Bezier Curves Method for Fourth-Order Integrodifferential Equations

Abstract

The Bezier curves method is applied to solve both linear and nonlinear BVPs for fourth-order integrodifferential equations. Also, the presented method is developed for solving BVPs which arise from the problems in calculus of variation. These BVPs result from the Euler-Lagrange equations which are the necessary conditions of the extremums of problems in calculus of variation. Some numerical examples demonstrate the validity and applicability of the technique.

1. Introduction

Recently, there has been much attention devoted to the search for reliable and more efficient solution methods for equations modelling physical phenomena in various fields of engineering (see [1, 2]). One of the methods which has received much concern is the Adomian decomposition method (ADM) (see [3, 4]). The ADM has been employed to solve various scientific models. In [5], Wazwaz's main objective was to obtain the exact solutions to two fourth-order integrodifferential equations. Hashim [4] determined the accuracy and efficiency of the ADM in solving integrodifferential equations. In the present work, we suggest a technique similar to the one which was used in [6–8] for solving both linear and nonlinear boundary value problems (BVPs) for fourth-order integrodifferential equations.

Now, we consider the following class of two-point BVPs for fourth-order integrodifferential equations 𝑦^(𝑖𝑣)(𝑡)(𝑡)(𝑡)+=𝑓+𝛾𝑦_𝑡0𝑔(𝑡)𝑦(𝑡)+ℎ(𝑡)𝐹𝑦(𝑡)𝑑𝑡,𝑡∈𝑡₀,𝑡_𝑓,𝑦𝑡₀=𝛼₀,𝑦^′′𝑡₀=𝛼₁,𝑦𝑡_𝑓=𝛽₀,𝑦^′′𝑡₀=𝛽₁, (1) where 𝐹 is a real nonlinear continuous function, 𝛾, 𝛼₀, 𝛼₁, 𝛽₀, and 𝛽₁ are real constants, and 𝑓, 𝑔, and ℎ are given and can be approximated by Taylor polynomials. The conditions for existence and uniqueness of solutions of (1) are given in [9].

The rest of this paper is organized as follows. In Section 2, we review the Beizer curves method. Several illustrative examples are provided in Section 3 for confirming the effectiveness of the presented method. Section 4 contains some conclusions and notations about the future works.

2. The Beizer Curves Method

Consider the problem (1). Divide the interval [𝑡₀,𝑡_𝑓] into a set of grid points such that 𝑡_𝑗=𝑡₀+𝑗ℎ,𝑗=0,1,…,𝑘, (2) where ℎ=(𝑡_𝑓−𝑡₀)/𝑘 and 𝑘 is a positive integer. Let 𝑆_𝑗=[𝑡_𝑗−1,𝑡_𝑗] for 𝑗=1,2,…,𝑘. Then, for 𝑡∈𝑆_𝑗, the problem (1) can be decomposed to the following problems: 𝑦_𝑗^(𝑖𝑣)(𝑡)=𝑓(𝑡)+𝛾𝑦_𝑗(𝑡)+^𝑡𝑗_𝑡𝑗−1𝑔(𝑡)𝑦_𝑗(𝑡)+ℎ(𝑡)𝐹𝑦_𝑗(𝑡)𝑡𝑑𝑡,𝑡∈_𝑗−1,𝑡_𝑗,𝑦𝑡₀=𝛼₀,𝑦^′′𝑡₀=𝛼₁,𝑦𝑡_𝑓=𝛽₀,𝑦^′′𝑡_𝑓=𝛽₁, (3) where 𝑦_𝑗(𝑡) is considered in 𝑡∈𝑆_𝑗. Let 𝑦(𝑡)=∑^𝑘_𝑗=1𝜒¹_𝑗(𝑡)𝑦_𝑗(𝑡) where 𝜒¹_𝑗(𝑡) is the characteristic function of 𝑦_𝑗(𝑡) for 𝑡∈[𝑡_𝑗−1,𝑡_𝑗]. It is trivial that [𝑡₀,𝑡_𝑓]=⋃^𝑘_𝑗=1𝑆_𝑗.

Our strategy is using Bezier curves to approximate the solutions 𝑦_𝑗(𝑡) by 𝑣_𝑗(𝑡) where 𝑣_𝑗(𝑡) is given below. Individual Bezier curves that are defined over the subintervals are joined together to form the Bezier spline curves. For 𝑗=1,2,…,𝑘, define the Bezier polynomials 𝑣_𝑗(𝑡) of degree 𝑛 that approximate the action of 𝑦_𝑗(𝑡) over the interval [𝑡_𝑗−1,𝑡_𝑗] as follows: 𝑣_𝑗(𝑡)=^𝑛_𝑟=0𝑎^𝑗_𝑟𝐵_𝑟,𝑛𝑡−𝑡_𝑗−1ℎ, (4) where 𝐵_𝑟,𝑛𝑡−𝑡_𝑗−1ℎ=𝑛𝑟1ℎ^𝑛𝑡_𝑗−𝑡^𝑛−𝑟𝑡−𝑡_𝑗−1^𝑟, (5) is the Bernstein polynomial of degree 𝑛 over the interval [𝑡_𝑗−1,𝑡_𝑗] and 𝑎^𝑗_𝑟 is the control points (see [7]). By substituting (4) in (3), one can define 𝑅_1,𝑗(𝑡) for 𝑡∈[𝑡_𝑗−1,𝑡_𝑗] as 𝑅_1,𝑗(𝑡)=𝑦_𝑗^(𝑖𝑣)(𝑡)−𝑓(𝑡)+𝛾𝑦_𝑗(𝑡)+^𝑡𝑗_𝑡𝑗−1𝑔(𝑡)𝑦_𝑗(𝑡)+ℎ(𝑡)𝐹𝑦_𝑗(𝑡)𝑑𝑡. (6) Let 𝑣(𝑡)=∑^𝑘_𝑗=1𝜒¹_𝑗(𝑡)𝐯_𝑗(𝑡) where 𝜒¹_𝑗(𝑡) is the characteristic function of 𝑣_𝑗(𝑡) for 𝑡∈[𝑡_𝑗−1,𝑡_𝑗]. Besides the boundary conditions on 𝑣(𝑡), at each node, we need to impose continuity condition on each successive pair of 𝑣_𝑗(𝑡) to guarantee the smoothness. Since the differential equation is of first order, the continuity of 𝑦 (or 𝑣) and its first derivative gives 𝐯_𝑗^(𝑠)𝑡_𝑗=𝐯^(𝑠)_𝑗+1𝑡_𝑗,𝑠=0,1,𝑗=1,2,…,𝑘−1, (7) where 𝑣_𝑗^(𝑠)(𝑡_𝑗) is the 𝑠th derivative 𝑣_𝑗(𝑡) with respect to 𝑡 at 𝑡=𝑡_𝑗. Thus, the vector of control points 𝑎^𝑗_𝑟 (𝑟=0,1,𝑛−1,𝑛) must satisfy (see [7]) 𝑎^𝑗_𝑛𝑡_𝑗−𝑡_𝑗−1^𝑛=𝑎₀^𝑗+1𝑡_𝑗+1−𝑡_𝑗^𝑛,𝑎^𝑗_𝑛−𝑎^𝑗_𝑛−1𝑡_𝑗−𝑡_𝑗−1^𝑛−1=𝑎₁^𝑗+1−𝑎₀^𝑗+1𝑡_𝑗+1−𝑡_𝑗^𝑛−1. (8) According to the definition of the 𝑡_𝑗=𝑡₀+𝑗ℎ, we get that 𝑡_𝑗−𝑡_𝑗−1=ℎ. Therefore 𝐚^𝑗_𝑛=𝐚₀^𝑗+1,𝐚^𝑗_𝑛−𝐚^𝑗_𝑛−1=𝐚₁^𝑗+1−𝐚₀^𝑗+1. (9) Ghomanjani et al. [7] proved the convergence of this method where ℎ→0.

Now, the residual function can be defined in 𝑆_𝑗 as follows: 𝑅_𝑗=^𝑡𝑗_𝑡𝑗−1𝑅_1,𝑗(𝑡)²𝑑𝑡, (10) where ‖⋅‖ is the Euclidean norm. Our aim is solving the following problem over 𝑆=⋃^𝑘_𝑗=1𝑆_𝑗: min^𝑘_𝑗=1𝑅_𝑗s.t.𝐚^𝑗_𝑛=𝐚₀^𝑗+1,𝐚^𝑗_𝑛−𝐚^𝑗_𝑛−1=𝐚₁^𝑗+1−𝐚₀^𝑗+1,𝑣₁𝑡₀=𝛼₀,𝑣₁^′′𝑡₀=𝛼₁,𝑣_𝑘𝑡_𝑓=𝛽₀,𝑣_𝑘^′′𝑡_𝑓=𝛽₁. (11) The mathematical programming problem (11) can be solved by many subroutine algorithms. Here, Maple 12 can solve this optimization problem.

Remark 1. Consider the following functional (see [10]): 𝐽𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑁(𝑡)=¹₀𝐺𝑥,𝑦₁(𝑡),…,𝑦_𝑁(𝑡),𝑦^′₁(𝑡),…,𝑦^′_𝑁(𝑡)𝑑𝑡, (12) to find the extreme value of 𝐽, the boundary points of the admissible curves are known as 𝑦_𝑖(0)=𝜃_𝑖𝑦,𝑖=1,2,…,𝑁,_𝑖(1)=𝛿_𝑖,𝑖=1,2,…,𝑁. (13)
To extremize the necessary condition, 𝐽[𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑁(𝑡)] is that it should satisfy the Euler-Lagrange equations 𝜕𝐺𝜕𝑦_𝑖−𝑑𝑑𝑡𝜕𝐺𝜕𝑦^′_𝑖=0,𝑖=1,2,…,𝑁, (14) with boundary conditions given in (13). The system of BVPs (14) does not always have a solution, and if the solution exists, it cannot be unique. Note that in many variational problems, the existence of a solution is obvious from the geometrical or physical meaning of the problem, and if the solution of the Euler equation satisfies the boundary conditions, it is unique. Also this unique extremal will be the solution of the given variational problem. Thus, another approach for solving the variational problem (12) is finding the solution of the system of ordinary differential equations (ODEs) (14) which satisfies the boundary conditions in (13) which were called systems of BVPs. The simplest form of the variational problem (12) is 𝐽𝑦(𝑡)=¹₀𝐺𝑡,𝑦(𝑡),𝑦^′(𝑡)𝑑𝑡, (15) with the given boundary conditions 𝑦(0)=𝜃,𝑦(1)=𝛿. (16) For the extremum of the functional (15), the necessary condition is to satisfy the following second-order differential equation: 𝜕𝐺−𝑑𝜕𝑦𝑑𝑡𝜕𝐺𝜕𝑦^′=0,𝑖=1,2,…,𝑁, (17) with boundary conditions given in (16).
One can emphasize that our method is to solve BVPs such as (14) and (17).

3. Numerical Results and Discussion

To demonstrate the accuracy of the decomposition method, we consider some examples with known exact solutions.

Example 1. First, we consider the linear fourth-order integrodifferential equation as in (1) with 𝑓(𝑡)=𝑡(1+𝑒^𝑡)+3𝑒^𝑡, 𝛾=1, 𝑔(𝑡)=−1, and ℎ(𝑡)=0; that is, 𝑦^(𝑖𝑣)(𝑡)=𝑡1+𝑒^𝑡+3𝑒^𝑡+𝑦(𝑡)−^𝑡₀𝑦(𝑡)𝑦(0)𝑑𝑡,0<𝑡<1,=1,𝑦^′′(0)𝑦(1)=2,=1+𝑒,𝑦^′′(1)=3𝑒, (18) where the exact solution is 𝑦(𝑡)=1+𝑡𝑒^𝑡, (19) with 𝑛=3 and 𝑘=3, and one can find the following approximated solution 𝑦(𝑡)=0.9999999997+1.004858451𝑡+0.9327855932𝑡²+0.7033912400𝑡³1,𝑡∈0,3,0.9935111555+1.063258052𝑡+0.7575867804𝑡²+0.8785900700𝑡³1,𝑡∈3,23,0.7908249981+1.975345738𝑡−0.6105447271𝑡²+1.562655820𝑡³2,𝑡∈3,1, (20) where the maximum absolute errors in [4] and presented method are, respectively, 0.008 and 0.0009. The graphs of approximated and exact solution are shown in Figure 1.

Figure 1 

Graphs of the exact and computed solution of the problem for Example 1.

Example 2. Now, consider the nonlinear fourth-order BVP (1) with 𝑓(𝑡)=1, 𝛾=0, 𝑔(𝑡)=0, and ℎ(𝑡)=𝑒^−𝑡, and 𝐹(𝑡)=𝑦²(𝑡), 𝑦^(𝑖𝑣)(𝑡)=1+^𝑡₀𝑒^−𝑡𝑦²(𝑡)𝑦(0)𝑑𝑡,0<𝑡<1,=1,𝑦^′′(0)𝑦(1)=1,=𝑒,𝑦^′′(1)=𝑒, (21) where the exact solution is 𝑦(𝑡)=𝑒^𝑡, (22) with 𝑛=3 and 𝑘=3, and one can find the following approximated solution: 𝑦(𝑡)=0.9999999997+1.001035251𝑡+0.4854296266𝑡²+0.2130969500𝑡³1,𝑡∈0,3,0.9983270381+1.016091895𝑡+0.4402597035𝑡²+0.2582668700𝑡³1,𝑡∈3,23,0.9556365363+1.208199192𝑡+0.1520987099𝑡²+0.4023473900𝑡³2,𝑡∈3,1, (23) where the maximum absolute errors in [4] and presented method are, respectively, 0.001 and 0.00018. The graphs of approximated and exact solution are shown in Figure 2.

Figure 2 

Graphs of the exact and computed solution of the problem for Example 2.

Example 3. Consider the problem of finding the minimum of the integral min¹₀𝑦^′(𝑡)²+𝑡𝑦^′(𝑡)+𝑦(𝑡)²𝑑𝑡, (24) with the boundary conditions 𝑦(0)(1)=1=0,𝑦4, (25) where the exact solution is 𝑦(𝑡)=𝑒^−𝑡−1𝑒−2𝑒²−2𝑒^𝑡+𝑒^𝑡+14𝑒².−1 (26) According to (17), the associated Euler-Lagrange equation is as follows: 𝑦(𝑡)−𝑦^′′(𝑡)−12=0, (27) with 𝑛=3 and 𝑘=10, and the computational results are shown in Table 1, where the computed values are compared with the values obtained from the analytical solution and Legendre polynomials in [11].

Table 1 

Legendre polynomials, presented method, exact, and absolute error of 𝑦(𝑡) for Example 3.

4. Conclusions

In this sequel, the Bezier curves method was employed to solve linear and nonlinear BVPs for fourth-order integrodifferential equations. The presented algorithm produced results which are of reasonable accuracy. Numerical examples show that the proposed method is efficient and very easy to use.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the referees for their valuable suggestions and comments that improved the paper, and they would like to express their sincere gratitude to Doctor Y. Damchi. The third author gratefully acknowledges the partial support from University Putra Malaysia.