New Numerical Method for Solving Tenth Order Boundary Value Problems

Akgül, Ali; Karatas Akgül, Esra; Baleanu, Dumitru; Inc, Mustafa

doi:10.3390/math6110245

Open AccessArticle

New Numerical Method for Solving Tenth Order Boundary Value Problems

¹

Department of Mathematics, Art and Science Faculty, Siirt University, TR-56100 Siirt, Turkey

²

Department of Mathematics, Faculty of Education, Siirt University, TR-56100 Siirt, Turkey

³

Department of Mathematics and Computer Sciences, Art and Science Faculty, Çankaya University, TR-06300 Ankara, Turkey

⁴

Department of Mathematics, Institute of Space Sciences, 077125 Bucharest, Romania

⁵

Department of Mathematics, Science Faculty, Firat University, 23100 Elazig, Turkey

^*

Author to whom correspondence should be addressed.

Mathematics 2018, 6(11), 245; https://doi.org/10.3390/math6110245

Submission received: 21 September 2018 / Revised: 4 November 2018 / Accepted: 5 November 2018 / Published: 8 November 2018

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)

Download Versions Notes

Abstract

In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables.

Keywords:

reproducing kernel Hilbert space method; tenth-order boundary value problems; reproducing kernel functions

AMS:

35C10; 46E22; 30E25

1. Introduction

A numerical approximation of tenth-order boundary value problems has been given in [1]. Usmani [2] has solved fourth order boundary value problems by using the quartic spline method. Twizell and Boutayeb [3] have improved the numerical approximations for higher order eigenvalue value problems. The approximation of second order boundary value problems has been presented by Alberg and Ito [4]. Siraj-ul-Islam et al. [5] has used a non-polynomial spline method to approximate the sixth-order boundary value problems. Papamichael and Worsey [6] have applied the cubic spline algorithm to solve linear fourth-order boundary value problems. Siddiqi and Twizell [7,8] have enhanced numerical approximations of tenth and twelfth-order boundary value problems.

We consider the following problem in the reproducing kernel Hilbert space:

\begin{matrix} q^{(x)} (τ) + a_{1} (τ) q^{(i x)} (τ) + a_{2} (τ) q^{(v i i i)} (τ) \\ + a_{3} (τ) q^{(v i i)} (τ) + a_{4} (τ) q^{(v i)} (τ) + a_{5} (τ) q^{(v)} (τ) \\ + a_{6} (τ) q^{(i v)} (τ) + a_{7} (τ) q^{‴} (τ) + a_{8} (τ) q^{″} (τ) \\ + a_{9} (τ) q^{'} (τ) + a_{10} (τ) q (τ) = H (τ), τ \in [a, b], \end{matrix}

with boundary conditions:

\begin{matrix} q (a) = α_{0}, q (b) = β_{0}, q^{″} (a) = α_{1}, q^{″} (b) = β_{1}, \\ q^{(i v)} (a) = α_{2}, q^{(i v)} (b) = β_{2}, q^{(v i)} (a) = α_{3}, \\ q^{(v i)} (b) = β_{3}, q^{(v i i i)} (a) = α_{4}, q^{(v i i i)} (b) = β_{4}, \end{matrix}

where

α_{j}, β_{j}, j = 0, 1, 2, 3, 4

are arbitrary fixed real constants, and

a_{j} (τ), j = 1, 2, \dots, 10

and

H (τ)

are continuous functions given on

[a, b] .

The notion of reproducing kernel has been presented by Zaremba [9]. Aronszajn has given a systematic reproducing kernel theory containing the Bergman kernel function [10]. For more details, see [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].

This paper is constructed as: Section 2 shows the reproducing kernel functions. The exact and approximate solutions of the problem are acquired in Section 3. Numerical results are given in Section 4. Conclusions are presented in the last section.

2. Reproducing Kernel Spaces and Their Reproducing Kernel Functions

We give some important reproducing kernel functions in this section. We define

V_{2}^{11} [0, 1]

by:

V_{2}^{11} [0, 1] = \{\begin{matrix} h, h^{^{'}}, h^{^{″}}, h^{^{‴}}, h^{(4)}, h^{(5)}, h^{(6)}, h^{(7)}, h^{(8)}, h^{(9)}, h^{(10)} \\ are absolutely continuous functions, h^{(11)} \in L^{2} [0, 1], \\ h (0) = h^{″} (0) = h^{(4)} (0) = h^{(6)} (0) = h^{(8)} (0) = 0 \\ h (1) = h^{″} (1) = h^{(4)} (1) = h^{(6)} (1) = h^{(8)} (1) = 0 . \end{matrix}\}

We define the inner product as:

\begin{matrix} {〈 h, P_{y} 〉}_{V_{2}^{11} [0, 1]} = \sum_{i = 0}^{10} h^{i} (0) P_{y}^{i} (0) + + \int_{0}^{1} h^{(11)} (τ) P_{y}^{(11)} (τ) d (τ) . \end{matrix}

We can obtain:

\begin{matrix} {〈 h, P_{y} 〉}_{V_{2}^{11} [0, 1]} & = & h (0) P_{y} (0) + h^{^{'}} (0) P_{y}^{^{'}} (0) + h^{^{″}} (0) P_{y}^{^{″}} (0) \\ + & h^{^{‴}} (0) P_{y}^{^{‴}} (0) + h^{(4)} (0) P_{y}^{(4)} (0) \\ + & h^{(5)} (0) P_{y}^{(5)} (0) + h^{(6)} (0) P_{y}^{(6)} (0) \\ + & h^{(7)} (0) P_{y}^{(7)} (0) + h^{(8)} (0) P_{y}^{(8)} (0) \\ + & h^{(9)} (0) P_{y}^{(9)} (0) + h^{(10)} (0) P_{y}^{(10)} (0) \\ + & \int_{0}^{1} h^{(11)} (τ) P_{y}^{(11)} (τ) d (τ) . \end{matrix}

P_{y} (τ)

is the reproducing kernel function of

V_{2}^{11} [0, 1] .

Therefore, we have

\begin{matrix} P_{y} (0) = P_{y}^{″} (0) = P_{y}^{(4)} (0) = P_{y}^{(6)} (0) = P_{y}^{(8)} (0) = 0, \\ P_{y} (1) = P_{y}^{″} (1) = P_{y}^{(4)} (1) = P_{y}^{(6)} (1) = P_{y}^{(8)} (1) = 0 . \end{matrix}

Thus, we get

\begin{matrix} {〈 h, P_{y} 〉}_{V_{2}^{11} [0, 1]} & = & h^{^{'}} (0) P_{y}^{^{'}} (0) + h^{^{‴}} (0) P_{y}^{^{‴}} (0) \\ + & h^{(5)} (0) P_{y}^{(5)} (0) + h^{(7)} (0) P_{y}^{(7)} (0) \\ + & h^{(9)} (0) P_{y}^{(9)} (0) + h^{(10)} (0) P_{y}^{(10)} (0) \\ + & \int_{0}^{1} h^{(11)} (τ) P_{y}^{(11)} (τ) d (τ) . \end{matrix}

We use integration by parts and we get

\begin{matrix} {〈 h, P_{y} 〉}_{V_{2}^{11} [0, 1]} & = & h^{^{'}} (0) P_{y}^{^{'}} (0) + h^{^{‴}} (0) P_{y}^{^{‴}} (0) \\ + & h^{(5)} (0) P_{y}^{(5)} (0) + h^{(7)} (0) P_{y}^{(7)} (0) \\ + & h^{(9)} (0) P_{y}^{(9)} (0) + h^{(10)} (0) P_{y}^{(10)} (0) \\ + & h^{(10)} (1) P_{y}^{(11)} (1) - h^{(10)} (0) P_{y}^{(11)} (0) \\ - & h^{(9)} (1) P_{y}^{(12)} (1) + h^{(9)} (0) P_{y}^{(12)} (0) \\ + & h^{(8)} (1) P_{y}^{(13)} (1) - h^{(8)} (0) P_{y}^{(13)} (0) \\ - & h^{(7)} (1) P_{y}^{(14)} (1) + h^{(4)} (1) P_{y}^{(17)} (1) \\ + & h^{^{″}} (1) P_{y}^{(19)} (1) + h^{(7)} (0) P_{y}^{(14)} (0) \\ + & h^{(6)} (1) P_{y}^{(15)} (1) + h (1) P_{y}^{(21)} (1) \\ - & h^{(6)} (0) P_{y}^{(15)} (0) - h^{(5)} (1) P_{y}^{(16)} (1) \\ + & h^{(5)} (0) P_{y}^{(16)} (0) - h^{(4)} (0) P_{y}^{(17)} (0) \\ - & h^{^{‴}} (1) P_{y}^{(18)} (1) + h^{^{‴}} (0) P_{y}^{(18)} (0) \\ - & h^{^{″}} (0) P_{y}^{(19)} (0) - h^{^{'}} (1) P_{y}^{(20)} (1) \\ + & h^{^{'}} (0) P_{y}^{(20)} (0) - h (0) P_{y}^{(21)} (0) \\ - & \int_{0}^{1} h (τ) P_{y}^{(22)} (τ) d (τ) . \end{matrix}

Therefore, we achieve

\begin{matrix} {〈 h, P_{y} 〉}_{V_{2}^{11} [0, 1]} & = & h^{^{'}} (0) P_{y}^{^{'}} (0) + h^{^{‴}} (0) P_{y}^{^{‴}} (0) \\ + & h^{(5)} (0) P_{y}^{(5)} (0) + h^{(7)} (0) P_{y}^{(7)} (0) \\ + & h^{(9)} (0) P_{y}^{(9)} (0) + h^{(10)} (0) P_{y}^{(10)} (0) \\ + & h^{(10)} (1) P_{y}^{(11)} (1) - h^{(10)} (0) P_{y}^{(11)} (0) \\ - & h^{(9)} (1) P_{y}^{(12)} (1) + h^{(9)} (0) P_{y}^{(12)} (0) \\ - & h^{(7)} (1) P_{y}^{(14)} (1) + h^{(7)} (0) P_{y}^{(14)} (0) \\ - & h^{(5)} (1) P_{y}^{(16)} (1) + h^{(5)} (0) P_{y}^{(16)} (0) \\ - & h^{^{‴}} (1) P_{y}^{(18)} (1) + h^{^{‴}} (0) P_{y}^{(18)} (0) \\ - & h^{^{'}} (1) P_{y}^{(20)} (1) + h^{^{'}} (0) P_{y}^{(20)} (0) \\ - & \int_{0}^{1} h (τ) P_{y}^{(22)} (τ) d (τ) . \end{matrix}

We choose

\begin{matrix} P_{y}^{^{'}} (0) + P_{y}^{(20)} (0) = 0, P_{y}^{‴} (0) + P_{y}^{(18)} (0) = 0, \\ P_{y}^{(5)} (0) + P_{y}^{(16)} (0) = 0, P_{y}^{(7)} (0) + P_{y}^{(14)} (0) = 0, \\ P_{y}^{(9)} (0) + P_{y}^{(12)} (0) = 0, P_{y}^{(10)} (0) - P_{y}^{(11)} (0) = 0, \\ P_{y}^{(11)} (1) = 0, P_{y}^{(12)} (1) = 0, P_{y}^{(14)} (1) = 0, \\ P_{y}^{(16)} (1) = 0, P_{y}^{(18)} (1) = 0, P_{y}^{(20)} (1) = 0 . \end{matrix}

Thus, we acquire

\begin{matrix} {〈 h, P_{y} 〉}_{V_{2}^{11} [0, 1]} = - \int_{0}^{1} h (τ) P_{y}^{(22)} (τ) d (τ) = h (y) . \end{matrix}

The reproducing kernel function

P_{y}

is obtained as

P_{y} (τ) = \{\begin{matrix} \sum_{i = 1}^{22} c_{i} τ^{i - 1}, τ \leq y, \\ \sum_{i = 1}^{22} d_{i} τ^{i - 1}, τ > y, \end{matrix}

by

\begin{matrix} P_{y}^{(22)} (τ) = - δ (τ - y) . \end{matrix}

Definition 1.

V_{2}^{1} [0, 1]

is given as:

V_{2}^{1} [0, 1] = {m \in A C [0, 1] : m^{'} \in L^{2} [0, 1]},

where

A C

shows the space of absolutely continuous functions.

{〈m, n〉}_{V_{2}^{1}} = \int_{0}^{1} (m (τ) n (τ) + m^{'} (τ) n^{'} (τ)) d τ, m, n \in V_{2}^{1} [0, 1]

(1)

and

{∥m∥}_{V_{2}^{1}} = \sqrt{{〈m, m〉}_{V_{2}^{1}}}, m \in V_{2}^{1} [0, 1],

(2)

are the inner product and the norm in

V_{2}^{1} [0, 1] .

T_{τ} (ς)

of

V_{2}^{1} [0, 1]

is obtained as [26]:

T_{τ} (ς) = \frac{1}{2 sinh (1)} [cosh (τ + ς - 1) + cosh (|τ - ς| - 1)] .

(3)

3. Solutions in $V_{2}^{11} [0, 1]$

The solution of the problem is considered in the

V_{2}^{11} [0, 1] .

We define

Y : V_{2}^{11} [0, 1] \to V_{2}^{1} [0, 1]

as

\begin{matrix} Y h (τ) = h^{(x)} (τ) + a_{1} (τ) h^{(i x)} (τ) + a_{2} (τ) h^{(v i i i)} (τ) + a_{3} (τ) h^{(v i i)} (τ) \\ + a_{4} (τ) h^{(v i)} (τ) + a_{5} (τ) h^{(v)} (τ) + a_{6} (τ) h^{(i v)} (τ) + a_{7} (τ) h^{‴} (τ) \\ + a_{8} (τ) h^{″} (τ) + a_{9} (τ) h^{'} (τ) + a_{10} (τ) h (τ) . \end{matrix}

The model problem changes to the following problem:

Y h = M (τ), τ \in [0, 1],

(4)

with the following boundary conditions:

\begin{matrix} h (a) = 0, h (b) = 0, h^{″} (a) = 0, h^{″} (b) = 0, \\ h^{(i v)} (a) = 0, h^{(i v)} (b) = 0, h^{(v i)} (a) = 0, \\ h^{(v i)} (b) = 0, h^{(v i i i)} (a) = 0, h^{(v i i i)} (b) = 0 . \end{matrix}

Theorem 1.

Y is a bounded linear operator.

Proof.

We need to show

{∥Y h∥}_{V_{2}^{1}}^{2} \leq M {∥h∥}_{V_{2}^{11}}^{2}

, where

M > 0

. We acquire

{∥Y h∥}_{V_{2}^{1}}^{2} = {〈Y h, Y h〉}_{V_{2}^{1}} = \int_{0}^{1} (Y h {(τ)}^{2} + Y h^{'} {(τ)}^{2}) d τ

by Equations (1) and (2). We obtain

h (τ) = {〈h (\cdot), P_{τ} (\cdot)〉}_{V_{2}^{11}}

and

Y h (τ) = {〈h (\cdot), Y P_{τ} (\cdot)〉}_{V_{2}^{11}},

by reproducing property. Thus, we reach

|Y h (τ)| \leq {∥h∥}_{V_{2}^{11}} {∥Y P_{τ}∥}_{V_{2}^{11}} = M_{1} {∥h∥}_{V_{2}^{11}},

where

M_{1} > 0 .

Therefore, we find

\int_{0}^{1} {[(Y h) (τ)]}^{2} d τ \leq M_{1}^{2} {∥h∥}_{V_{2}^{11}}^{2} .

Since

{(Y h)}^{'} (τ) = {〈h (\cdot), {(Y P_{η})}^{'} (\cdot)〉}_{V_{2}^{11}},

then

|{(Y h)}^{'} (η)| \leq {∥h∥}_{V_{2}^{11}} {∥{(Y P_{τ})}^{'}∥}_{V_{2}^{11}} = M_{2} {∥f∥}_{V_{2}^{11}},

where

M_{2} > 0 .

Therefore, we reach

{[{(Y h)}^{'} (τ)]}^{2} \leq M_{2}^{2} {∥h∥}_{V_{2}^{11}}^{2}

and

\int_{0}^{1} {[{(Y h)}^{'} (τ)]}^{2} d τ \leq M_{2}^{2} {∥h∥}_{V_{2}^{11}}^{2} .

Finally, we obtain

\begin{matrix} {∥Y h∥}_{V_{2}^{1}}^{2} & \leq & \int_{0}^{1} ({[(Y h) (τ)]}^{2} + {[{(Y h)}^{'} (τ)]}^{2}) d τ \\ \leq & (M_{1}^{2} + M_{2}^{2}) {∥h∥}_{V_{2}^{11}}^{2} = M {∥h∥}_{V_{2}^{11}}^{2}, \end{matrix}

where

M = M_{1}^{2} + M_{2}^{2} > 0 .

☐

The Main Results

Let

φ_{i} (τ) = T_{τ_{i}} (τ)

and

ψ_{i} (τ) = Y^{*} φ_{i} (τ)

,

Y^{*}

is an adjoint operator of Y. The orthonormal system

{\{{\bar{Ψ}}_{i} (τ)\}}_{i = 1}^{\infty}

of

V_{2}^{11} [0, 1]

can be obtained as:

{\bar{ψ}}_{i} (τ) = \sum_{k = 1}^{i} β_{i k} ψ_{k} (τ), (β_{i i} > 0, i = 1, 2, \dots) .

(5)

Theorem 2.

Let

{\{τ_{i}\}}_{i = 1}^{\infty}

be dense in

[0, 1]

and

ψ_{i} (τ) = {Y_{ς} T_{τ} (ς)|}_{ς = τ_{i}}

. Then, the sequence

{\{ψ_{i} (τ)\}}_{i = 1}^{\infty}

is a complete system in

V_{2}^{11} [0, 1]

.

Proof.

We get

\begin{matrix} ψ_{i} (τ) = (Y^{*} φ_{i}) (τ) = 〈(Y^{*} φ_{i}) (ς), T_{τ} (ς)〉 \\ = 〈φ_{i}) (ς), Y ς T_{τ} (ς)〉 = {Y_{ς} T_{τ} (ς)|}_{ς = τ_{i}} . \end{matrix}

We know that

ψ_{i} (τ) \in V_{2}^{11} [0, 1]

. For each fixed

h (τ) \in V_{2}^{11} [0, 1]

, let

〈h (τ), ψ_{i} (η)〉 = 0, (i = 1, 2, \dots),

〈h (τ), (Y^{*} φ_{i}) (τ)〉 = 〈Y h (\cdot), φ_{i} (\cdot)〉 = (Y h) (τ_{i}) = 0 .

{\{τ_{i}\}}_{i = 1}^{\infty}

is dense in

[0, 1]

. Thus,

(Y h) (τ) = 0

.

h \equiv 0

by the

Y^{- 1}

. ☐

Theorem 3.

If

h (τ)

is the exact solution of Equation (4), then

h = \sum_{i = 1}^{\infty} \sum_{k = 1}^{i} β_{i k} M (τ_{k}) {\bar{ψ}}_{i} (τ),

(6)

where

{(τ_{i})}_{i = 1}^{\infty}

is dense in

[0, 1]

.

Proof.

We obtain

\begin{matrix} h (τ) & = & \sum_{i = 1}^{\infty} {〈h (τ), {\bar{ψ}}_{i} (τ)〉}_{V_{2}^{11}} {\bar{ψ}}_{i} (τ) \\ = & \sum_{i = 1}^{\infty} \sum_{k = 1}^{i} β_{i k} {〈h (τ), Ψ_{k} (τ)〉}_{V_{2}^{11}} {\bar{ψ}}_{i} (τ) \\ = & \sum_{i = 1}^{\infty} \sum_{k = 1}^{i} β_{i k} {〈h (τ), Y^{*} φ_{k} (τ)〉}_{V_{2}^{11}} {\bar{ψ}}_{i} (τ) \\ = & \sum_{i = 1}^{\infty} \sum_{k = 1}^{i} β_{i k} {〈Y h (τ), φ_{k} (η)〉}_{V_{2}^{1}} {\bar{ψ}}_{i} (τ) \\ = & \sum_{i = 1}^{\infty} \sum_{k = 1}^{i} β_{i k} Y h (τ_{k}) {\bar{ψ}}_{i} (τ) \\ = & \sum_{i = 1}^{\infty} \sum_{k = 1}^{i} β_{i k} M (τ_{k}) {\bar{ψ}}_{i} (η), \end{matrix}

by Equation (5) and the uniqueness of solution of Equation (4). ☐

The approximate solution

h_{n}

can be obtained by:

h_{n} = \sum_{i = 1}^{n} \sum_{k = 1}^{i} β_{i k} M (τ_{k}) {\bar{ψ}}_{i} (τ) .

(7)

4. Numerical Results

Example 1.

We consider the following tenth-order equation as:

\begin{matrix} q^{(10)} (τ) = - (80 + 19 τ + τ^{2}) exp (τ), 0 \leq τ \leq 1, \end{matrix}

with boundary conditions:

\begin{matrix} q (0) = 0, & q (1) = 0, \\ q^{″} (0) = 0, & q^{″} (1) = - 4 exp (1), \\ q^{(i v)} (0) = - 8, & q^{(i v)} (1) = - 16 exp (1), \\ q^{(v i)} (0) = - 24, & q^{(v i)} (1) = - 36 exp (1), \\ q^{(v i i i)} (0) = - 48, & q^{(v i i i)} (1) = - 64 exp (1) . \end{matrix}

The exact solution to the above boundary value problem is given by [1]:

\begin{matrix} q (τ) = τ (1 - τ) exp (τ) . \end{matrix}

We need to homogenize the boundary conditions to apply the reproducing kernel Hilbert space method:

\begin{matrix} r_{1} (x) = c_{1} + c_{2} x + c_{3} x^{2} + c_{4} x^{3} + c_{5} x^{4} \\ + c_{6} x^{5} + c_{7} x^{6} + c_{8} x^{7} + c_{9} x^{8} + c_{10} x^{9} \\ s_{1} = c_{1} = 0, \\ s_{2} = 2 c_{3} = 0, \\ s_{3} = 24 c_{5} = - 8, \\ s_{4} = 720 c_{7} = - 24, \\ s_{5} = 40, 320 c_{9} = - 48, \\ s_{6} = c_{1} + c_{2} + c_{3} + c_{4} + c_{5} + c_{6} + c_{7} + c_{8} + c_{9} + c_{10} = 0, \\ s_{7} = 72 c_{10} + 56 c_{9} + 42 c_{8} + 30 c_{7} + 20 c_{6} + 12 c_{5} + 6 c_{4} + 2 c_{3} = - 4 e, \\ s_{8} = 3024 c_{10} + 1680 c_{9} + 840 c_{8} + 360 c_{7} + 120 c_{6} + 24 c_{5} = - 16 e, \\ s_{9} = 60, 480 c_{10} + 20, 160 c_{9} + 5040 c_{8} + 720 c_{7} = - 36 e, \\ s_{10} = 362, 880 c_{10} + 40, 320 c_{9} = - 64 e, \end{matrix}

\begin{matrix} \{\begin{matrix} c_{1} = 0, c_{2} = - \frac{24}{175} + \frac{1123}{2700} e, c_{3} = 0, \\ c_{4} = \frac{352}{945} - \frac{719}{2268}, c_{5} = - \frac{1}{3}, c_{6} = \frac{28}{225} - \frac{253}{2700} e, \\ c_{7} = - \frac{1}{30}, c_{8} = \frac{1}{126} - \frac{19}{3780} e, \\ c_{9} = - \frac{1}{840}, c_{10} = \frac{1}{7560} - \frac{1}{5670} e . \end{matrix}\} \end{matrix}

Therefore, we obtain

\begin{matrix} r_{1} (x) = - \frac{1}{113, 400} x (20 x^{8} e - 15 x^{8}, \\ + 570 x^{6} e + 135 x^{7} - 900 x^{6} + 10, 626 x^{4} e, \\ + 3780 x^{5} - 14, 112 x^{4} + 35, 950 x^{2} e, \\ + 37, 800 x^{3} - 42, 240 x^{2} - 47, 166 e + 15, 552) . \end{matrix}

We obtain Table 1 by our accurate technique.

5. Conclusions

In this paper, we used an accurate technique for investigating tenth order boundary value problems. An example was chosen to prove the computational accuracy. As shown in Table 1, our method is very accurate. We acquired some significant reproducing kernel functions to get approximate solutions and absolute errors.

Author Contributions

These authors contributed equally to this work.

Funding

This research was supported by 2017-SİÜFED-39, 2017-SİÜFEB-40 and 2018-SİÜFEB-012.

Conflicts of Interest

The authors declare no conflict of interest.

References

Iqbal, M.J.; Rehman, S.; Pervaiz, A.; Hakeem, A. Approximations for linear tenth-order boundary value problems through polynomial and non-polynomial cubic spline techniques. Proc. Pakistan Acad. Sci. 2015, 52, 389–396. [Google Scholar]
Usmani, R.A. The use of quartic splines in the numerical solution of a fourth-order boundary value problem. J. Comput. Appl. Math. 1992, 44, 187–199. [Google Scholar] [CrossRef]
Twizell, E.H.; Boutayeb, A.; Djidjeli, K. Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability. Adv. Comput. Math. 1994, 2, 407–436. [Google Scholar] [CrossRef]
Ahlberg, J.H.; Ito, T. A collocation method for two-point boundary value problems. Math. Comp. 1975, 29, 761–776. [Google Scholar] [CrossRef]
Islam, S.U.; Tirmizi, I.A.; Haq, F.I.; Khan, M.A. Non-polynomial splines approach to the solution of sixth-order boundary-value problems. Appl. Math. Comput. 2008, 195, 270–284. [Google Scholar]
Papamichael, N.; Worsey, A.J. A cubic spline method for the solution of a linear fourth-order two-point boundary value problem. J. Comput. Appl. Math. 1981, 7, 187–189. [Google Scholar] [CrossRef]
Siddiqi, S.S.; Twizell, E.H. Spline solutions of linear twelfth-order boundary-value problems. J. Comput. Appl. Math. 1997, 78, 371–390. [Google Scholar] [CrossRef]
Siddiqi, S.S.; Twizell, E.H. Spline solutions of linear tenth-order boundary-value problems. Int. J. Comput. Math. 1998, 68, 345–362. [Google Scholar] [CrossRef]
Zaremba, S. Sur le calcul numérique des fonctions demandées dan le probléme de dirichlet et le probleme hydrodynamique. Bulletin International l’Académia des Sciences de Cracovie 1908, 68, 125–195. [Google Scholar]
Aronszajn, N. Theory of reproducing kernels. Trans. Am. Math. Soc. 1950, 68, 337–404. [Google Scholar] [CrossRef]
Arqub, O.A. Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm. Calcolo 2018, 55, 31. [Google Scholar] [CrossRef]
Akgül, E.K. Reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation. Int. J. Optim. Control. Theor. Appl. 2018, 8, 145–151. [Google Scholar] [CrossRef]
Al-Azzawi, S.N.; Momani, S.; Al-Deen Jameel, H.E. Full details of solving initial value problems by reproducing kernel Hilbert space method. Math. Theor. Model. 2015, 5, 11–19. [Google Scholar]
Chavan, S.; Podder, S.; Trivedi, S. Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces. J. Math. Anal. Appl. 2018, 466, 1337–1358. [Google Scholar] [CrossRef]
Chen, L.; Cheng, Y.M. The complex variable reproducing kernel particle method for bending problems of thin plates on elastic foundations. Comput. Mech. 2018, 62, 67–80. [Google Scholar] [CrossRef]
Farzaneh Javan, S.; Abbasbandy, S.; Fariborzi Araghi, M.A. Application of reproducing kernel Hilbert space method for solving a class of nonlinear integral equations. Math. Probl. Eng. 2017, 2017, 1–10. [Google Scholar] [CrossRef]
Geng, F.; Tang, Z.; Zhou, Y. Reproducing kernel method for singularly perturbed one-dimensional initial-boundary value problems with exponential initial layers. Qual. Theory Dyn. Syst. 2018, 17, 177–187. [Google Scholar] [CrossRef]
Geng, F.J. A new reproducing kernel Hilbert space method for solving nonlinear fourthorder boundary value problems. Appl. Math. Comput. 2009, 213, 163–169. [Google Scholar]
Javadi, S.; Babolian, E.; Moradi, E. New implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations. Commun. Numer. Anal. 2014, 2014, 1–7. [Google Scholar] [CrossRef]
Li, X.; Wu, B. A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions. Appl. Math. Lett. 2018, 86, 194–199. [Google Scholar] [CrossRef]
Mohammadi, M.; Zafarghandi, F.S.; Babolian, E.; Jvadi, S. A local reproducing kernel method accompanied by some different edge improvement techniques: application to the Burgers’ equation. Iran. J. Sci. Technol. Trans. A Sci. 2018, 42, 857–871. [Google Scholar] [CrossRef]
Momani, Z.; Al Shridah, M.; Abu Arqub, O.; Al-Momani, M.; Momani, S. Modeling and analyzing neural networks using reproducing kernel Hilbert space algorithm. Appl. Math. Inf. Sci. 2018, 12, 89–99. [Google Scholar] [CrossRef]
Moradi, E.; Yusefi, A.; Abdollahzadeh, A.; Tila, E.J. New implementation of reproducing kernel Hilbert space method for solving a class of third-order differential equations. J. Math. Comput. Sci. 2014, 12, 253–262. [Google Scholar] [CrossRef]
Wang, Y.-L.; Du, M.-J.; Temuer, C.-L.; Tian, D. Using reproducing kernel for solving a class of time-fractional telegraph equation with initial value conditions. Int. J. Comput. Math. 2018, 95, 1609–1621. [Google Scholar] [CrossRef]
Xu, M.; Niu, J.; Lin, Y. An efficient method for fractional nonlinear differential equations by quasi-Newton’s method and simplified reproducing kernel method. Math. Methods Appl. Sci. 2018, 41, 5–14. [Google Scholar] [CrossRef]
Cui, M.; Lin, Y. Nonlinear Numerical Analysis in the Reproducing Kernel Space; Nova Science Publishers Inc.: New York, NY, USA, 2009. [Google Scholar]

Table 1. Exact solution (ES) and absolute errors (AE) of Example 1.

$τ$	ES	AE (NPCSM) [1]	AE (PCSM) [1]	AE (RKHSM)
$0.2$	$0.195424441$	$2.433 \times 10^{- 7}$	$3.982 \times 10^{- 4}$	$3.330 \times 10^{- 8}$
$0.4$	$0.358037927$	$3.986 \times 10^{- 7}$	$6.663 \times 10^{- 4}$	$7.031 \times 10^{- 8}$
$0.6$	$0.358037927$	$4.428 \times 10^{- 7}$	$7.598 \times 10^{- 4}$	$6.076 \times 10^{- 8}$
$0.8$	$0.356086549$	$3.328 \times 10^{- 7}$	$5.885 \times 10^{- 4}$	$2.682 \times 10^{- 8}$

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Akgül, A.; Karatas Akgül, E.; Baleanu, D.; Inc, M. New Numerical Method for Solving Tenth Order Boundary Value Problems. Mathematics 2018, 6, 245. https://doi.org/10.3390/math6110245

AMA Style

Akgül A, Karatas Akgül E, Baleanu D, Inc M. New Numerical Method for Solving Tenth Order Boundary Value Problems. Mathematics. 2018; 6(11):245. https://doi.org/10.3390/math6110245

Chicago/Turabian Style

Akgül, Ali, Esra Karatas Akgül, Dumitru Baleanu, and Mustafa Inc. 2018. "New Numerical Method for Solving Tenth Order Boundary Value Problems" Mathematics 6, no. 11: 245. https://doi.org/10.3390/math6110245

APA Style

Akgül, A., Karatas Akgül, E., Baleanu, D., & Inc, M. (2018). New Numerical Method for Solving Tenth Order Boundary Value Problems. Mathematics, 6(11), 245. https://doi.org/10.3390/math6110245

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

New Numerical Method for Solving Tenth Order Boundary Value Problems

Abstract

1. Introduction

2. Reproducing Kernel Spaces and Their Reproducing Kernel Functions

3. Solutions in $V_{2}^{11} [0, 1]$

The Main Results

4. Numerical Results

5. Conclusions

Author Contributions

Funding

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Article Menu

New Numerical Method for Solving Tenth Order Boundary Value Problems

Abstract

1. Introduction

2. Reproducing Kernel Spaces and Their Reproducing Kernel Functions

3. Solutions in V 2 11 [ 0 , 1 ]

The Main Results

4. Numerical Results

5. Conclusions

Author Contributions

Funding

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

3. Solutions in $V_{2}^{11} [0, 1]$