Saad A. Manaa

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<strong>Saad</strong> A. <strong>Manaa</strong>∞u= limϕn=∑u i(15)n→∞i=03. The Adomian decomposition method applied to Newell-Whitehead modelThe nonlinear wave equation with dissipation and nonlinear transport termis given as: [15], [16]Where are distinct real numbers and , are constants.For, Eq. (16) reduces to the nonlinear reaction- diffusion formfor different choices of the parameters a1, a2, a3 Eq. (17) reduces to the well knownnonlinear reaction diffusion equations appearing in many different branches ofsciences: when we get the Newell-Whitehead equation [20]∂u2= ∆u+ u(1− u )∂t( t,x)∈(0,∞)× Ωwith the initial and boundary conditionsu( x,0)= u0( x),x∈ Ωu(x,t)= 0, t > 0 , x = 0 and∂u= 0∂xat x = 0 andx = L.x = L(16)(17)(18)(18a)which arises after carrying out a suitable normalization in the study of thermalconvection of a fluid heated from below. Considering the Perturbation from a stationarystate, the equation describes the evolution of the amplitude of the vertical velocity if thisis a slowly varying function of time t and position x.The equation (18) written in an operator formLu= u + u − N(u)(19)txxwhere ∂3L t= and N ( u)= u is nonlinear term. Applying the inverse operator to the∂ tequation (19) and using the initial data (18a) yields−1−1−1u(x,t)= u ( x)+ L u + L u − L N()(20)0 xxuThe ADM suggests the solution u( x,t)componentsbe decomposed by infinite series of= ∑ ∞ u ( x,t)un(x,t)(21)n=0and the nonlinear operator N(u) by the infinite series of the Adomian polynomials= ∑ ∞ N ( u)A n(22)n =0The first four components of Adomain polynomials according to (13) read174
An Approximate Solution to The Newell-Whitehead Equation by …From eq. (21) and eq. (22), the iterates are determined by the following recursive way:u0= u0(x)(24)−1−1un= L ( ∆un−1+ un−1)− L ( An−1),n ≥ 1.The decomposition method provides a reliable technique that requires less work ifcompared with traditional techniques.4. Application and Numerical ResultsTo give a clear overview of the methodology, the following example will bediscussed. All the results are calculated by using the MATLAB 7.4 software.Consider the Newell-Whitehead equation [11]∂u= ∆u+ u(1− u∂twith the initial conditions2)(0..5( 2a) x)( −0.5(2a) x)a c1e − c2eu ( x,0)= −(25)(0.5( 2a) x)( −0.5(2a) x)b ( c e + c e + 1)And exact Solutionu(x,t)= −ab1( c e1c1e(0.5( 2a) x)2(0..5( 2a) x)+ c2e− c2e( −0.5(2a) x)( −0.5(2a) x)+ c3e( −3 / 2) a tSubstitution the initial condition eq. (25) into eq. (24) and using eq. (23) tocalculate the Adomian polynomials, yields the following recursive relation)(23)u0= −ab( cce(0..5( 2a) x)1(0.5( 2a) x)1eWe will takeseries are given by:− c e+ c( −0.5(2a) x)2( −0.5(2a) x)2e+ 1)(26). The first few terms of the decomposition(27)175
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Saad A. Manaa

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