Efficient Method for Solving Fourth Order PDEs

Efficient Method for Solving Fourth Order PDEs

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Efficient Method for Solving Fourth Order PDEs

L. N. M. Tawfiq1 _{and A. Q. Ibrahim Abed}

Department of Mathematics, College of Education for Pure Science Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq.

Email: 1_{[email protected]}

Abstract. The aim of the article is to implement the LA Transform with HPM to solve the 4th order non-linear PDEs arising in mathematical physics and astrophysics. This method is based on a combination of LA Transformation with HPM without any discretization or restrictive assumptions. Some well-known 4th order non-linear PDEs are solved as illustrative examples to demonstrate the accuracy, efficiency and easy implementation of this technique.

Keywords: Forth order PDEs, HPM, LA-transform, He's polynomials. 1. Introduction

Differential equations (DEs) have been very important role in all fields of engineering and science. In science for example, contamination, heat, mass, flow and wave all phenomena are well described by PDEs [1–3]. So, it is a useful tool for describing models for natural phenomena in real life. Hence being of fundamental importance to find suitable method for each model. Various efficient methods have been proposed by many researchers for obtaining analytic, approximate and numerical solutions.

Among these are perturbation techniques [4] and Hirota’s bilinear method"[5]. Perturbation techniques were generated useful solutions in describing both quantitative and qualitative properties of the problem, which is an advantage compared to numerical solutions. However, some drawbacks were obvious for complex equations due to either such parameters cause a divergence of solutions as the quantities increase/decrease, or the non-existence of small or large perturbation parameters. In problems where these quantities do not exist, the parameter has to be artificially introduced which may lead to incorrect results [6]. Perturbation techniques are therefore found to be mainly suitable for weakly nonlinear problems.

"In recent years, many research proposed various efficient methods to solve non-linear PDEs. Among these are the ADM [7, 8], HPM" [9, 10], HAM [11], Cubic Trigonometric B-Spline method [12], Laplace decomposition method [13, 14], VIM [15], parallel processing technique [16-18] and semi analytic technique [19-22].

2

2. LA- Transformation

Here we define LA- transform as a function f(t) by

(1)

Where u is a real number, for those values of u which the improper integral is finite. For more details about LA-Transform see [23]

3. LA-transform Homotopy Perturbation Method

This method defined as combining LA-transform with HPM and denoted by LATHPM, we consider a general form of nonlinear PDE with the initial conditions as:

(2)

where L is the linear differential operator of 4rth _{order, R is also linear differential operator but} of less order than L; N is a nonlinear differential operator and g(x, t) is the inhomogeneous term.

Take LA-transform for both sides of Eq. (2) to have:

(3) Using the property of differentiation for LA-transform, we have

(4) Operating two sides of Eq. (4) by the inverse of LA-transform to get:

𝑢(𝑥, 𝑡) = 𝐺(𝑥, 𝑡) − 𝕋−1[_𝑣1₂𝕋[𝑅𝑢(𝑥, 𝑡) − 𝑁𝑢(𝑥, 𝑡)]] (5) where G(x,t) represents the part arising from the source part and the prescribed initial conditions. Herein we applied HPM:

(6) The nonlinear part can be decomposed as

(7) Where Hn (u) is He's polynomials (see [24]) that are defined by

(8) Substituting Eq. (8), (7) and (6) in Eq. (5) to obtain:

∑∞𝑛=0𝑝𝑛𝑢𝑛(𝑥, 𝑡)= 𝐺(𝑥, 𝑡) − 𝑝 (𝕋−1[ 1 𝑣2𝕋[𝑅 ∑ 𝑝 𝑛_𝑢 𝑛(𝑥, 𝑡) ∞ 𝑛=0 + 𝑁 ∑∞𝑛=0𝑝𝑛𝐻𝑛(𝑢)]]) (9)

which is the LATHPM, now using He's polynomials and comparing the like powers of p for parameters, so we the get following:

,

, (10) ,

and so on. Commonly substitute the above values in Eq. (6) to get required solution.

4. Applications

In this section, we used LATHPM presented in previous section to solve different types of non-linear 4th _{order PDEs with variable coefficients.}

Example 1

Consider the following 4th _{order linear automatous PDE}

Subject to the IC:

Using the LATHPM we have

𝐻(𝑢, 𝑝) = (1 − 𝑝)(𝑢𝑡− 𝑢0𝑡) + 𝑝(𝑢𝑡+ 2𝑢𝑥𝑥𝑥𝑥− 𝑢𝑥𝑥𝑡) = 𝑢𝑡− 𝑢0𝑡+ 𝑝(𝑢0𝑡+ 2𝑢𝑥𝑥𝑥𝑥− 𝑢𝑥𝑥𝑡) = 0 Where the is a polynomial of the following form

using induction, to get:

4

we see that . Hence

Example 2

Consider the following nonlinear 4th _{order PDE with variable coefficients}

with the ICs:

By applying LATHPM, we get

Where and are He's polynomials. First we compute He's polynomials as:

and so on and so on

Comparing the coefficients for different powers of p, we have

Hence the series solution is:

That is closed to the exact solution.

5. The Advantages of the LA- Transformation

The LA-transformation has many interesting properties which make it rival to the other transform. Some of these properties are:

The domain of the LA is wider than the domain of other transform as illustrate in [23]. This feature makes the LA more widely used in most problems.

Depending on [23] the LA has the duality with Laplace transform (LT), therefore, the LA can be solve all the problems which be solved by LT.

The unit step function in the t-domain is transformed to unity in the u-domain.

The differentiation and integration in the t‐domain are equivalent to multiplication and division of the transformed function F(u) by u in the u‐domain .

6. Convergence of the Series Solution for Linear Case

Here, we show the convergence of the series form to the exact form for linear PDEs.

Lemma 1

For any continues function f the following hold

Proof

"Suppose that

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝑐

Assume that 𝑥 = 𝑡 − 𝜏 then 𝑑𝑥 = −𝑑𝜏 then 𝜕 𝜕𝑡∫ 𝑓(𝑡 − 𝜏)𝑑𝜏 𝑡 0 = − 𝜕 𝜕𝑡∫ 𝑓(𝑥)𝑑𝑥 0 𝑡 = 𝜕 𝜕𝑡∫ 𝑓(𝑥)𝑑𝑥 𝑡 0 = 𝜕 𝜕𝑡[𝐹(𝑥)|0 𝑡_{] =} 𝜕 𝜕𝑡[𝐹(𝑡) − 𝐹(0)] = 𝜕 𝜕𝑡𝐹(𝑡) − 𝜕 𝜕𝑡𝐹(0) = 𝑓(𝑡) 𝑠𝑜, 𝜕 𝜕𝑡∫ 𝑓(𝑡 − 𝜏)𝑑𝜏 𝑡 0 = 𝑓(𝑡) Lemma 2

Let 𝕋 is LA-transform. Then

Proof

Depending on property 2 and 3 of LA-transform, and lemma (1), we get

6

Theorem 3

"If the series form given in equation (6) with p = 1, i.e., 𝑢(𝑥, 𝑦, 𝑧, 𝑡) = ∑ 𝑢𝑛(𝑥, 𝑦, 𝑧, 𝑡)

∞

𝑛=0

(11)2.30

is convergent. Then the limit point converges to the exact solution of equation (1), where un (n = 0, 1, …)are computed by LATHPM, i.e.,

Proof

Suppose that equation (11) converges to the limit point say as 𝑤(𝑥, 𝑦, 𝑧, 𝑡) = ∑ 𝑢𝑛(𝑥, 𝑦, 𝑧, 𝑡)

∞

𝑛=0

Now, from right hand side of equation (1) we have:

𝛼𝜕𝑤 𝜕𝑡 = 𝛼 𝜕 𝜕𝑡 ∑ 𝑢𝑛(𝑥, 𝑦, 𝑧, 𝑡) ∞ 𝑛=0 = 𝛼 𝜕 𝜕𝑡 [𝑢0+ 𝑢1+ ∑ 𝑢𝑛(𝑥, 𝑦, 𝑧, 𝑡) ∞ 𝑛=2 ] = 𝛼 𝜕 𝜕𝑡 [𝕋 −1_{{𝑓 +} 1 𝑣𝛼 𝕋 {−𝑅[𝑢0]}} − ∑ 𝕋 −1_{ 1 𝑣𝛼[𝕋{𝑅[𝑢𝑛−1]}]} ∞ 𝑛=2 ] = 𝛼 𝜕𝑓 𝜕𝑡− 𝑅[𝑢0] − 𝜕 𝜕𝑡 (∑ 𝕋 −1_{1 𝑣[𝕋{𝑅[𝑢𝑛]}]} ∞ 𝑛=2 ) = 0 − 𝑅[𝑢0] − ∑ 𝜕 𝜕𝑡 (𝕋 −1_{1 𝑣[𝕋{𝑅[𝑢𝑛]}]}) ∞ 𝑛=0 (12) .31 By lemma (2), equation (12) becomes

𝛼𝜕𝑤 𝜕𝑡 = − ∑ 𝑅[𝑢𝑛] ∞ 𝑛=0 = − 𝑅 [∑ 𝑢𝑛 ∞ 𝑛=0 ] = −𝑅𝑤 = 𝑤𝑥𝑥+ 𝑤𝑦𝑦+ 𝑤𝑧𝑧

Then w(x, y, z, t) satisfies equation (1). So, it is exact solution.

7. Convergence of the Solution for Nonlinear Case

Here, we show the convergence of series solution to the exact solution when we used the LATHPM, the solution is given in equation (13), where un,, (n= 0, 1, …),are calculated by LA-transform, i.e., 𝑢(𝑥, 𝑦, 𝑧, 𝑡) = 𝑢0(𝑥, 𝑦, 𝑧, 𝑡) + 𝑢1(𝑥, 𝑦, 𝑧, 𝑡) + ⋯ = ∑ 𝑢𝑛(𝑥, 𝑦, 𝑧, 𝑡) ∞ 𝑛=0 (13) 𝑢0= 𝑓(𝑥, 𝑦, 𝑧) 𝑢𝑛= −𝕋−1{ 𝕋{𝑅[𝑢_𝑛−1] + 𝐴𝑛−1} 𝑣 } , 𝑛 ≥ 1 } (14) and An, (n = 0, 1, …), are defined as

𝐴𝑛= 𝑢𝑛 𝜕𝑢0 𝜕𝑧 + 𝑢𝑛−1 𝜕𝑢1 𝜕𝑧 + ⋯ + 𝑢0 𝜕𝑢𝑛 𝜕𝑧 = ∑ 𝑢𝑘 𝜕𝑢𝑛−𝑘 𝜕𝑧 𝑛 𝑘=0 (15) Now we proof the convergence in the following theorem

Theorem 4 (Convergence Theorem)

If the series form (13) which was computed by LATHPM is convergent to the exact form

𝑤(𝑥, 𝑦, 𝑧, 𝑡) = ∑ 𝑢𝑛(𝑥, 𝑦, 𝑧, 𝑡) ∞

𝑛=0

Now, from left hand side of nonlinear equation we get:

"[13] = 𝜕𝑓 𝜕𝑡− ∑ 𝜕 𝜕𝑡[𝕋 −1_{𝕋{𝑅[𝑢𝑛]} 𝑣 }] ∞ 𝑛=0 − ∑ 𝜕 𝜕𝑡[𝕋 −1_{𝕋{𝐴𝑛} 𝑣 }] ∞ 𝑛=0 (16) By lemma (2) and equation (16) we get:

𝜕𝑤 𝜕𝑡 = 0 − ∑ 𝑅[𝑢𝑛] ∞ 𝑛=0 − ∑ 𝐴_𝑛 ∞ 𝑛=0 (17)

then, from equation (15) we get:

Then substitute equation (18) in equation (17) to get:

8

8. Conclusions

In this article, new coupled method is used to solve 4th _{order nonlinear PDEs to get exact} analytical solution, where numerical method are used in other articles to solve the same examples but cannot be getting exact analytical solution. Moreover, the illustrated applications of the LATHPM was thoroughly investigated to confirm the accuracy of the method. So, this approach is very efficient, easy implementation and rapid convergence to the exact solutions.

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