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An accelerated solution for some classes of nonlinear partial differential equations

Abstract

In this paper, we apply an accelerated version of the Adomian decomposition method for solving a class of nonlinear partial differential equations. This version is a smart recursive technique in which no differentiation for computing the Adomian polynomials is needed. Convergence analysis of this version is discussed, and the error of the series solution is estimated. Some numerical examples were solved, and the numerical results illustrate the effectiveness of this version.

Background

Many physics and engineering problems are modeled by partial differential equations (PDEs). In many instances, these equations are nonlinear and the exact solutions are difficult to be obtained. Several methods were developed over some time to find approximate solutions to these nonlinear equations, such as homotopy analysis method (HAM) [1,2,3,4], homotopy perturbation method (HPM) [1, 5, 6], and Adomian decomposition method (ADM) [7,8,9,10,11,12,13,14,15]. In this paper, we introduce an accelerated version of the ADM for solving some classes of NPDEs. In ADM, the nonlinear term is replaced by a series of what are called Adomian polynomials which were introduced by Adomian and his colleagues have so far. Some other authors have suggested different formulas for computing Adomian polynomials [16,17,18,19,20,21,22,23]. This work aims to apply the accelerated formula proposed by El-Kalla in [21] for solving some classes of nonlinear partial differential equations. The main advantages of this accelerated version of Adomian polynomials can be summarized in the following main three points:

  1. 1.

    It is recursive and does not have derivative terms so, it is easy in programming, and, on the same processor, it saves time compared with the traditional formula;

  2. 2.

    Solution using it converges faster than the traditional Adomian polynomials;

  3. 3.

    It is used in convergence analysis and consequently in estimating the maximum absolute truncated error of the series solution.

The paper is organized as follows. In “The method” section, the standard ADM and the accelerated version of ADM are introduced. In “Convergence analysis” section, the convergence analysis of the accelerated version is introduced, while in “Numerical examples” section, some examples are solved to illustrate the effectiveness of this version.

The method

Consider the nonlinear partial differential equation given in the operator form:

$$L_{t} u(x,t) + R(u(x,t)) + N(u(x,t)) = g(x,t),$$
(1)

where \(L_{t} (.) = \frac{{\partial^{k} (.)}}{{\partial t^{k} }}\), \(R\) is the linear remainder operator that could include partial derivatives with respect to \(x\), \(N\) is the nonlinear operator, and \(g\) is the nonhomogeneous term.

Put (1) in the following form

$$L_{t} u(x,t) = g(x,t) - R(u(x,t)) - N(u(x,t)).$$
(2)

Applying \(L_{t}^{ - 1}\) to both sides of (2), we obtain

$$u(x,t) = \Phi (x,t) + L_{t}^{ - 1} g(x,t) - L_{t}^{ - 1} R(u(x,t))\, - L_{t}^{ - 1} N(u(x,t)),$$
(3)

where \(\Phi (x,t)\) is the solution of \(L_{t} u(x,t) = 0\) satisfied by the given initial conditions and \(L_{t}^{ - 1} (.) = \int\limits_{0}^{t} {\ldots k{\text{-fold}}\ldots \int\limits_{0}^{t} {(.){{\rm d}}t...{{\rm d}}t} } .\)

ADM assumes that the solution \(u\) can be decomposed into infinite series

$$u(x,t) = \sum\limits_{n = 0}^{\infty } {u_{n} (x,t),}$$
(4)

and the nonlinear term \(Nu\) by:

$$Nu = \sum\limits_{n = 0}^{\infty } {A_{n} .}$$
(5)

The components \(u_{n} ,\,n \ge 0\) of the solution \(u\) can be determined by using the recursive relation:

$$\begin{aligned} u_{0} & = \Phi (x,t) + L_{t}^{ - 1} g(x,t), \\ u_{n + 1} & = - L_{t}^{ - 1} (Ru_{n} ) - L_{t}^{ - 1} (A_{n} ),\quad n \ge 0, \\ \end{aligned}$$
(6)

where \(A_{n} = A_{n} (u_{0} ,u_{1} , \ldots ,u_{n} )\) is Adomian polynomials that can be determined by the traditional polynomials formula,

$$A_{n} = \frac{1}{n!}\frac{{{\text{d}}^{n} }}{{{\text{d}}\lambda^{n} }}\left[ {N\left( {\sum\limits_{i = 0}^{\infty } {\lambda^{i} u_{i} } } \right)} \right]_{\lambda = 0} ,\quad n = 0,1,2, \ldots,$$
(7)

or by El-Kalla formula [21],

$$\overline{A}_{n} = N(S_{n} ) - \sum\limits_{i = 0}^{n - 1} {\overline{A}_{i} } ,\quad n = 0,1,2, \ldots,$$
(8)

where the partial sum \(S_{n} = \sum\nolimits_{i = 0}^{n} {u_{i} (x,t)}\).

For example, Table 1 shows the first four polynomials of the nonlinear term \(u^{2}\) generated by both the traditional polynomials formula (7) and El-Kalla polynomials formula (8).

Table 1 The first four Adomian polynomials and the first four El-Kalla polynomials of the nonlinear term u2

Clearly, the first four polynomials generated by El-Kalla formula (8) include the first four polynomials generated by the traditional formula (7) in addition to other terms that should appear in \(A_{4} ,A_{5} , \ldots\) using formula (7). Thus, the solution obtained using El-Kalla polynomials converges faster than the solution obtained using the traditional polynomials.

Convergence analysis

Theorem 1

(Uniqueness theorem) Assume that \(R\) and \(N\) are Lipschitzian with respect to \(u\) such that \(\left| {R(u) - R(v)} \right| \le L_{1} \left| {u - v} \right|\) and \(\left| {N(u) - N(v)} \right| \le L_{2} \left| {u - v} \right|\). Let \(E = (C\left[ \Omega \right],\left\| . \right\|)\) denote the Banach space of all continuous functions on the domain of \(x\) and \(t\); \(\Omega = \left[ {0,x} \right] \times \left[ {0,T} \right]\) with the norm \(\left\| {u(x,t)} \right\| = \mathop {\max }\nolimits_{\Omega } \left| {u(x,t)} \right|\). Then, problem (1) has a unique solution whenever \(0 < \alpha < 1,\,\alpha = \frac{{(L_{1} + L_{2} )T^{k} }}{k!}\).

Proof

Define a mapping \(F:E \to E\) such that \(F(u) = \Phi (x,t) + L_{t}^{ - 1} g(x,t) - L_{t}^{ - 1} R(u(x,t)) - L_{t}^{ - 1} N(u(x,t))\) and let \(u,\,u^{ * } \in E\). Then,

$$\begin{aligned} \left\| {Fu - Fu^{ * } } \right\| & = \mathop {\max }\limits_{\Omega } \left| {L_{t}^{ - 1} (Ru - Ru^{ * } ) + L_{t}^{ - 1} (Nu - Nu^{ * } )} \right| \\ & \le \mathop {\max }\limits_{\Omega } L_{t}^{ - 1} \left( {L_{1} \left| {u - u^{ * } } \right| + L_{2} \left| {u - u^{ * } } \right|} \right) \\ & \le (L_{1} + L_{2} )\mathop {\max }\limits_{\Omega } \left| {u - u^{ * } } \right|\int\limits_{0}^{t} { \ldots k{\text{-fold}} \ldots \int\limits_{0}^{t} {{\text{d}}t \ldots {\text{d}}t} } \\ & \le \frac{{(L_{1} + L_{2} )T^{k} }}{k!}\left\| {u - u^{ * } } \right\| \\ & \le \alpha \left\| {u - u^{ * } } \right\|. \\ \end{aligned}$$

Under the condition \(0 < \alpha < 1\), the mapping \(F\) is contraction; therefore, by the Banach fixed-point theorem for contraction, there exists a unique solution to problem (1).

Theorem 2

(Convergence theorem) The series solution (4) of problem (1) using ADM converges whenever \(0 < \alpha < 1\) and \(\left| {u(x,t)} \right| < \infty\) on \(\Omega\).

Proof

Let, \(S_{n}\) and \(S_{m}\) be arbitrary partial sums with \(n > m\). We are going to prove that \(\left\{ {S_{n} } \right\}\) is a Cauchy sequence in Banach space \(E\). From Theorem 1, we write

$$\left\| {S_{m + 1} - S_{m} } \right\| \le \alpha \left\| {S_{m} - S_{m - 1} } \right\| \le \alpha^{2} \left\| {S_{m - 1} - S_{m - 2} } \right\| \le \cdots \le \alpha^{m} \left\| {S_{1} - S_{0} } \right\|.$$

Using the triangle inequality, we have

$$\begin{aligned} \left\| {S_{n} - S_{m} } \right\| & \le \left\| {S_{m + 1} - S_{m} } \right\| + \left\| {S_{m + 2} - S_{m + 1} } \right\| + \cdots + \left\| {S_{n} - S_{n - 1} } \right\| \\ & \le \left[ {\alpha^{m} + \alpha^{m + 1} + \cdots + \alpha^{n - 1} } \right]\left\| {S_{1} - S_{0} } \right\| \\ & \le \alpha^{m} \left[ {1 + \alpha + \alpha^{2} + \cdots + \alpha^{n - m - 1} } \right]\left\| {S_{1} - S_{0} } \right\| \\ & \le \alpha^{m} \left( {\frac{{1 - \alpha^{n - m} }}{1 - \alpha }} \right)\left\| {u_{1} (x,t)} \right\|. \\ \end{aligned}$$

Since \(0 < \alpha < 1\) so, \(1 - \alpha^{n - m} < 1\), then

$$\left\| {S_{n} - S_{m} } \right\| \le \frac{{\alpha^{m} }}{1 - \alpha }\mathop {\max }\limits_{\Omega } \left| {u_{1} (x,t)} \right|,$$
(9)

but \(\;\mathop {\max }\nolimits_{\Omega } \left| {u_{1} (x,t)} \right| < \infty\) then \(\;\left\| {S_{n} - S_{m} } \right\| \to \infty\) as \(m \to \infty\), then we conclude that \(\left\{ {S_{n} } \right\}\) is a Cauchy sequence in \(E\) and the series \(\sum\nolimits_{i = 0}^{\infty } {u_{i} (x,t)}\) converges to the unique solution \(u(x,t)\).

Theorem 3

(Error estimate) An estimate for the truncation error of the series solution (4) to problem (1) is given by:

$$\mathop {\max }\limits_{\Omega } \left| {u(x,t) - \mathop {\mathop \sum \limits_{i = 0} }\limits^{m} u_{i} (x,t)} \right| \le \frac{{\alpha^{m} }}{1 - \alpha }\mathop {\max }\limits_{\Omega } \left| {u_{1} (x,t)} \right|.$$

Proof

From (9) in Theorem 2, we have

$$\left\| {S_{n} - S_{m} } \right\| \le \frac{{\alpha^{m} }}{1 - \alpha }\mathop {\max }\limits_{\Omega } \left| {u_{1} (x,t)} \right|,$$

as \(n \to \infty\) then \(S_{n} \to u(x,t)\) so, we have

$$\;\left\| {u(x,t) - S_{m} } \right\| \le \frac{{\alpha^{m} }}{1 - \alpha }\mathop {\max }\limits_{\Omega } \left| {u_{1} (x,t)} \right|.$$

Finally, the truncation error in the region \(\Omega\) is estimated to be

$$\mathop {\max }\limits_{\Omega } \left| {u(x,t) - \mathop {\mathop \sum \limits_{i = 0} }\limits^{m} u_{i} (x,t)} \right| \le \frac{{\alpha^{m} }}{1 - \alpha }\mathop {\max }\limits_{\Omega } \left| {u_{1} (x,t)} \right|.$$

Numerical examples

In this section, we present some numerical examples to illustrate the effectiveness of the proposed version of ADM. All the results are calculated using Mathematica 11.

Example 1

Consider the following nonlinear partial differential equation:

$$\frac{\partial u}{{\partial t}} + uu_{x} = x + xt^{2} ,$$
(10)

with initial condition

$$u\left( {x,0} \right) = 0,$$
(11)

which has exact solution \(u\left( {x,t} \right) = xt.\)

Solution Equation (10) is rewritten in the form:

$$L_{t} u = x + xt^{2} - uu_{x} ,$$
(12)

where \(Nu = uu_{x} \;\;{\text{and}}\;\;L_{t} = \frac{\partial }{\partial t}.\)

Applying \(L_{t}^{ - 1} (.) = \int\limits_{0}^{t} {(.){\text{d}}t}\) to (12), we get

$$u = u(x,0) + L_{t}^{ - 1} \left( {x + xt^{2} } \right) - L_{t}^{ - 1} \left( {Nu} \right).$$
(13)

Based on the recurrence relation (6) and substituting the initial value, we get

$$\begin{gathered} u_{0} = xt + \frac{1}{3}xt^{3} , \hfill \\ u_{n + 1} = - L_{t}^{ - 1} \left( {A_{n} } \right),n \ge 0, \hfill \\ \end{gathered}$$
(14)

using the traditional polynomials formula (7),

$$\begin{aligned} A_{0} &= u_{0} u_{0x} , \hfill \\ A_{1} &= u_{0} u_{1x} + u_{1} u_{0x} , \hfill \\ A_{2} &= u_{0} u_{2x} + u_{1} u_{1x} + u_{2} u_{0x} . \hfill \\ \end{aligned}$$
(15)

Then, from (14) and (15) we get

$$\begin{aligned} u_{0} & = xt + \frac{1}{3}xt^{3} , \\ u_{1} & = - \frac{{t^{3} x}}{3} - \frac{{2t^{5} x}}{15} - \frac{{t^{7} x}}{63}, \\ u_{2} & = \frac{{17t^{6} x}}{45} + \frac{{20t^{8} x}}{63} + \frac{{206t^{10} x}}{2025} + \frac{{1412t^{12} x}}{93555} + \frac{{13t^{14} x}}{14553}, \\ \end{aligned}$$
(16)

and using El-Kalla polynomials formula (8),

$$\begin{aligned} \overline{A}_{0} & = u_{0} u_{0x} , \\ \overline{A}_{1} & = u_{0} u_{1x} + u_{1} u_{0x} + u_{1} u_{1x} , \\ \overline{A}_{2} & = u_{0} u_{2x} + u_{1} u_{2x} + u_{2} u_{0x} + u_{2} u_{1x} + u_{2} u_{2x} . \\ \end{aligned}$$
(17)

Then, from (14) and (17) we get

$$\begin{aligned} u_{0} & = xt + \frac{1}{3}xt^{3} , \\ u_{1} & = - \frac{{t^{3} x}}{3} - \frac{{2t^{5} x}}{15} - \frac{{t^{7} x}}{63}, \\ u_{2} & = \frac{{2t^{5} x}}{15} + \frac{{17t^{7} x}}{315} + \frac{{2t^{9} x}}{567} - \frac{{4t^{11} x}}{2475} - \frac{{4t^{13} x}}{12285} - \frac{{t^{15} x}}{59535}. \\ \end{aligned}$$
(18)

Table 2 shows the absolute relative error (ARE) for the sixth-order approximate solution using the proposed version of ADM and the seventh-order approximate solution using the standard ADM at \(t = 1\) for some values of \(x\) in Example 1.

Table 2 The absolute relative error for Example 1

Example 2

Consider a nonlinear partial differential equation:

$$\frac{{\partial^{2} u}}{{\partial t^{2} }} - \frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\pi^{2} }}{4}u + u^{2} = x^{2} \sin^{2} \frac{\pi t}{2},$$
(19)

with initial condition

$$u(x,0) = 0, \,u_{t} (x,0) = \frac{\pi x}{2}.$$
(20)

This problem was solved in [24] by using the standard Adomian decomposition method. Now, we will apply the proposed accelerated version of ADM and compare the results in Table 3.

Table 3 The absolute relative error for Example 2

Solution Equation (19) is rewritten in the form:

$$L_{t} u = x^{2} \sin^{2} \frac{\pi t}{2} + \frac{{\partial^{2} u}}{{\partial x^{2} }} - \frac{{\pi^{2} }}{4}u - u^{2} ,$$
(21)

where \(Nu = u^{2} \;\;{\text{and}}\;\;L_{t} = \frac{{\partial^{2} }}{{\partial t^{2} }}.\)

Applying \(L_{t}^{ - 1} (.) = \int\limits_{0}^{t} {\int\limits_{0}^{t} {(.){\text{d}}t} } {\text{d}}t\) to (21), we get

$$u = u(x,0) + tu_{t} (x,0) + L_{t}^{ - 1} \left( {x^{2} \sin^{2} \frac{\pi t}{2}} \right) + L_{t}^{ - 1} \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }}} \right) - L_{t}^{ - 1} \left( {\frac{{\pi^{2} }}{4}u} \right) - L_{t}^{ - 1} (Nu).$$
(22)

Based on the recurrence relation (6) and substituting the initial value, we get

$$\begin{aligned} u_{0} & = \frac{\pi xt}{2} - \frac{{x^{2} }}{{2\pi^{2} }} + \frac{{x^{2} }}{4}\left( {t^{2} + \frac{2}{{\pi^{2} }}\cos \left( {\pi t} \right)} \right), \\ u_{n + 1} & = L_{t}^{ - 1} \left( {\frac{{\partial^{2} u_{n} }}{{\partial x^{2} }}} \right) - L_{t}^{ - 1} \left( {\frac{{\pi^{2} }}{4}u_{n} } \right) - L_{t}^{ - 1} A_{n} ,n \ge 0, \\ \end{aligned}$$
(23)

using El-Kalla polynomials formula (8),

$$\begin{aligned} \overline{A}_{0} & = u_{0}^{2} , \\ \overline{A}_{1} & = 2u_{0} u_{1} + u_{1}^{2} , \\ \overline{A}_{2} & = 2u_{0} u_{2} + 2u_{1} u_{2} + u_{2}^{2} . \\ \end{aligned}$$
(24)

Then, from (23) and (24) we get

\(u_{0} = \frac{\pi xt}{2} - \frac{{x^{2} }}{{2\pi^{2} }} + \frac{{x^{2} }}{4}\left( {t^{2} + \frac{2}{{\pi^{2} }}\cos \left( {\pi t} \right)} \right)\),

$$\begin{aligned} u_{1} & = - \frac{{t^{2} }}{{2\pi^{2} }} + \frac{{t^{4} }}{24} + \frac{{1 - \cos \left( {\pi t} \right)}}{{\pi^{4} }} - \frac{{x(4\pi^{5} t^{3} + 24x - 12\pi^{2} t^{2} x + \pi^{4} t^{4} x - 24x\cos \left( {\pi t} \right))}}{{192\pi^{2} }} \\ & \quad - \,\frac{1}{{480\pi^{6} }}x^{2} (10\pi^{8} t^{4} - 240\pi^{3} tx - 40\pi^{5} t^{3} x + 6\pi^{7} t^{5} x - 945x^{2} + 90\pi^{2} t^{2} x^{2} \\ & \quad - \,10\pi^{4} t^{4} x^{2} + \pi^{6} t^{6} x^{2} - 120x(2\pi^{3} t - 8x + \pi^{2} t^{2} x)\cos (\pi t) \\ & \quad - \,15x^{2} \cos (2\pi t) + 480\pi^{2} x\sin (\pi t) + 480\pi tx^{2} \sin (\pi t)). \\ \end{aligned}$$
$$\begin{aligned} u_{2} & = \frac{143}{{8\pi^{10} }} + \frac{1}{{2\pi^{4} }} - \frac{{3t^{2} }}{{4\pi^{8} }} - \frac{{t^{2} }}{{4\pi^{2} }} + \frac{{t^{4} }}{48} + \frac{{t^{4} }}{{12\pi^{6} }} - \frac{{t^{6} }}{{90\pi^{4} }} - \frac{{\pi^{2} t^{6} }}{480} + \frac{{t^{8} }}{{1344\pi^{4} }} - \frac{{t^{10} }}{51840} \\ & \quad - \,\frac{18\cos (\pi t)}{{\pi^{10} }} - \frac{\cos (\pi t)}{{2\pi^{4} }} + \frac{{4t^{2} \cos (\pi t)}}{{\pi^{8} }} - \,\frac{{t^{4} \cos (\pi t)}}{{12\pi^{6} }} + \,\frac{\cos (2\pi t)}{{8\pi^{10} }} - \,\frac{12t\sin (\pi t)}{{\pi^{9} }} \\ & \quad + x\left( { - \frac{41t}{{4\pi^{5} }} + \frac{{t^{3} }}{{3\pi^{3} }} + \frac{{5t^{5} }}{96\pi } + \frac{{\pi^{5} t^{5} }}{3840} - \frac{{11\pi t^{7} }}{3360} + \frac{{\pi^{3} t^{9} }}{41472} - \frac{19t\cos (\pi t)}{{4\pi^{5} }} + \frac{{t^{3} \cos (\pi t)}}{{24\pi^{3} }} + \frac{15\sin (\pi t)}{{\pi^{6} }} - \frac{{t^{2} \sin (\pi t)}}{{4\pi^{4} }}} \right) \\ & \quad + \,x^{2} \left( { - \frac{353}{{4\pi^{8} }} - \frac{1}{{32\pi^{2} }} + \frac{{t^{2} }}{64} + \frac{{51t^{2} }}{{4\pi^{6} }} - \frac{{7t^{4} }}{768} + \frac{{t^{6} }}{{45\pi^{2} }} + \frac{{7\pi^{4} t^{6} }}{7680} - \frac{{37t^{8} }}{26880} - \frac{{\pi^{6} t^{8} }}{129024} + \frac{{\pi^{2} t^{10} }}{41472}} \right. \\ & \quad \left. { + \,\frac{177\cos (\pi t)}{{2\pi^{8} }} + \frac{\cos (\pi t)}{{32\pi^{2} }} - \frac{{8t^{2} \cos (\pi t)}}{{\pi^{6} }} + \frac{{5t^{4} \cos (\pi t)}}{{48\pi^{4} }} - \frac{\cos (2\pi t)}{{4\pi^{8} }} + \frac{49t\sin (\pi t)}{{\pi^{7} }} - \frac{{5t^{3} \sin (\pi t)}}{{6\pi^{5} }}} \right) \\ & \quad + x^{3} \left( {\frac{139t}{{8\pi^{9} }} + \frac{21t}{{32\pi^{3} }} - \frac{{t^{3} }}{{12\pi^{7} }} + \frac{{t^{5} }}{{60\pi^{5} }} - \frac{{7\pi t^{5} }}{1280} + \frac{{t^{7} }}{{630\pi^{3} }} + \frac{{9\pi^{3} t^{7} }}{8960} - \frac{{7t^{9} }}{25920\pi } - \frac{{5\pi^{5} t^{9} }}{331776} + \frac{{\pi t^{11} }}{105600}} \right. \\ & \quad + \frac{64t\cos (\pi t)}{{\pi^{9} }} + \frac{23t\cos (\pi t)}{{32\pi^{3} }} - \frac{{16t^{3} \cos (\pi t)}}{{3\pi^{7} }} - \frac{{5t^{3} \cos (\pi t)}}{192\pi } + \frac{{t^{5} \cos (\pi t)}}{{15\pi^{5} }} - \frac{t\cos (2\pi t)}{{8\pi^{9} }} - \frac{82\sin (\pi t)}{{\pi^{10} }} \\ & \quad \left. { - \frac{11\sin (\pi t)}{{8\pi^{4} }} + \frac{{24t^{2} \sin (\pi t)}}{{\pi^{8} }} + \frac{{5t^{2} \sin (\pi t)}}{{32\pi^{2} }} - \frac{{3t^{4} \sin (\pi t)}}{{4\pi^{6} }} + \frac{3\sin (2\pi t)}{{8\pi^{10} }}} \right) \\ & \quad + x^{4} \left( {\frac{2703479}{{9216\pi^{12} }} + \frac{3811}{{256\pi^{6} }} - \frac{{95t^{2} }}{{32\pi^{10} }} - \frac{{45t^{2} }}{{128\pi^{4} }} + \frac{{83t^{4} }}{{384\pi^{8} }} - \frac{{5t^{4} }}{{192\pi^{2} }} - \frac{{13t^{6} }}{2880} - \frac{{151t^{6} }}{{11520\pi^{6} }} + \frac{{13t^{8} }}{{17920\pi^{4} }}} \right. \\ & \quad + \,\frac{{151\pi^{2} t^{8} }}{258048} - \frac{{11t^{10} }}{{259200\pi^{2} }} - \frac{{221\pi^{4} t^{10} }}{16588800} + \frac{{t^{12} }}{760320} - \frac{9415\cos (\pi t)}{{32\pi^{12} }} - \frac{477\cos (\pi t)}{{32\pi^{6} }} + \frac{{723t^{2} \cos (\pi t)}}{{8\pi^{10} }} \\ & \quad + \,\frac{{23t^{2} \cos (\pi t)}}{{8\pi^{4} }} - \frac{{85t^{4} \cos (\pi t)}}{{24\pi^{8} }} - \frac{{41t^{4} \cos (\pi t)}}{{768\pi^{2} }} + \frac{{t^{6} \cos (\pi t)}}{{40\pi^{6} }} + \frac{897\cos (2\pi t)}{{1024\pi^{12} }} + \frac{5\cos (2\pi t)}{{256\pi^{6} }} \\ & \quad - \,\frac{{39t^{2} \cos (2\pi t)}}{{512\pi^{10} }} + \frac{{t^{4} \cos (2\pi t)}}{{1536\pi^{8} }} - \frac{\cos (3\pi t)}{{288\pi^{12} }} - \frac{467t\sin (\pi t)}{{2\pi^{11} }} - \frac{159t\sin (\pi t)}{{16\pi^{5} }} + \frac{{65t^{3} \sin (\pi t)}}{{3\pi^{9} }} \\ & \quad \left. { + \frac{{15t^{3} \sin (\pi t)}}{{32\pi^{3} }} - \frac{{23t^{5} \sin (\pi t)}}{{60\pi^{7} }} + \frac{51t\sin (2\pi t)}{{128\pi^{11} }} - \frac{{t^{3} \sin (2\pi t)}}{{384\pi^{9} }}} \right) \\ \end{aligned}$$
$$\begin{aligned} & \quad + x^{5} \left( { - \frac{39119t}{{2048\pi^{7} }} - \frac{{53t^{3} }}{{192\pi^{5} }} + \frac{{47t^{5} }}{{15360\pi^{3} }} - \frac{{3t^{7} }}{1792\pi } + \frac{{29\pi t^{9} }}{138240} - \frac{{17\pi^{3} t^{11} }}{2534400} - \frac{249t\cos (\pi t)}{{4\pi^{7} }}} \right.^{5} \\ & \quad + \frac{{37t^{3} \cos (\pi t)}}{{8\pi^{5} }} - \frac{{5t^{5} \cos (\pi t)}}{{96\pi^{3} }} + \frac{179t\cos (2\pi t)}{{2048\pi^{7} }} - \frac{{t^{3} \cos (2\pi t)}}{{3072\pi^{5} }} + \frac{327\sin (\pi t)}{{4\pi^{8} }} - \frac{{22t^{2} \sin (\pi t)}}{{\pi^{6} }} \\ & \quad \left. { + \frac{{59t^{4} \sin (\pi t)}}{{96\pi^{4} }} - \frac{249\sin (2\pi t)}{{1024\pi^{8} }} + \frac{{t^{2} \sin (2\pi t)}}{{1024\pi^{6} }}} \right) \\ & \quad + \,x^{6} \left( { - \frac{20131027}{{73728\pi^{10} }} + \frac{{411t^{2} }}{{256\pi^{8} }} - \frac{{511t^{4} }}{{3072\pi^{6} }} + \frac{{499t^{6} }}{{92160\pi^{4} }} - \frac{{457t^{8} }}{{1290240\pi^{2} }} + \frac{{103t^{10} }}{2073600} - \frac{{61\pi^{2} t^{12} }}{30412800}} \right. \\ & \quad + \,\frac{70115\cos (\pi t)}{{256\pi^{10} }} - \frac{{5663t^{2} \cos (\pi t)}}{{64\pi^{8} }} + \frac{{235t^{4} \cos (\pi t)}}{{64\pi^{6} }} - \frac{{9t^{6} \cos (\pi t)}}{{320\pi^{4} }} - \frac{6917\cos (2\pi t)}{{8192\pi^{10} }} \\ & \quad + \,\frac{{323t^{2} \cos (2\pi t)}}{{4096\pi^{8} }} - \frac{{5t^{4} \cos (2\pi t)}}{{12288\pi^{6} }} + \frac{5\cos (3\pi t)}{{2304\pi^{10} }} + \frac{3567t\sin (\pi t)}{{16\pi^{9} }} - \frac{{175t^{3} \sin (\pi t)}}{{8\pi^{7} }} \\ & \quad \left. { + \frac{{199t^{5} \sin (\pi t)}}{{480\pi^{5} }} - \frac{447t\sin (2\pi t)}{{1024\pi^{9} }} + \frac{{5t^{3} \sin (2\pi t)}}{{3072\pi^{7} }}} \right) \\ & \quad + \,x^{7} \left( {\frac{1549169t}{{18432\pi^{11} }} - \frac{{21t^{3} }}{{64\pi^{9} }} - \frac{{17t^{5} }}{{1280\pi^{7} }} + \frac{{229t^{7} }}{{161280\pi^{5} }} - \frac{{47t^{9} }}{{82944\pi^{3} }} + \frac{{t^{11} }}{126720\pi } - \frac{{\pi t^{13} }}{2995200}} \right. \\ & \quad + \,\frac{32631t\cos (\pi t)}{{64\pi^{11} }} - \frac{{923t^{3} \cos (\pi t)}}{{16\pi^{9} }} + \frac{{121t^{5} \cos (\pi t)}}{{80\pi^{7} }} - \frac{{t^{7} \cos (\pi t)}}{{120\pi^{5} }} - \frac{2091t\cos (2\pi t)}{{2048\pi^{11} }} \\ & \quad + \,\frac{{109t^{3} \cos (2\pi t)}}{{3072\pi^{9} }} - \frac{{t^{5} \cos (2\pi t)}}{{5120\pi^{7} }} + \frac{t\cos (3\pi t)}{{576\pi^{11} }} - \frac{4765\sin (\pi t)}{{8\pi^{12} }} + \frac{{214t^{2} \sin (\pi t)}}{{\pi^{10} }} - \frac{{263t^{4} \sin (\pi t)}}{{24\pi^{8} }} \\ & \quad \left. { + \,\frac{{7t^{6} \sin (\pi t)}}{{48\pi^{6} }} + \frac{2817\sin (2\pi t)}{{2048\pi^{12} }} - \frac{{149t^{2} \sin (2\pi t)}}{{512\pi^{10} }} + \frac{{t^{4} \sin (2\pi t)}}{{1024\pi^{8} }} - \frac{\sin (3\pi t)}{{216\pi^{12} }}} \right) \\ & \quad + \,x^{8} \left( {\frac{641700505}{{884736\pi^{14} }} - \frac{{12035t^{2} }}{{4096\pi^{12} }} + \frac{{63t^{4} }}{{1024\pi^{10} }} - \frac{{19t^{6} }}{{3840\pi^{8} }} + \frac{{41t^{8} }}{{143360\pi^{6} }} - \frac{{7t^{10} }}{{518400\pi^{4} }} + \frac{{t^{12} }}{{1520640\pi^{2} }}} \right. \\ & \quad - \,\frac{{t^{14} }}{41932800} - \frac{46525\cos (\pi t)}{{64\pi^{14} }} + \frac{{33807t^{2} \cos (\pi t)}}{{128\pi^{12} }} - \frac{{465t^{4} \cos (\pi t)}}{{32\pi^{10} }} + \frac{{121t^{6} \cos (\pi t)}}{{480\pi^{8} }} - \frac{{t^{8} \cos (\pi t)}}{{960\pi^{6} }} \\ & \quad + \,\frac{13609\cos (2\pi t)}{{8192\pi^{14} }} - \frac{{2119t^{2} \cos (2\pi t)}}{{4096\pi^{12} }} + \frac{{109t^{4} \cos (2\pi t)}}{{12288\pi^{10} }} - \frac{{t^{6} \cos (2\pi t)}}{{30720\pi^{8} }} - \frac{17\cos (3\pi t)}{{1728\pi^{14} }} + \frac{{t^{2} \cos (3\pi t)}}{{1152\pi^{12} }} \\ & \quad + \,\frac{\cos (4\pi t)}{{32768\pi^{14} }} - \frac{4989t\sin (\pi t)}{{8\pi^{13} }} + \frac{{1747t^{3} \sin (\pi t)}}{{24\pi^{11} }} - \frac{{263t^{5} \sin (\pi t)}}{{120\pi^{9} }} + \frac{{t^{7} \sin (\pi t)}}{{48\pi^{7} }} + \frac{2845t\sin (2\pi t)}{{2048\pi^{13} }} \\ & \quad \left. { - \,\frac{{149t^{3} \sin (2\pi t)}}{{1536\pi^{11} }} + \frac{{t^{5} \sin (2\pi t)}}{{5120\pi^{9} }} - \frac{t\sin (3\pi t)}{{216\pi^{13} }}} \right). \\ \end{aligned}$$

Table 3 shows the absolute relative error (ARE) for the third-order approximate solution using the proposed version of ADM and the third-order approximate solution using the standard ADM at \(t = 0.2\) for some values of \(x\) in Example 2. The exact solution of the partial differential Eq. (19) is given in [24] by \(u(x,t) = x\sin \left( {\frac{\pi t}{2}} \right).\)

Conclusion

An accelerated technique based on ADM is proposed. In this proposed technique, there is no need for differentiation in calculations of the Adomian polynomials. Consequently, it makes programming easier and saves much time on the same processor compared with the calculations using traditional Adomian polynomials. Convergence analysis of this version is discussed, and the error analysis of the series solution is estimated. Results of numerical examples show the effectiveness of the proposed technique. Accordingly, in the future, this accelerated version is recommended for solving nonlinear equations with different complicated piece-wise differentiable nonlinearity terms.

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Abbreviations

ADM:

Adomian decomposition method

PDEs:

Partial differential equations

NPDEs:

Nonlinear partial differential equations

HAM:

Homotopy analysis method

HPM:

Homotopy perturbation method

ARE:

Absolute relative error

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El-Kalla, I.L., Mohamed, E.M. & El-Saka, H.A.A. An accelerated solution for some classes of nonlinear partial differential equations. J Egypt Math Soc 29, 7 (2021). https://doi.org/10.1186/s42787-021-00116-9

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Keywords

  • Adomian decomposition method
  • Adomian polynomials
  • Nonlinear partial differential equations

Mathematics Subject Classification

  • 35A01
  • 35A02
  • 35A35