An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics

Abstract

Variational iteration method (VIM) is applied to solve linear and nonlinear boundary value problems with particular significance in structural engineering and fluid mechanics. These problems are used as mathematical models in viscoelastic and inelastic flows, deformation of beams, and plate deflection theory. Comparison is made between the exact solutions and the results of the variational iteration method (VIM). The results reveal that this method is very effective and simple, and that it yields the exact solutions. It was shown that this method can be used effectively for solving linear and nonlinear boundary value problems.

1. Introduction

This paper discusses the analytical approximate solution for fourth-order equations with nonlinear boundary conditions involving third-order derivatives. The general form of the equation for a fixed positive integer 𝑛,𝑛≥2, is a differential equation of order 2𝑛: 𝑦^(2𝑛)+𝑓(𝑥,𝑦)=0 (1.1) subject to the boundary conditions 𝑦^(2𝑗)(𝑎)=𝐴_2𝑗,𝑦^(2𝑗)(𝑏)=𝐵_2𝑗,𝑗=0(1)𝑛−1, (1.2) where −∞<𝑎≤𝑥≤𝑏<∞,𝐴_2𝑗,𝐵_2𝑗,𝑗=0(1)𝑛−1 are finite constants.

It is assumed that 𝑦 is sufficiently differentiable and that a unique solution of (1.1) exists. Problems of this kind are commonly encountered in plate-deflection theory and in fluid mechanics for modeling viscoelastic and inelastic flows [1–3]. Usmani [1, 2] discussed sixth order methods for the linear differential equation 𝑦⁽⁴⁾+𝑃(𝑥)𝑦=𝑞(𝑥) subject to the boundary conditions 𝑦(𝑎)=𝐴₀, 𝑦^′′(𝐴)=𝐴₂, 𝑦(𝑏)=𝐵₀,𝑦^′′(𝑏)=𝐵₂. The method described in [1] leads to five diagonal linear systems and involves 𝑝^′,𝑝^′′,𝑞^′,𝑞^′′ at 𝑎 and 𝑏, while the method described in [2] leads to nine diagonal linear systems.

Ma and Silva [4] adopted iterative solutions for (1.1) representing beams on elastic foundations. Referring to the classical beam theory, they stated that if 𝑢=𝑢(𝑥) denotes the configuration of the deformed beam, then the bending moment satisfies the relation 𝑀=−𝐸𝐼𝑢^///, where 𝐸 is the Young modulus of elasticity and 𝐼 is the inertial moment. Considering the deformation caused by a load 𝑓=𝑓(𝑥), they deduced, from a free-body diagram, that 𝑓=−𝑣^/ and 𝑣=𝑀^/=−𝐸𝐼𝑢^///, where 𝑣 denotes the shear force. For u representing an elastic beam of length 𝐿=1, which is clamped at its left side 𝑥=0, and resting on an elastic bearing at its right side 𝑥 =1, and adding a load 𝑓 along its length to cause deformations (Figure 1), Ma and Silva [4] arrived at the following boundary value problem assuming an 𝐸𝐼=1: 𝑢^(𝑖𝑣)(𝑥)=𝑓(𝑥,𝑢(𝑥)),0<𝑥<1, (1.3) the boundary conditions were taken as 𝑢(0)=𝑢^/𝑢(0)=0,_//(1)=0,𝑢^///(1)=𝑔(𝑢(1)), (1.4)(1.5) where 𝑓∈𝐶([0,1]×ℝ) and 𝑔∈𝐶(ℝ) are real functions. The physical interpretation of the boundary conditions is that 𝑢^///(1) is the shear force at 𝑥=1, and the second condition in (1.5) means that the vertical force is equal to 𝑔(𝑢(1)), which denotes a relation, possibly nonlinear, between the vertical force and the displacement 𝑢(1). Furthermore, since 𝑢^//(1)=0 indicates that there is no bending moment at 𝑥=1, the beam is resting on the bearing 𝑔.

Figure 1 

Beam on elastic bearing.

Solving (1.3) by means of iterative procedures, Ma and Silva [4] obtained solutions and argued that the accuracy of results depends highly upon the integration method used in the iterative process.

With the rapid development of nonlinear science, many different methods were proposed to solve differential equations, including boundary value problems (BVPS). These two methods are the homotopy perturbation method (HPM) [5–7] and the variational iteration method (VIM) [8–17]. In this paper, it is aimed to apply the variational iteration method proposed by He [14] to different forms of (1.1) subject to boundary conditions of physical significance.

2. Basic Idea of He’s Variational Iteration Method

To clarify the basic ideas of He's VIM, the following differential equation is considered: 𝐿[𝑢(𝑡)]+𝑁[𝑢(𝑡)]=𝑔(𝑡), (2.1) where L is a linear operator, N is a nonlinear operator, and 𝑔(𝑡) is an inhomogeneous term. According to VIM, a correction functional could be written as follows:𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡₀𝜆(𝜏)(𝐿𝑢_𝑛(𝜏)+𝑁𝑢^{0𝑥002𝑑𝑐}_𝑛(𝜏)−𝑔(𝜏))𝑑𝜏, (2.2)where 𝜆 is a general Lagrange multiplier which can be identified optimally via the variational theory. The subscript 𝑛 indicates the 𝑛th approximation and ^0𝑥0𝑢^02𝑑𝑐_𝑛 is considered as a restricted variation, that is, 𝛿𝑢^{0𝑥002𝑑𝑐}_𝑛=0.

For fourth-order boundary value problem with suitable boundary conditions, Lagrangian multiplier can be identified by substituting the problem into (2.2), upon making it stationary leads to the following: 𝑑⁴𝑑𝜏⁴𝜆=0,−𝜆^′′′+1|_𝜏=𝑥𝜆=0,_′′|_𝜏=𝑥=0. (2.3)Solving the system of (2.3) yields1𝜆=6(𝜏−𝑥)³ (2.4) and the variational iteration formula is obtained in the form 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢_𝑛⁽⁴⁾(𝜏)+𝑓(𝜏,𝑢_𝑛,𝑢^′_𝑛,𝑢_𝑛^′′,𝑢_𝑛^′′′))𝑑𝜏. (2.5)

3. The Applications of VIM Method

In this section, the VIM is applied to different forms of the fourth-order boundary value problem introduced in through (1.1).

Example 3.1. Consider the following linear boundary value problem:𝑢⁽⁴⁾(𝑥)=4𝑒^𝑥+𝑢(𝑥),0<𝑥<1, (3.1)subject to the boundary conditions𝑢(0)=1,𝑢^′(0)=2,𝑢(1)=2𝑒,𝑢^′(1)=3𝑒. (3.2) The exact solution for this problem is 𝑢(𝑥)=(1+𝑥)𝑒^𝑥. (3.3) According to (2.5), the following iteration formulation is achieved:𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢_𝑛⁽⁴⁾(𝜏)−𝑢_𝑛(𝜏)−4𝑒^𝜏)𝑑𝜏. (3.4) Now it is assumed that an initial approximation has the form 𝑢₀(𝑥)=𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑, (3.5) where 𝑎,𝑏,𝑐, and 𝑑 are unknown constants to be further determined.
By the iteration formula (3.4), the following first-order approximation may be written: 𝑢₁(𝑥)=𝑢₀(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢₀⁽⁴⁾(𝜏)−𝑢₀(𝜏)−4𝑒^𝜏)𝑑𝜏=𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑+^𝑥₀16(𝜏−𝑥)³(−𝑎𝜏³−𝑏𝜏²−𝑐_𝜏−𝑑−4𝑒^𝜏=1)𝑑𝜏840𝑎𝑥⁷+1360𝑏𝑥⁶+1120𝑐𝑥⁵+124𝑑𝑥⁴2+(−3+𝑎)𝑥³+(𝑏−2)𝑥²+(𝑐−4)𝑥+4𝑒^𝑥+𝑑−4. (3.6) Incorporating the boundary conditions (3.2), into 𝑢₁(𝑥), the following coefficients can be obtained:𝑎=−2289756+301681916440301681𝑒,𝑏=4575063−3016811516680301681𝑒,𝑐=2,𝑑=1. (3.7) Therefore, the following first-order approximate solution is derived: 𝑢₁(𝑥)=(−27259+30168101091301681𝑒)𝑥⁷+(1525021−362017204213301681𝑒)𝑥⁶+1𝑥60⁵+1𝑥24⁴+(−7472630+905043916440301681𝑒)𝑥³+(3971701−3016811516680301681𝑒)𝑥²−2𝑥−3+4𝑒^𝑥. (3.8) Comparison of the first-order approximate solution with exact solution is tabulated in Table 1, showing a remarkable agreement.
Similarly, the following second-order approximation is obtained: 𝑢₂(𝑥)=𝑢₁(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢₁⁽⁴⁾(𝜏)−𝑢₁(𝜏)−4𝑒^𝜏=1)𝑑𝜏6652800𝑎𝑥¹¹[phone]𝑏𝑥¹⁰+1362880𝑐𝑥⁹+140320𝑑𝑥⁸1+(1840𝑎−1260)𝑥⁷1+(1360𝑏−180)𝑥⁶1+(−+130120𝑐)𝑥⁵1+(−6+124𝑑)𝑥⁴4+(−3+𝑎)𝑥³+(𝑏−4)𝑥²+(𝑐−8)𝑥−8+8𝑒^𝑥+𝑑,𝑎=−12706529114180+6816288623918553568161600012042109902241𝑒,𝑐=2,𝑏=8416302814865−22720962079715745272661440012042109902241𝑒,𝑑=1. (3.9) Therefore, the second-order approximate solution may be written as 𝑢₂(𝑥)=(−57756950519+206124567987038401285709512042109902241𝑒)𝑥¹¹+(1683260562973−824498271948153608677950112042109902241𝑒)𝑥¹⁰+1𝑥181440⁹+1𝑥40320⁸+(−731163797543+3180934691158010182819240012042109902241𝑒)𝑥⁷+(7961883573271−8179546348692043736868504012042109902241𝑒)𝑥⁶−1𝑥60⁵−18𝑥⁴+(−13615367597368+6816288623918553568161600012042109902241𝑒)𝑥³+(7507464331677−22720962079715745272661440012042109902241𝑒)𝑥²−6𝑥−7+8𝑒^𝑥. (3.10) Again, the obtained solution is of distinguishing accuracy, as indicated in Table 2 and Figure 2.

Table 1 

Comparison of the first-order approximate solution with exact solution.

Table 2 

Comparison of the second-order approximate solution with exact solution.

Figure 2 

Comparison between different solutions.

Example 3.2. Consider the following linear boundary value problem:𝑢⁽⁴⁾(𝑥)=𝑢(𝑥)+𝑢^′′(𝑥)+𝑒^𝑥(𝑥−3),0<𝑥<1, (3.11) subject to the boundary conditions𝑢(0)=1,𝑢^′(0)=0,𝑢(1)=0,𝑢^′(1)=−𝑒. (3.12) The exact solution for this problem is𝑢(𝑥)=(1−𝑥)𝑒^𝑥. (3.13) According to (2.5), the iteration formulation may be written as 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢_𝑛⁽⁴⁾(𝜏)−𝑢_𝑛(𝜏)−𝑢_𝑢^′′(𝜏)−𝑒^𝜏(𝜏−3))𝑑𝜏. (3.14) Now it is assumed that an initial approximation has the form𝑢₀(𝑥)=𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑. (3.15) Where 𝑎,𝑏,𝑐, and 𝑑 are unknown constants to be further determined.
By the iteration formula (3.14), the following first-order approximation is developed:𝑢₁(𝑥)=𝑢₀(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢₀⁽⁴⁾(𝜏)−𝑢₀(𝜏)−𝑢₀^′′(𝜏)−𝑒^𝜏(𝜏−3))𝑑𝜏=𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑+^𝑥₀16(𝜏−𝑥)³(−𝑎𝜏³−𝑏𝜏²−(6𝑎+𝑐)𝜏−2𝑏−𝑑−𝑒^𝜏=1(𝜏−3))𝑑𝜏840𝑎𝑥⁷+1360𝑏𝑥⁶1+(120𝑎+120𝑐)𝑥⁵1+(112𝑏+24𝑑)𝑥⁴2+(3+𝑎)𝑥³5+(𝑏+2)𝑥²+(𝑒^𝑥+6+𝑐)𝑥−7𝑒^𝑥+7+𝑑. (3.16) Incorporating the boundary conditions (3.12), into 𝑢₁(𝑥), it can be written as𝑎=7904470−3231492950080323149𝑒,𝑏=−12770295+3231494640400323149𝑒,𝑐=0,𝑑=1. (3.17) Therefore, the following first-order approximate solution is obtained:𝑢₁(𝑥)=(112921−38777883512323149𝑒)𝑥⁷+(−851353+775557612890323149𝑒)𝑥⁶+(790447−646298147504323149𝑒)𝑥⁵+(−25217441+7755576386700323149𝑒)𝑥⁴+(24359708−9694472950080323149𝑒)𝑥³+(−23924845+6462984640400323149𝑒)𝑥²+(6+𝑒^𝑥)𝑥+8−7𝑒^𝑥. (3.18) Comparison of the first-order approximate solution with exact solution is tabulated in Table 3, again showing a clear agreement. Even higher accurate solutions could be obtained without any difficulty.
Similarly, the following second-order approximation can be written as𝑢₂(𝑥)=𝑢₁(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢₁⁽⁴⁾(𝜏)−𝑢₁(𝜏)−𝑢₁^′′(𝜏)−𝑒^𝜏=1(𝜏−3))𝑑𝜏6652800𝑎𝑥¹¹[phone]𝑏𝑥¹⁰1+(1362880𝑐+30240𝑎)𝑥⁹1+(110080𝑏+40320𝑑)𝑥⁸1+(15040𝑐+1420𝑎+1260)𝑥⁷1+(1720𝑑++1144180𝑏)𝑥⁶1+(+112120𝑎+120𝑐)𝑥⁵1+(2+1124𝑑+12𝑏)𝑥⁴+(3+𝑎)𝑥³+(𝑏+212)𝑥²+(24+3𝑒^𝑥+𝑐)𝑥+27−27𝑒^𝑥+𝑑. (3.19) Incorporating the boundary conditions, (3.12), into 𝑢₂(𝑥), yields 𝑎=381804789300110−42897120046671409850288000004289712004667𝑒,𝑐=0,𝑏=−629495301082065+42897120046672307900373632004289712004667𝑒,𝑑=1. (3.20) The following second-order approximate solution is then achieved in the following form:𝑢₂(𝑥)=(3470952630001−2594417820422601606357550012869136014001𝑒)𝑥¹¹+(−41966353405471+51888356408452032038159728412869136014001𝑒)𝑥¹⁰+(38180478930011−129720891021130081398661000012869136014001𝑒)𝑥⁹+(−2513691492323593+172961188028173440228958370404289712004667𝑒)𝑥⁸+(1149704079904997−54050371258804203356786400004289712004667𝑒)𝑥⁷+(−415373822050043+51476544056004012821668742404289712004667𝑒)𝑥⁶+(233372585584733−5147654405600470492514400004289712004667𝑒)𝑥⁵+(−1203224346103459+102953088112008192325031136004289712004667𝑒)𝑥⁴+(394673925314111−42897120046671409850288000004289712004667𝑒)𝑥³+(−1168906650066123+85794240093342307900373632004289712004667𝑒)𝑥²+(3𝑒^𝑥+24)𝑥+28−27𝑒^𝑥. (3.21) The obtained solution is of evident accuracy, as shown in Table 4 and Figure 3.

Table 3 

Comparison of the first-order approximate solution with exact solution.

Table 4 

Comparison of the second-order approximate solution with exact solution.

Figure 3 

Comparison between different solutions.

Example 3.3. Consider the following nonlinear boundary value problem:𝑢⁽⁴⁾(𝑥)=𝑢²(𝑥)+𝑔(𝑥),0<𝑥<1, (3.22) subject to the boundary conditions𝑢(0)=0,𝑢^′(0)=0,𝑢(1)=1,𝑢^′(1)=1, (3.23)where𝑔(𝑥)=−𝑥¹⁰+4𝑥⁹−4𝑥⁸−4𝑥⁷+8𝑥⁶−4𝑥⁴+120𝑥−48. (3.24) The exact solution for this problem is𝑢(𝑥)=𝑥⁵−2𝑥⁴+2𝑥². (3.25) According to (2.5), the iteration formulation is written as follows: 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢_𝑛⁽⁴⁾(𝜏)−𝑢²_𝑛(𝜏)−𝑔(𝜏))𝑑𝜏. (3.26) Now it is assumed that an initial approximation has the form𝑢₀(𝑥)=𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑, (3.27) where 𝑎,𝑏,𝑐, and 𝑑 are unknown constants to be further determined.
By the iteration formula (3.26), the following first-order approximation is obtained:𝑢₁(𝑥)=𝑢₀(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢₀⁽⁴⁾(𝜏)−𝑢²₀(𝜏)+𝜏¹⁰−4𝜏⁹+4𝜏⁸+4𝜏⁷−8𝜏⁶+4𝜏⁴1−120𝜏+48)𝑑𝜏=−𝑥24024¹⁴+1𝑥4290¹³−1𝑥2970¹²−1𝑥1980¹¹1+(𝑎5040²+1630)𝑥¹⁰+11512𝑎𝑏𝑥⁹1+(−+1420𝑏1680²+1840𝑎𝑐)𝑥⁸1+(1420𝑏𝑐+420𝑎𝑑)𝑥⁷1+(1180𝑏𝑑+𝑐360²)𝑥⁶1+(1+60𝑐𝑑)𝑥⁵1+(𝑑24²−2)𝑥⁴+𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑. (3.28) Incorporating the boundary conditions (3.23), into 𝑢₁(𝑥), results in the following values: 𝑎=−0.006871650809;𝑏=2.005929593;𝑐=0,𝑑=0. (3.29) The following first-order approximate solution is then achieved:𝑢₁(𝑥)=−4.162504162×10⁻⁵𝑥¹⁴+2.331002331×10⁻⁴𝑥¹³−3.367003367×10⁻⁴𝑥¹²−5.050505050×10⁻⁴𝑥¹¹+1.587310956×10⁻³𝑥¹⁰−9.116433669×10⁻⁶𝑥⁹+1.4139007×10⁻⁵𝑥⁸+𝑥⁵−2𝑥⁴−6.871650809×10⁻³𝑥³+2.005929593𝑥². (3.30) Comparison of the first-order approximate solution with exact solution is tabulated in Table 5, which once again shows an excellent agreement.
Similarly, the following second-order approximation may be written:𝑢₂(𝑥)=𝑢₁(𝑥)+^𝑥₀16(𝜏−𝑥)³(𝑢₁⁽⁴⁾(𝜏)−𝑢²₁(𝜏)+𝜏¹⁰−4𝜏⁹+4𝜏⁸+4𝜏⁷−8𝜏⁶+4𝜏⁴−120𝜏+48)𝑑𝜏. (3.31) Incorporating the boundary conditions, (3.23), into 𝑢₂(𝑥), yields 𝑎=−8.269548014𝐸−7;𝑏=2.000000763;𝑐=0,𝑑=0. (3.32) The following second-order approximate solution is obtained:𝑢₂(𝑥)=−1.093855974×10⁻⁹𝑥⁹+1.817×10⁻⁹𝑥⁸−2𝑥⁴−1.117934793×10⁻⁸𝑥²¹+1.463705892×10⁻⁹𝑥²⁰+6.586694874×10⁻⁸𝑥¹⁹+2.000000763𝑥²−8.269548014×10⁻⁷𝑥³−1.047931585×10⁻⁷𝑥¹⁸−3.536760165×10⁻⁸𝑥¹⁷+1.453571773×10⁻⁷𝑥¹⁶−5.173598972×10⁻¹³𝑥²⁸−2.569735395×10⁻¹⁴𝑥³¹+1.252296566×10⁻¹³𝑥³⁰−2.016131906×10⁻¹³𝑥²⁹+2.007605778×10⁻¹⁵𝑥³²+2.564345160×10⁻¹²𝑥²⁷+3.603899741×10⁻⁹𝑥²²+3.025×10⁻¹³𝑥¹⁴−1.392179800×10⁻¹⁰𝑥¹²+6.103539401×10⁻¹⁰𝑥¹¹+𝑥⁵+9.879565106×10⁻¹²𝑥²⁴−2.268156651×10⁻¹²𝑥²⁶−5.281071651×10⁻¹²𝑥²⁵−3.917282540×10⁻¹⁰𝑥²³−1.335600908×10⁻¹³𝑥¹⁵−6.059998643×10⁻¹⁰𝑥¹⁰. (3.33) The obtained solution is once again of remarkable accuracy, as shown in Table 6 and Figure 4.

Table 5 

Comparison of the first-order approximate solution with exact solution.

Table 6 

Comparison of the second-order approximate solution with exact solution.

Figure 4 

Comparison between different solutions.

4. Conclusion

This study showed that the variational iteration method is remarkably effective for solving boundary value problems. A fourth-order differential equation with particular engineering applications was solved using the VIM in order to prove its effectiveness. Different forms of the equation having boundary conditions of physical significance were considered. Comparison between the approximate and exact solutions showed that one iteration is enough to reach the exact solution. Therefore the VIM is able to solve partial differential equations using a minimum calculation process. This method is a very promoting method, which promises to find wide applications in engineering problems.