Two computational approaches for solving a fractional obstacle system in Hilbert space
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Abstract
The primary motivation of this paper is to extend the application of the reproducingkernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, \(0<\alpha\leq1\), concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and noninteger (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducingkernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a wellposed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.
Keywords
Reproducingkernel method Residual power series method Inner product spaces Obstacle problems Caputofractional derivative1 Introduction
Many obstacle problems of integer order have been studied and solved using several numerical approaches such as variation of parameters, collocation, finite difference, spline and residual power series techniques. For instance, the collocation technique was applied in [4] for solving the obstacle BVPs (1.1) and (1.2) utilizing Bcubic splines basis functions that yields approximate solution of firstorder accuracy. Later, similar results were drawn to obstacle BVPs (1.1) and (1.2) utilizing finite difference and spline methods [4, 7]. While the Numerov technique was modified and employed in solving the obstacle system by AlSaid and Noor [9]. Further, the quadratic and cubic splines techniques were implemented and analyzed by AlSaid [8] to produce a smooth approximate solution of the obstacle BVPs (1.1) and (1.2) over the domain [\(a,b\)]. Also, quarticspline functions were presented to get some consistent relations that utilized in developing a numeric approach in finding smooth approximate solutions for the proposed obstacle system [10].
However, in the last few decades, fractional calculus has attracted the attention of many researchers for its considerable importance in many applications in fluid dynamics, viscoelasticity, physics, entropy theory and engineering. Therefore, many boundary differential equations and initial differential equations of integer order were generalized to fractional order and several powerful methods were modified to approximate their solutions. The Adomian decomposition method [12], the variational iteration method [13], the differential transformation method [14], the finite difference method [15], the homotopy analysis method [16], and the homotopy perturbation method [17] are some of these methods. Also, MOLGPS and theta methods have been applied for solving Burgers equation [18], and fractional telegraph differential equation [19], respectively. For using Riesz Riemann–Liouville, Riesz–Caputo, and other fractional concepts, we refer to [20, 21].
The reproducingkernel approach has been developed as an efficacious numericanalytic method in treating different type of ordinary and partial differential, integral, integrodifferential equations with singularity, fuzziness, nonlocal, and nonclassical constraint conditions [22, 23, 24, 25, 26, 27]. Recently, the RKM has been improved and successfully applied in obtaining approximations of solutions for many initial and boundary problems that appear in natural sciences and engineering. The RKM was successfully used for solving the Thomas–Fermi equation [28], the Poisson–Boltzmann equation for semiconductor devices [29], variableorder fractional differential equations [30] and secondorder partial differential equations [31] and others [32, 33, 34]. Moreover, Cui and Lin [24] have efficiently solved obstacle thirdorder BVP using RKM. It should be noted here that the obstacle system of fractional order has not been solved using RKM before. The RPSM is an analytical as well as an approximate method for handling several kinds of FDEs. This method has been applied effectively to construct a fractional power series solution for numerous linear and nonlinear equations without linearization, perturbation, or discretization [35]. The FRPS method is basically utilized for the residual functions and the generalized Taylor expansion by selecting a proper initial guess approximation to introduce a suitable analytical solution.
The present analysis extends the application of the RKM and RPSM for finding approximate solutions of fractional obstacle system in the Caputo sense. The structure of the present article is organized as follows. In Sect. 2, some necessary definitions and mathematical preliminaries are introduced, including several reproducingkernel spaces required in establishing the results of our analysis and the generalized Taylor’s expansion. In Sect. 3, short description of the RK and RPS techniques for solving fractional obstacle system (1.4) and (1.5) is given. In Sect. 4, numerical application is presented to show the capability and validity of the kernel method. This analysis ends with Sect. 5 with some conclusions.
2 Basic concepts and fundamentals
In this section, we have given some basic definitions and theorems regarding the reproducingkernel spaces and the generalized power series representations. For more details about these definitions and properties, one can refer to [22, 23, 24, 25, 26, 27]. Throughout the current paper, \(L^{2} [ a,b ]\) stands for the set of all square integrable functions on \([ a,b ]\) while \(AC [ a,b ]\) stands for the set of all absolutely continuous functions on \([ a,b ]\) such that \(AC [ a,b ] = \{ u: [ a,b ]\rightarrow\mathbb{ R}: u\mbox{ is absolutely continuous on }[ a,b ] \}\).
Definition 2.1
 (a)
For each τ in Λ, we have \(\psi ( {\cdot},\tau ) \in H\),
 (b)
For each τ in Λ and any function ϕ in H, we have \(\langle \phi ( {\cdot} ),\psi ( {\cdot},\tau ) \rangle_{H} =\phi ( \tau )\).
The last condition indicates that function’s value ϕ at any τ in Λis reproduced through the inner product for ϕ and \(\psi ( {\cdot},\tau )\), where the function ψ is called the reproducingkernel function of H that possesses some important properties such as being unique, conjugate symmetric and positivedefinite.
Consequently, two RKHSs are introduced as follows.
Definition 2.2
The inner product for \(\varphi,\vartheta\in \mathcal{W}_{1} [ a,b ]\) is given by \(\langle \varphi,\vartheta \rangle_{\mathcal{W}_{1}} = \int_{a}^{b} ( \varphi ' (\xi) \vartheta ' (\xi)+\varphi(\xi)\vartheta(\xi) ) \,d\xi\), and the norm of φ is \(\Vert \varphi \Vert _{\mathcal{W}_{1}} = \sqrt{ \langle \varphi ( \xi ), \varphi (\xi) \rangle_{\mathcal{W}_{1}}}\).
Theorem 2.1
Definition 2.3
The unique representation to reproducingkernel function \(\mathcal{Q}_{x} ( {v} )\) within the space \(\mathcal{W}_{2} [ a,b ]\) can be formulated as in the following theorem.
Theorem 2.2
Proof
To construct \(\mathcal{Q}_{x} ( {v} )\), let \(u ( x ) \in \mathcal{W}_{2} [ a,b ]\). Then, \(\langle u ( {v} ), \mathcal{Q}_{x} ( {v} ) \rangle_{\mathcal{W}_{2}} = u ' ( a ) \mathcal{Q}_{x} ' ( a ) + \int_{a}^{b} u^{\prime\prime\prime} ( \xi ) \mathcal{Q}_{x}^{\prime\prime\prime} ( \xi ) \,d\xi\). By doing integrations by parts for \(\int_{a}^{b} u^{\prime\prime\prime} ( \xi ) \mathcal{Q}_{x}^{\prime\prime\prime} (\xi)\,d\xi\), we obtain \(\langle u ( {v} ), \mathcal{Q}_{x} ( {v} ) \rangle_{\mathcal{W}_{2}} = u ' ( a ) [ \mathcal{Q}_{x} ' ( a ) + \mathcal{Q}_{x}^{(4)} ( a ) ]  u^{\prime\prime} ( a ) \mathcal{Q}_{x}^{ ( 3 )} (a)+ u^{\prime\prime} ( b ) \mathcal{Q}_{x}^{ ( 3 )} (b) u ' ( b ) \mathcal{Q}_{x}^{ ( 4 )} (b) \int_{a}^{b} u ( \xi ) \mathcal{Q}_{x}^{(6)} (\xi)\,d\xi\).
Definition 2.4
([35])
Theorem 2.3
([35])
If\(u(x)\in C[ x_{0}, x_{0} +R)\), and\(D^{i \alpha} u ( x ) \in C( x_{0}, x_{0} +R)\), for\(i=0,1,2,\dots\), then coefficients\(a_{i}\)will be in the form\(a_{i} = \frac{\mathcal{D}_{0}^{i \alpha} u ( x_{0} )}{\varGamma ( n \alpha +1 )}\), whereRis radius of convergence, and\(\mathcal{D}^{i \alpha} = \mathcal{D}^{\alpha} {\cdot} \mathcal{D}^{\alpha} \cdots \mathcal{D}^{\alpha}\) (itimes).
3 The application of fractional RKM and RPSM
In this section, the iterative reproducingkernel method will be executed to handle the fractional obstacle system in the complete Hilbert space \(\mathcal{W}_{2} [ a,b ]\) and the procedure of the RPSM is presented. Meanwhile, description of the modified RKM, solution formula and error analysis are introducing in the same Hilbert space.
Theorem 3.1
Let\(\{ x_{i} \}_{i=1}^{\infty}\)be a dense subset of\([ a, b ]\), then the system\(\{ \overline{\psi}_{i} ( x ) \}_{i=1}^{\infty}\)will be a complete normal basis to\(\mathcal{W}_{2} [ a, b ]\)with\(\psi_{i} ( x ) = \mathcal{P}_{{v}} \mathcal{Q}_{x} ( {v} ) _{{v}= x_{i}}\), where the subscriptvby the operator\(\mathcal{P}\)indicate that\(\mathcal{P}\)employ directly to the function ofv.
Proof
For fixed \(u ( x )\) in \(\mathcal{W}_{2} [ a, b ]\), it follows that \(\langle u(x), \psi_{i} ( x ) \rangle =0,i\in \mathbb{N}\), which mean that \(\langle u(x), \mathcal{P}^{*} {w}_{i} ( x ) \rangle_{\mathcal{W}_{2}} = \langle \mathcal{P} u(x), {w}_{i} ( x ) \rangle_{\mathcal{W}_{1}} = ( \mathcal{P} u ) ( x_{i} ) =0\). But\(\{ x_{i} \}_{i=1}^{\infty}\) is dense on the compact interval \([ a, b ]\) that leads to \(( \mathcal{P} u ) ( x ) =0\). Since \(\mathcal{P} \) is invertible operator, \(u ( x ) \equiv0\). Further and in terms of the reproducing property, it follows that \(\psi_{i} ( x ) = ( \mathcal{P}^{*} {w}_{i} ) ( x ) = \langle \mathcal{P}^{*} {w}_{i} ( {v} ), \mathcal{Q}_{x} ( {v} ) \rangle_{\mathcal{W}_{2}} = \langle {w}_{i} ( {v} ), \mathcal{P}_{{v}} \mathcal{Q}_{x} ( {v} ) \rangle_{\mathcal{W}_{2}} = \mathcal{P}_{{v}} \mathcal{Q}_{x} ( {v)} _{{v} = x_{i}}\). □
Theorem 3.2
Proof
Theorem 3.3
Let\({r}_{n} (x)\)be the actual error between the analytical, \(u ( x )\), and the approximate, \(u_{n} (x)\)solutions. Then, \({r}_{n} (x)\)be a monotonic decreasing function within\(\Vert {\cdot} \Vert _{\mathcal{W}_{2}}\)and\(\Vert {r}_{n} (x) \Vert _{\mathcal{W}_{2}}\)approaches 0.
Proof
From Theorem 3.2, the proof of monotone decreasing of \({r}_{n} (x)\) is straightforward as follows: \(\Vert {r}_{n} (x) \Vert _{\mathcal{W}_{2}}^{2} = \Vert u ( x )  u_{n} (x) \Vert _{\mathcal{W}_{2}}^{2} = \Vert \sum_{i=n+1}^{\infty} \sum_{k=1}^{i} \sigma_{ik} \mathcal{ S}( x_{k}, u( x_{k} )) \overline{\psi_{i}} ( x ) \Vert _{\mathcal{W}_{2}}^{2} = \sum_{i=n+1}^{\infty} ( \sum_{k=1}^{i} \sigma_{ik} \mathcal{ S}( x_{k}, u( x_{k} )) )^{2} = \sum_{i=n+1}^{\infty} \mathcal{E}_{i}^{2}, \mathcal{E}_{i} = \sum_{k=1}^{i} \sigma_{ik} \mathcal{ S}( x_{k}, u( x_{k} ))\).
Consequently, \(\{ {r}_{n} (x) \}\) is monotone decreasing function in the space \(\mathcal{W}_{2} [ a, b ]\). Also, since the series \(\sum_{i=1}^{\infty} \mathcal{E}_{i} \overline{\psi}_{i} ( x )\) is convergent, then \(\Vert {r}_{n} (x) \Vert _{\mathcal{W}_{2}} \rightarrow0\). □
Corollary 3.1
For\(u ( x ) \in \mathcal{W}_{2} [ a, b ]\), the approximation\(u_{n}^{ ( i )} ( x ),i=0,1,2\), are uniformly converging to analytic solution\(u^{ ( i )} ( x ),i=0,1,2\), as soon as\(n\rightarrow\infty\).
Proof
Hence, when as \(\lim_{n\rightarrow\infty} \Vert u_{n} ( x ) u ( x ) \Vert _{\mathcal{W}_{2}} =0\), the approximation \(u_{n}^{ ( i )} ( x ), n=0,1,2\), will be converging to analytical solutions \(u ( x )\) and its derivative, respectively, uniformly.

Case one, the RPS solution, \(u_{1} ( x )\), on \([ a,c ]\) can be presented as follows:
Consequently, we need to minimize \(\operatorname{Res}_{u_{1}}^{k} ( x )\) and utilize the relation \(D_{a}^{(k2)\alpha} \operatorname{Res}_{u_{1}}^{k} ( x ) _{x=a} =0, k=2,3,\dots\), to determine the unknown coefficients \(c_{n},n=2,3, \dots,k\), of Eq. (3.7). At this point, we note that the value of \(c_{1} =A\) will be determined later by using the continuity conditions of Eq. (1.4).

Case two, the RPS solution, \(u_{2} ( x )\), on \([ c,d ]\) can be presented as follows:

Case three, the RPS solution, \(u_{3} ( x )\), on \([ d,b ]\) can be presented as follows:
Hence, the kth RPSapproximate solution is completely constructed for the BVPs (1.4) and (1.5). □
4 Numerical outcomes
To test simplicity, applicability and accuracy of the proposed RK and RPS algorithms, the numerical experiment is presented in this section. The methodology is directly employed without using discretization, transformation, and restrictive assumptions. The appropriateness and effectiveness of the proposed methods are evident when we compare it with each other for different values of α. Numerical comparison between the RKM, RPSM and other wellknown methods are also presented for \(\alpha=1\). The motivation of the current section is to obtain the RKsolution and RPSsolution for the obstacle BVP (1.4) and (1.5).
Numerical results of BVP (4.3) at \(\alpha=1\) using RKM
\(x_{i}\)  Exact solution  Approximation  \( u ( x )  u_{10} ( x ) \) 

π/10  0.135983977  0.13533462  6.494 × 10^{−4} 
π/5  0.271967954  0.27065256  1.315 × 10^{−3} 
3π/10  0.400072208  0.39882699  1.245 × 10^{−3} 
2π/5  0.476916995  0.47644324  4.738 × 10^{−4} 
π/2  0.501709552  0.50192359  2.140 × 10^{−4} 
3π/5  0.476916995  0.47780544  8.884 × 10^{−4} 
7π/10  0.400072208  0.40168706  1.615 × 10^{−3} 
4π/5  0.271967954  0.27361739  1.649 × 10^{−3} 
9π/10  0.135983977  0.13681750  8.335 × 10^{−4} 
The maximum absolute error \( u' ( x_{i} )  u '_{n} ( x_{i} ) \) at \(\alpha=1\) using RKM
\(x_{i}\)  n = 20  n = 40  n = 100  n = 200 

π/4  0.0000  0.0000  0.0000  0.0000 
π/3  1.249 × 10^{−4}  2.164 × 10^{−5}  2.516 × 10^{−6}  5.486 × 10^{−7} 
π/2  5.703 × 10^{−5}  8.945 × 10^{−6}  8.858 × 10^{−7}  1.744 × 10^{−7} 
2π/3  2.544 × 10^{−5}  9.454 × 10^{−6}  1.902 × 10^{−6}  5.118 × 10^{−7} 
3π/4  9.562 × 10^{−5}  2.732 × 10^{−5}  4.726 × 10^{−6}  1.212 × 10^{−6} 
Numerical comparison via maximum absolute error at \(\alpha=2\)
h  \(\frac{\pi}{20}\)  \(\frac{\pi}{40}\)  \(\frac{\pi}{80}\) 

RKM  2.07 × 10^{−6}  1.06 × 10^{−6}  5.37 × 10^{−6} 
RPSM  7.35 × 10^{−7}  92.4 × 10^{−5}  1.66 × 10^{−5} 
Spline [5]  6.43 × 10^{−4}  1.83 × 10^{−4}  4.87 × 10^{−5} 
Cubic splines [11]  1.26 × 10^{−3}  3.29 × 10^{−4}  8.43 × 10^{−5} 
Spline [6]  1.94 × 10^{−3}  4.99 × 10^{−4}  1.27 × 10^{−4} 
Spline [10]  2.20 × 10^{−3}  5.87 × 10^{−4}  1.51 × 10^{−4} 
Finite difference [7]  2.50 × 10^{−2}  1.29 × 10^{−2}  6.58 × 10^{−3} 
Cubic splines [4]  1.40 × 10^{−2}  7.71 × 10^{−3}  4.04 × 10^{−3} 
Numerical results of BVP (4.3) for different values of α
\(x_{i}\)  Approximation  RKM solution  RPSM solution  

α = 1  α = 0.8  α = 0.6  α = 0.8  α = 0.6  
π/10  0.1353346202  0.227355418  0.294857345  0.144088806  0.152676695 
π/5  0.2706525605  0.641687896  0.913634455  0.278361226  0.284904787 
3π/10  0.3988269892  1.189220758  1.763838308  0.402107443  0.403056686 
2π/5  0.4764432429  1.762740706  2.685475497  0.480094928  0.477335183 
π/2  0.5019235931  2.273677267  3.523536976  0.520431513  0.517187458 
3π/5  0.4778054404  2.609863040  4.085579443  0.537613446  0.535307921 
7π/10  0.4016870606  2.608549642  4.108581183  0.548667613  0.544985720 
4π/5  0.2736173911  2.140881895  3.391789601  0.320958940  0.367743677 
9π/10  0.1368175023  1.206815918  1.916014914  0.197961504  0.256800410 
5 Conclusion
The main goal of this analysis is to implement reliable numericanalytic techniques that depend on the use of reproducingkernel theory and the generalized Taylor expansion for the solution of BVPs of fractional order associated with obstacle in the Caputo sense. This goal has been achieved by improving the RK and RPS algorithms to handle such class of obstacle problems. A numerical investigation has been presented to demonstrate the approximate solution of a wellknown example in the literature. The solution behavior of approximation of some values of the fractional order α is shown quantitatively and graphically. Anyway, results acquired explicitly show the full reliability and regularity of the proposed methods. The applications of BVPs fractional order with the Riesz derivative, which is a twosided spacefractional derivative, should be investigated for future work.
Notes
Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.
Authors’ contributions
All authors contributed equally, read and approved the manuscript.
Funding
No funding sources to be declared.
Competing interests
The authors declare that they have no competing interests.
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