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the series solution. On selecting an appropriate value of β , the reader is referred to Liao [ 39 ]. In that work, Liao presents a HAM solution to the Blasius flow problem and finds admissible values of β to be β > β c , where β ≈ 2.5. For computational purposes, we set β = 3 and minimize the error function (26) to obtain the optimal value of ℏ . We consider the following three sets of values for the parameters: Mn = 15, C = 0.001, λ 1 = 0.3, γ = 1.0, and δ = –2.0, Mn = 20, C = 0.01, λ 1 = 0.2, γ = 0.5, and δ = –0.5, Mn = 10, C = 0

), a ∈ (0, 1), it follows f ( ax 1 + (1 − a ) x 2 ) < af ( x 1 ) + (1 − a ) f ( x 2 ).If the function is continuous and twice differentiable i.e. C 2 on the open interval ( a , b ), then this function is concave on ( a , b ) if: ∀ x ∈ ( a , b ), f ″( x ) < 0. Or a C 2 function: g : A → R n on the open and convex set A ⊂ R n is concave if and only if ∂ 2 f ( x ) < 0 and is semidefinite for all x , then f is strictly concave. In the literature of this king very important term is marginal cost pricing equilibrium which is a family of

application for Caputo fractional derivatives only [ 25 ]. His modifications are of minor importance and did not increased overall accuracy capabilities of the method. Maximum of available computational accuracy of fractional order derivatives and integrals calculations applying formulas ( 5 ) and ( 6 ) and presently available methods of numerical integration is presented in the tables 1 - 3 . Table 1 D (1/2) f ( t ), f ( t ) = t , t ∈ (0,1), relative error in % N GL NCm Diet Odiba 8 8.11 10.68 0.0004 0.0024 15 4.25 7.81 0.0004 0.0023 21 3.02 6.6 0.0004 0.0023 61 1.03

will be considered later. 2.1 Bayesian Network Model Let us consider a finite set X = { X 1 , X 2 … X n of discrete random variables and P a joint probability distribution over the variables in X . A Bayesian network (BN) is a directed acyclic graph that encodes a joint probability distribution over the set of random variables X . Formally, a BN for X is a pair B = ( G , Θ) where G is the graph whose vertexes are the random variables X 1 , X 2 … X n , and the Θ represents the set of parameters that qualify the network. The graph encodes

Solutions of (2+1)-Dimensional Time Fractional Coupled Burger Equations, Modelling and Simulation in Engineering, vol. 2014. Ray S.S. 2014 A New Coupled Fractional Reduced Differential Transform Method for the Numerical Solutions of (2+1)-Dimensional Time Fractional Coupled Burger Equations Modelling and Simulation in Engineering 2014 [25] Sasso, M., Palmieri, G., & Amodio, D. (2011). Application of fractional derivative models in linear viscoelastic problems. Mechanics of Time-Dependent Materials, 15(4), 367-387. Sasso M. Palmieri G. Amodio D. 2011 Application of

a formal first integral of the form H = y 2 + F ( x , y ), where F starts with terms of order higher than two. (b) If the system has a local analytic first integral defined at the origin, then it has a local analytic first integral of the form H = y 2 + F ( x , y ), where F starts with terms of order higher than two. (c) If X = yf ( x , y 2 ) and Y = g ( x , y 2 ), then the system has a local analytic first integral of the form H = y 2 + F ( x , y ), where F starts with terms of order higher than two. We note that statement (b) provides a tool

is the new value of pressure obtained in the iteration. The load balance condition can be achieved by modifying the rigid film thickness H 0 as follows. H ¯ 0 = H ~ 0 + c 2 [ G Δ − Δ π ∑ j = 1 N − 1 ( P j + P j + 1 ) ] , $$\begin{array}{} \displaystyle \overline{H}_0=\widetilde{H}_0+c_2\left[G^\Delta-\frac{\Delta}{\pi}\sum_{j=1}^{N-1}(P_j+P_{j+1})\right], \end{array} $$ where c 2 is the relaxation factor and G Δ is the nondimensional load on the coarsest grid. Following procedure is followed for the determination of pressure cavitation point [ 25 ]. Let X c

and only if there exists an integer m ≥ 0 and a causal continuous function g : ℝ → X with polynomial growth at infinity such that h = g ( m ) , with differentiation understood in the sense of tempered distributions. Moreover, if H : ℂ + → X ℂ is a holomorphic function satisfying (24) and (25) (with the conditions given for ψ ), then H = 𝓛{ h } for some h ∈ TD( X ). If h ∈ TD(ℝ) and a ∈ X , then the tensor product a ⊗ h defines a distribution in TD( X ). If X and Y are Banach spaces and h ∈ TD(𝓑( X , Y )), then the

1 σ ′ , π 2 σ ′ and π 1s govern the solution pattern of the problem. Table 3 Verification of the dimensionless groups for the extended Davis and Raymond model with both non-constant c v and 1+e and variable dz. Case K o (m/yr) e o I c σ ′ o (N/m 2 ) H o (m) σ ′ f (N/m 2 ) c vo (m 2 /yr) τ o, σ ′ (yr) τ o,s (yr) π 1 σ ′ π 2 σ ′ π 1s 01 0.02 1.5 0.45 30000 1 60000 0.783 0.4941 0.4328 0.967 2.0 0.847 02 0.04 1.5 0.45 15000 1 30000 0.783 0.4941 0.4328 0.967 2.0 0.847 03 0.02 0.25 0.1125 30000 1 60000 1.566 0.4941 0.4328 0.967 2.0 0.847 04 0.04 1.5 0

the indicators in the two cases: Fig. 4 GDP simulation value(One hundred million yuan) Fig. 5 Energy consumption simulation value(10,000 tons of standard coal) 3.3 Analysis of simulation results The simulation of situation 1 shows that in 2030, the GDP will reach 211.046 billion yuan, the amount of GDP growth will be 8.775 billion yuan, the industrial output value will be 25.732 billion yuan, the air pollution index will be 94.4925, the industrial solid waste pollution index will be 102.921, and the urban sewage pollution index will be 101.578. In this situation