not applicable.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors declare that they have no competing interests.

  4. Data availability statement: Not applicable.

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Received: 2022-08-11
Revised: 2022-09-25
Accepted: 2022-12-06
Published Online: 2023-05-05

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2022-0267
Keywords for this article
Caputo derivative; nonlinear time-fractional delayed partial differential equations; proportional delay; new integral transform
Creative Commons
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