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| The '''Adomian decomposition method (ADM)''' is a semi-analytical method for solving [[ordinary differential equations|ordinary]] and [[partial differential equations|partial]] [[nonlinear]] [[differential equation]]s. The method was developed from the 1970s to the 1990s by [[George Adomian]], chair of the Center for Applied Mathematics at the [[University of Georgia]].<ref name="Adomian94">{{cite book |title=Solving Frontier problems of Physics: The decomposition method|first=G.|last=Adomian|publisher=Kluwer Academic Publishers|year=1994|isbn=|page=}}</ref>
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you have any inquiries concerning where and the best ways to make use of [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC], you could contact us at the page. |
| It is further extensible to [[stochastic system]]s by using the [[Ito integral]].<ref>{{cite book |title=Nonlinear Stochastic Operator Equations | |
| |first=G.|last=Adomian|publisher=Kluwer Academic Publishers|year=1986|isbn=0-12-044375-9|page=}} [http://www.amazon.com/Nonlinear-Stochastic-Operator-Equations-Adomian/dp/0120443759]</ref>
| |
| The aim of this method is towards a unified theory for the solution of [[partial differential equation]]s (PDE); an aim which has been superseded by the more general theory of the [[homotopy analysis method]].<ref>{{citation | last=Liao | first=S.J.|authorlink=Liao Shijun | title=Homotopy Analysis Method in Nonlinear Differential Equation| publisher=Springer & Higher Education Press| location=Berlin & Beijing | year=2012 | isbn=978-3642251313}} [http://www.amazon.com/Homotopy-Analysis-Nonlinear-Differential-Equations/dp/3642251315]</ref>
| |
| The crucial aspect of the method is employment of the "Adomian polynomials" which allow for solution convergence of the nonlinear portion of the equation, without simply linearizing the system. These [[polynomial]]s mathematically generalize to a [[Maclaurin series]] about an arbitrary external parameter; which gives the solution method more flexibly than direct [[Taylor series]] expansion.<ref>{{cite book |title=Partial Differential Equations and Solitary Waves Theory|first=Abdul-Majid|last=Wazwaz|publisher=Higher Education Press|year=2009|isbn=90-5809-369-7|page=15}}</ref>
| |
| | |
| == Ordinary Differential Equations ==
| |
| | |
| Adomian method is well suited to solve [[Cauchy problem]]s, an important class of problems which include [[initial value problem|initial condition]]s problems.
| |
| | |
| === Application to a first order nonlinear system ===
| |
| | |
| An example of initial condition problem for an Ordinary Differential Equation is the following:
| |
| | |
| :<math>
| |
| y^\prime(t) + y^{2}(t) = -1,
| |
| </math> | |
| | |
| :<math> | |
| y(0) = 0.
| |
| </math>
| |
| | |
| To solve the problem, the highest degree differential operator (written here as ''L'') is put on the left side, in the following way:
| |
| | |
| :<math>
| |
| Ly = -1 - y^{2},
| |
| </math>
| |
| | |
| with ''L'' = d/d''t'' and <math>L^{-1}=\int_{0}^{t}() </math>. Now the solution is assumed to be an infinite series of contributions: | |
| | |
| :<math>
| |
| y = y_{0} + y_{1} + y_{2} + y_{3} + \cdots.
| |
| </math>
| |
| | |
| Replacing in the previous expression, we obtain:
| |
| | |
| :<math>
| |
| (y_{0} + y_{1} + y_{2} + y_{3} + \cdots) = y(0) + L^{-1}[-1 - (y_{0} + y_{1} + y_{2} + y_{3} + \cdots)^{2}].
| |
| </math>
| |
| | |
| Now we identify ''y''<sub>0</sub> with some explicit expression on the right, and ''y''<sub>''i''</sub>, ''i'' = 1, 2, 3, ..., with some expression on the right containing terms of lower order than ''i''. For instance:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| &y_{0} &=&\ y(0)+ L^{-1}(-1) &=& -t \\
| |
| &y_{1} &=& -L^{-1}(y_{0}^{2}) =-L^{-1}(t^{2}) &=& -t^{3}/3 \\
| |
| &y_{2} &=& -L^{-1}(2y_{0}y_{1}) &=& -2t^{5}/15 \\
| |
| &y_{3} &=& -L^{-1}(y_{1}^{2}+2y_{0}y_{2}) &=& -17t^{7}/315.
| |
| \end{align}
| |
| </math>
| |
| | |
| In this way, any contribution can be explicitly calculated at any order. If we settle for the four first terms, the approximant is the following:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| y &= y_{0} + y_{1} + y_{2} + y_{3} + \cdots \\
| |
| & = -\left[ t + \frac{1}{3} t^{3} + \frac{2}{15} t^{5} + \frac{17}{315} t^{7} + \cdots \right]
| |
| \end{align}
| |
| </math> | |
| | |
| === Application to Blasius equation ===
| |
| | |
| A second example, with more complex boundary conditions is the [[Blasius boundary layer|Blasius Equation]] for a flow in a [[boundary layer]]:
| |
| | |
| :<math>
| |
| \frac{\mathrm{d}^{3} u}{\mathrm{d} x^{3}} + \frac{1}{2} u \frac{\mathrm{d}^{2} u}{\mathrm{d}x^{2}} = 0
| |
| </math>
| |
| | |
| With the following conditions at the boundaries:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| u(0) &= 0 \\
| |
| u^{\prime}(0) &= 0 \\
| |
| u^{\prime}(x) &\to 1, \qquad x \to \infty
| |
| \end{align}
| |
| </math>
| |
| | |
| Linear and non-linear operators are now called <math>L = \frac{\mathrm{d}^{3} }{\mathrm{d} x^{3}}</math> and <math>Nu =\frac{1}{2} u \frac{\mathrm{d}^{2} u}{\mathrm{d}x^{2}}</math>, respectively. Then, the expression becomes:
| |
| | |
| :<math>
| |
| L u + N u = 0
| |
| </math>
| |
| | |
| and the solution may be expressed, in this case, in the following simple way:
| |
| | |
| :<math>
| |
| u = \alpha + \beta x + \gamma x^{2}/2 - L^{-1} N u
| |
| </math>
| |
| | |
| where: <math>L^{-1} \xi (x) = \int dx \int \mathrm{d}x \int \mathrm{d}x \;\; \xi(x) </math> If:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| u &= u^{0} + u^{1} + u^{2} + \cdots + u^{N} \\
| |
| &=\alpha + \beta x + \gamma x^{2}/2 - \frac{1}{2} L^{-1} (u^{0}+u^{1}+u^{2}+\cdots+u^{N}) \frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}(u^{0} + u^{1} + u^{2} + \cdots + u^{N})
| |
| \end{align}
| |
| </math>
| |
| | |
| and:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| &u^{0} &=& \alpha + \beta x + \gamma x^{2}/2 \\
| |
| &u^{1} &=& -\frac{1}{2} L^{-1}(u^{0}u^{0''}) &=& -L^{-1} A_{0} \\
| |
| &u^{2} &=& -\frac{1}{2} L^{-1}(u^{1}u^{0''}+u^{0}u^{1''}) &=& -L^{-1} A_{1} \\
| |
| &u^{3} &=& -\frac{1}{2} L^{-1}(u^{2}u^{0''}+u^{1}u^{1''}+u^{0}u^{2''}) &=& -L^{-1} A_{2} \\
| |
| &&\cdots&
| |
| \end{align}
| |
| </math>
| |
| | |
| Adomian’s polynomials to linearize the non-linear term can be obtained systematically by using the following rule:
| |
| | |
| :<math>
| |
| A_{n} = \frac{1}{n!} \frac{\mathrm{d}^{n}}{\mathrm{d}\lambda^{n}} f(u(\lambda))\mid_{\lambda=0},
| |
| </math>
| |
| | |
| where: <math>\frac{\mathrm{d}^{n}}{\mathrm{d}\lambda^{n}} u(\lambda)\mid_{\lambda=0} = n! u_{n}</math>
| |
| | |
| Boundary conditions must be applied, in general, at the end of each approximation. In this case, the integration constants must be grouped into three final independent constants. However, in our example, the three constants appear grouped from the beginning in the form shown in the formal solution above. After applying the two first boundary conditions we obtain the so-called Blasius series:
| |
| | |
| :<math>
| |
| u = \frac{\gamma}{2} x^2 - \frac{\gamma^2}{2}\left(\frac{x^5}{5!}\right) + \frac{11 \gamma^{3}}{4}\left(\frac{x^{8}}{8!}\right) - \frac{375 \gamma^{4}}{8} \left(\frac{x^{11}}{11!}\right) + \cdots
| |
| </math> | |
| | |
| To obtain γ we have to apply boundary conditions at ∞, which may be done by writing the series as a Padé approximant:
| |
| | |
| :<math>
| |
| f(z) = \sum_{n=0}^{L+M} c_{n} z^{n} = \frac{a_{0} + a_{1}z + \cdots + a_{L}z^{L}}{b_{0} + b_{1} z + \cdots + b_{M}z^{M}}
| |
| </math>
| |
| | |
| where ''L'' = ''M''. The limit at <math>\infty</math> of this expression is ''a''<sub>''L''</sub>/''b''<sub>''M''</sub>.
| |
| | |
| If we choose ''b''<sub>0</sub> = 1, ''M'' linear equations for the ''b'' coefficients are obtained:
| |
| | |
| :<math>
| |
| \left[ \begin{array}{cccc}
| |
| c_{L-M+1} & c_{L-M+2} & \cdots & c_{L} \\
| |
| c_{L-M+2} & c_{L-M+3} & \cdots & c_{L+1} \\
| |
| \vdots & \vdots & & \vdots \\
| |
| c_{L} & c_{L+1} & \cdots & c_{L+M-1}
| |
| \end{array} \right]
| |
| \left[ \begin{array}{c}
| |
| b_{M} \\ b_{M-1} \\ \vdots \\ b_{1}
| |
| \end{array} \right]
| |
| =
| |
| - \left[ \begin{array}{c}
| |
| c_{L+1} \\ c_{L+2} \\ \vdots \\ c_{L+M}
| |
| \end{array} \right]
| |
| </math>
| |
| | |
| Then, we obtain the ''a'' coefficients by means of the following sequence:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| a_{0} &= c_{0} \\
| |
| a_{1} &= c_{1} + b_{1} c_{0} \\
| |
| a_{2} &= c_{2} + b_{1}c_{1}+b_{2}c_{0} \\
| |
| &\cdots \\
| |
| a_{L} &= c_{L} + \sum_{i=1}^{\min(L,m)} b_{i} c_{L-i}.
| |
| \end{align}
| |
| </math>
| |
| | |
| In our example: | |
| | |
| :<math>
| |
| u'(x) = \gamma x - \frac{\gamma^{2}}{2} \left(\frac{x^{4}}{4!}\right) + \frac{11 \gamma^{3}}{4} \left(\frac{x^7}{7!}\right) - \frac{375 \gamma^{4}}{8} \left(\frac{x^{10}}{10!}\right)
| |
| </math>
| |
| | |
| Which when γ = 0.0408 becomes:
| |
| | |
| :<math>
| |
| u'(x)
| |
| =
| |
| \frac{
| |
| 0.0204 + 0.0379\, z
| |
| - 0.0059\, z^{2}
| |
| - 0.00004575\, z^{3}
| |
| + 6.357 \cdot 10^{-6} z^{4}
| |
| -1.291\cdot 10^{-6} z^{5}
| |
| }{
| |
| 1 - 0.1429\, z
| |
| - 0.0000232\, z^{2}
| |
| +0.0008375\, z^{3}
| |
| - 0.0001558\, z^{4}
| |
| - 1.2849\cdot 10^{-6} z^{5}
| |
| },
| |
| </math>
| |
| | |
| with the limit:
| |
| | |
| :<math>
| |
| \lim_{x \to \infty} u'(x) = 1.004.
| |
| </math>
| |
| | |
| Which is approximately equal to 1 (from boundary condition (3)) with an accuracy of 4/1000.
| |
| | |
| == Partial Differential Equations ==
| |
| | |
| === Application to a rectangular system with nonlinearity ===
| |
| | |
| One of the most frequent problems in physical sciences is to obtain the solution of a (linear or nonlinear) partial differential equation which satisfies a set of functional values on a rectangular boundary. For instance, let us consider the following problem:
| |
| | |
| :<math>
| |
| \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} - b \frac{\partial u^2}{\partial x} = \rho(x, y) \qquad (1)
| |
| </math>
| |
|
| |
| with the following boundary conditions defined on a rectangle:
| |
| | |
| :<math>
| |
| u(x=0) = f_{1}(y)
| |
| \quad\text{and}\quad
| |
| u(x=x_{l}) = f_{2}(y) \qquad \text{(1-a)}
| |
| </math>
| |
| | |
| :<math>
| |
| u(y=-y_{l}) = g_{1}(x)
| |
| \quad\text{and}\quad
| |
| u(y=y_{l}) = g_{2}(x) \qquad \text{(1-b)}
| |
| </math>
| |
| | |
| This kind of partial differential equation appears frequently coupled with others in [[science]] and [[engineering]]. For instance, in the [[incompressible fluid]] flow problem, the [[Navier–Stokes equations]] must be solved in parallel with a [[Poisson equation]] for the pressure.
| |
| | |
| ==== Decomposition of the system ====
| |
| | |
| Let us use the following notation for the problem (1):
| |
| | |
| :<math>
| |
| L_{x} u + L_{y} u + N u = \rho(x, y) \qquad (2)
| |
| </math> | |
| | |
| where ''L''<sub>x</sub>, ''L''<sub>y</sub> are double derivate operators and ''N'' is a non-linear operator.
| |
| | |
| The formal solution of (2) is:
| |
| | |
| :<math>
| |
| u = a(y) + b(y) x + L_{x}^{-1} \rho(x, y) - L_{x}^{-1} L_{y} u - L_{x}^{-1} N u \qquad (3)
| |
| </math>
| |
| | |
| Expanding now u as a set of contributions to the solution we have:
| |
| | |
| :<math>
| |
| u = u_{0} + u_{1} + u_{2} + u_{3} + \cdots
| |
| </math>
| |
|
| |
| By substitution in (3) and making a one-to-one correspondence between the contributions on the left side and the terms on the right side we obtain the following iterative scheme:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| u_{0} &= a_{0}(y) + b_{0}(y) x + L_{x}^{-1} \rho(x, y) \\
| |
| u_{1} &= a_{1}(y) + b_{1}(y) x - L_{x}^{-1} L_{y} u_{0} + b \int dx A_{0} \\
| |
| &\cdots \\
| |
| u_{n} &= a_{n}(y) + b_{n}(y) x - L_{x}^{-1} L_{y} u_{n-1} + b \int dx A_{n-1} \quad 0 < n < \infty
| |
| \end{align}
| |
| </math>
| |
| | |
| where the couple {''a''<sub>''n''</sub>(''y''), ''b''<sub>''n''</sub>(''y'')} is the solution of the following system of equations:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \varphi^{n}(x=0) &= f_{1}(y) \\
| |
| \varphi^{n}(x=x_{l}) &= f_{2}(y),
| |
| \end{align}
| |
| </math>
| |
| | |
| here <math>\varphi^{n} \equiv \sum_{i=0}^{n} u_{i}</math> is the ''n''<sup>th</sup>-order approximant to the solution and ''N u'' has been consistently expanded in Adomian polynomials:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| N u &= -b \partial_{x} u^{2} = -b \partial_{x} (u_{0} + u_{1} + u_{2} + u_{3} + \cdots)(u_{0} + u_{1} + u_{2} + u_{3} + \cdots) \\
| |
| &= -b \partial_{x} (u_{0} u_{0} + 2 u_{0} u_{1} + u_{1} u_{1} + 2 u_{0} u_{2} + \cdots) \\
| |
| &= -b \partial_{x} \sum_{n=1}^{\infty} A(n-1),
| |
| \end{align}
| |
| </math>
| |
| | |
| where <math>A_{n} = \sum_{\nu=1}^{n} C(\nu, n) f^{(\nu)}(u_{0})</math> and ''f''(''u'') = ''u''<sup>2</sup> in the example (1).
| |
| | |
| Here ''C''(ν, ''n'') are products (or sum of products) of ν components of ''u'' whose subscripts sum up to ''n'', divided by the factorial of the number of repeated subscripts. It is only a thumb-rule to order systematically the decomposition to be sure that all the combinations appearing are utilized sooner or later.
| |
| | |
| The <math>\sum_{n=0}^{\infty} A_{n}</math> is equal to the sum of a generalized Taylor series about ''u''<sub>0</sub>.<ref name="Adomian94"/>
| |
| | |
| For the example (1) the Adomian polynomials are:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| A_{0} &= u_{0}^{2} \\
| |
| A_{1} &= 2 u_{0} u_{1} \\
| |
| A_{2} &= u_{1}^{2} + 2 u_{0} u_{2} \\
| |
| A_{3} &= 2 u_{1} u_{2} + 2 u_{0} u_{3} \\
| |
| & \cdots
| |
| \end{align}
| |
| </math>
| |
| | |
| Other possible choices are also possible for the expression of ''A''<sub>''n''</sub>.
| |
| | |
| ==== Series solutions ====
| |
| | |
| Cherruault established that the series terms obtained by Adomian's method approach zero as 1/(''mn'')! if ''m'' is the order of the highest linear differential operator and that <math>\lim_{n \to \infty} \varphi^{n} = u</math>.<ref>{{citation | last=Cherruault| first=Y. | title=Convergence of Adomian's Method| journal=Kybernetes | volume=18 | publisher=Emerald| location=Bingley, U.K.| year=1989 | pages=31–38}}</ref> With this method the solution can be found by systematically integrating along any of the two directions: in the ''x''-direction we would use expression (3); in the alternative ''y''-direction we would use the following expression:
| |
| | |
| :<math>
| |
| u = c(x) + d(x) y + L_{y}^{-1} \rho(x, y) - L_{y}^{-1} L_{x} u - L_{y}^{-1} N u
| |
| </math> | |
| | |
| where: ''c''(''x''), ''d''(''x'') is obtained from the boundary conditions at ''y'' = - ''y''<sub>''l''</sub> and ''y'' = ''y''<sub>''l''</sub>:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| u(y=-y_{l}) &= g_{1}(y) \\
| |
| u(y=y_{l}) &= g_{2}(y)
| |
| \end{align}
| |
| </math>
| |
| | |
| If we call the two respective solutions ''x-partial solution'' and ''y-partial solution'', one of the most interesting consequences of the method is that the ''x-partial solution'' uses only the two boundary conditions (1-a) and the ''y-partial solution'' uses only the conditions (1-b). | |
| | |
| Thus, one of the two sets of boundary functions {''f''<sub>1</sub>, ''f''<sub>2</sub>} or {''g''<sub>1</sub>, ''g''<sub>2</sub>} is redundant, and this implies that a partial differential equation with boundary conditions on a rectangle cannot have arbitrary boundary conditions on the borders, since the conditions at ''x'' = ''x''<sub>1</sub>, ''x'' = ''x''<sub>2</sub> must be consistent with those imposed at ''y'' = ''y''<sub>1</sub> and ''y'' = ''y''<sub>2</sub>.
| |
| | |
| An example to clarify this point is the solution of the Poisson problem with the following boundary conditions:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| u(x=0) &= f_{1}(y) = 0 \\
| |
| u(x=x_{l}) &= f_{2}(y) = 0
| |
| \end{align}
| |
| </math>
| |
| | |
| By using Adomian's method and a symbolic processor (such as [[Mathematica]] or [[Maple]]) it is easy to obtain the third order approximant to the solution. This approximant has an error lower than 5×10<sup>−16</sup> in any point, as it can be proved by substitution in the initial problem and by displaying the absolute value of the residual obtained as a function of (''x'', ''y'').<ref name="Garcia03a"/>
| |
| | |
| The solution at ''y'' = -0.25 and ''y'' = 0.25 is given by specific functions that in this case are:
| |
|
| |
| :<math>
| |
| g_{1}(x) = 0.0520833\, x -0.347222\, x^{3} + 9.25186 \times 10^{-17} x^{4} + 0.833333\, x^{5} -0.555556\, x^{6}
| |
| </math>
| |
| | |
| and ''g''<sub>2</sub>(''x'') = ''g''<sub>1</sub>(''x'') respectively.
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| If a (double) integration is now performed in the ''y''-direction using these two boundary functions the same solution will be obtained, which satisfy ''u''(''x''=0, ''y'') = 0 and ''u''(''x''=0.5, ''y'') = 0 and cannot satisfy any other condition on these borders.
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| Some people are surprised by these results; it seems strange that not all initial-boundary conditions must be explicitly used to solve a differential system. However, it is a well established fact that any [[Elliptic partial differential equation|elliptic equation]] has one and only one solution for any functional conditions in the four sides of a rectangle provided there is no discontinuity on the edges.
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| The cause of the misconception is that scientists and engineers normally think in a boundary condition in terms of [[weak convergence (Hilbert space)|weak convergence]] in a [[Hilbert space]] (the distance to the boundary function is small enough to practical purposes). In contrast, Cauchy problems impose a point-to-point convergence to a given boundary function and to all its derivatives (and this is a quite strong condition!). | |
| For the first ones, a function satisfies a boundary condition when the area (or another functional distance) between it and the true function imposed in the boundary is so small as desired; for the second ones, however, the function must tend to the true function imposed in any and every point of the interval.
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| The commented Poisson problem does not have a solution for any functional boundary conditions ''f''<sub>1</sub>, ''f''<sub>2</sub>, ''g''<sub>1</sub>, ''g''<sub>2</sub>; however, given ''f''<sub>1</sub>, ''f''<sub>2</sub> it is always possible to find boundary functions ''g''<sub>1</sub><sup>*</sup>, ''g''<sub>2</sub><sup>*</sup> so close to ''g''<sub>1</sub>, ''g''<sub>2</sub> as desired (in the weak convergence meaning) for which the problem has solution. This property makes it possible to solve Poisson's and many other problems with arbitrary boundary conditions but never for analytic functions exactly specified on the boundaries.
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| The reader can convince himself (herself) of the high sensitivity of PDE solutions to small changes in the boundary conditions by solving this problem integrating along the ''x''-direction, with boundary functions slightly different even though visually not distinguishable. For instance, the solution with the boundary conditions:
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| :<math>
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| f_{1,2}(y) = 0.00413682 - 0.0813801\, y^{2} + 0.260416\, y^{4} - 0.277778\, y^{6}
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| </math>
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| at ''x'' = 0 and ''x'' = 0.5, and the solution with the boundary conditions:
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| :<math>
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| \begin{align}
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| f_{1,2}(y) = 0.00413683 &- 0.00040048\, y - 0.0813802\, y^{2} + 0.0101279\, y^{3} + 0.260417\, y^{4} \\
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| &- 0.0694455\, y^{5} - 0.277778\, y^{6} + 0.15873\, y^{7} + \cdots
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| \end{align}
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| </math>
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| at ''x'' = 0 and ''x'' = 0.5, produce lateral functions with different sign convexity even though both functions are visually not distinguishable.
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| Solutions of elliptic problems and other partial differential equations are highly sensitive to small changes in the boundary function imposed when only two sides are used. And this sensitivity is not easily compatible with models that are supposed to represent real systems, which are described by means of measurements containing experimental errors and are normally expressed as initial-boundary value problems in a Hilbert space.
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| ==== Improvements to the decomposition method ====
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| At least three methods have been reported
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| <ref name="Garcia03a">{{citation | last=García-Olivares | first=A. | title=Analytic solution of partial differential equations with Adomian's decomposition| journal=Kybernetes | volume=32 | publisher=Emerald| location=Bingley, U.K.| year=2003 | pages=354–368}} [http://www.emeraldinsight.com/journals.htm?articleid=1454508&show=abstract]</ref>
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| <ref name="Garcia02">{{citation | last=García-Olivares | first=A. | title=Analytical approximants of time-dependent partial differential equations with tau methods| journal=Mathematics and Computers in Simulation | volume=61 | publisher=Elsevier| location=Amsterdam, Netherlands| year=2002 | pages=35–45}} [http://dx.doi.org/10.1016/S0378-4754(02)00133-7]</ref>
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| <ref name="Garcia03b">{{citation | last=García-Olivares | first=A. | title=Analytical solution of nonlinear partial differential equations of physics| journal=Kybernetes | volume=32 | publisher=Emerald| location=Bingley, U.K.| year=2003 | pages=548–560}} [DOI: 10.1108/03684920310463939] [http://www.emeraldinsight.com/journals.htm?articleid=876024&show=abstract]</ref>
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| to obtain the boundary functions ''g''<sub>1</sub><sup>*</sup>, ''g''<sub>2</sub><sup>*</sup> that are compatible with any lateral set of conditions {''f''<sub>1</sub>, ''f''<sub>2</sub>} imposed. This makes it possible to find the analytical solution of any PDE boundary problem on a closed rectangle with the required accuracy, so allowing to solve a wide range of problems that the standard Adomian's method was not able to address.
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| The first one perturbs the two boundary functions imposed at ''x'' = 0 and ''x'' = ''x''<sub>1</sub> (condition 1-a) with a ''N''<sup>th</sup>-order polynomial in ''y'': ''p''<sub>1</sub>, ''p''<sub>2</sub> in such a way that: ''f''<sub>1</sub>' = ''f''<sub>1</sub> + ''p''<sub>1</sub>, ''f''<sub>2</sub>' = ''f''<sub>2</sub> + ''p''<sub>2</sub>, where the norm of the two perturbation functions are smaller than the accuracy needed at the boundaries. These ''p''<sub>1</sub>, ''p''<sub>2</sub> depend on a set of polynomial coefficients ''c''<sub>''i''</sub>, ''i'' = 1, ..., ''N''. Then, the Adomian method is applied and functions are obtained at the four boundaries which depend on the set of ''c''<sub>''i''</sub>, ''i'' = 1, ..., ''N''. Finally, a boundary function ''F''(''c''<sub>1</sub>, ''c''<sub>2</sub>, ..., ''c''<sub>''N''</sub>) is defined as the sum of these four functions, and the distance between ''F''(''c''<sub>1</sub>, ''c''<sub>2</sub>, ..., ''c''<sub>''N''</sub>) and the real boundary functions ((1-a) and (1-b)) is minimized. The problem has been reduced, in this way, to the global minimization of the function ''F''(''c''<sub>1</sub>, ''c''<sub>2</sub>, ..., ''c''<sub>''N''</sub>) which has a global minimum for some combination of the parameters ''c''<sub>''i''</sub>, ''i'' = 1, ..., ''N''. This minimum may be found by means of a genetic algorithm or by using some other optimization method, as the one proposed by Cherruault (1999).<ref>{{cite book |title=Optimization, Méthodes locales et globales|first=Y.|last=Cherruault|publisher=Presses Universitaires de France|year=1999|isbn=2-13-049910-4}}</ref>
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| A second method to obtain analytic approximants of initial-boundary problems is to combine Adomian decomposition with spectral methods.<ref name="Garcia02"/>
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| Finally, the third method proposed by García-Olivares is based on imposing analytic solutions at the four boundaries, but modifying the original differential operator in such a way that it is different to the original one only in a narrow region close to the boundaries, and it forces the solution to satisfy exactly analytic conditions at the four boundaries.<ref name="Garcia03b"/>
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| ==Gallery==
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| [[File:Dym equation Adomian cos plot.gif|Adomian plot of [[Dym equation]]]]
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| [[File:Burgers Fisher equation tanh Adomian plot.gif|Adomian plot of Burgers-Fisher equation]]
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| [[File:Kuramoto-Sivashinsky equation Adomian solution sin plot.gif|Kuramoto-Sivashinsky equation Adomian solution sin plot]]
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| ==References==
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| <references />
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| [[Category:Differential equations]]
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