Research Article High Order Projection Plane Method for Evaluation of Supersingular Curved Boundary Integrals in BEM
|
|
- Annabel Hampton
- 5 years ago
- Views:
Transcription
1 Mathematical Problems in Engineering Volume 06, Article ID , 8 pages Research Article High Order Projection Plane Method for Evaluation of Supersingular Curved Boundary Integrals in BEM Miao Cui, Wei-zhe Feng, Xiao-wei Gao, and Kai Yang School of Aeronautics and Astronautics, Faculty of Vehicle Engineering and Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 604, China Correspondence should be addressed to Kai Yang; kyang@dlut.edu.cn Received 7 December 05; Revised February 06; Accepted 7 February 06 Academic Editor: Ivano Benedetti Copyright 06 Miao Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Boundary element method (BEM) is a very promising approach for solving various engineering problems, in which accurate evaluation of boundary integrals is required. In the present work, the direct method for evaluating singular curved boundary integrals is developed by considering the third-order derivatives in the projection plane method when expanding the geometry quantities at the field point as Taylor series. New analytical formulas are derived for geometry quantities defined on the curved line/plane, and unified expressions are obtained for both two-dimensional and three-dimensional problems. For the twodimensional boundary integrals, analytical expressions for the third-order derivatives are derived and are employed to verify the complex-variable-differentiation method (CVDM) which is used to evaluate the high order derivatives for three-dimensional problems. A few numerical examples are given to show the effectiveness and the accuracy of the present method.. Introduction Boundary element method (BEM) [] is a very promising approach for solving various engineering problems, such as heat transfer problems [ 5] in energy science and engineering, in which accurate evaluation of boundary integrals [6 0] is required. Weak, strong, or hypersingularities are involved in these boundary integrals, if the source point is located on the element under integration []. These issues have been focused on by researchers, and many techniques, such as Gauss logarithmic quadrature rule for weak singularities [], rigid body displacement algorithm for strong singularities [], and regularization for hypersingularities [3], have been proposed to remove the singularities. Each method has its advantage and disadvantage [ 9]. A comprehensive review has been given in [] on the treatment of singularities. In general, the efficient methods for singular curved line and surfaceintegralsarethosebasedontheoperationoverthe intrinsic coordinate system [4 6]. Recently, Gao et al. [] proposed another efficient direct method for numerically evaluating all kinds of singular curved boundary integrals for two-dimensional (D) and three-dimensional (3D) BEM analyses, and numerical examples were given to demonstrate the stability of the proposed method. However, only quadratic terms were truncated when expanding the coordinate of the field point as Taylor series, which means the method will be invalid if the curved boundary is beyond the quadratic, such as the cubed curve. In the present work, the direct method in [] for evaluating arbitrary high order singular curved boundary integrals is developed, by considering the third-order derivatives when expanding the geometry quantities at the field point as Taylor series. Corresponding analytical expressions are derived. The first innovation is that the present method will extend the application range of the method in []. That is, the singular integrals over higher order elements, such as cubed curve, can be directly evaluated. Another innovation is that the complex-variable-differentiation method [0] is introduced to determine the third-order derivatives. For D boundary integral, the third-order derivatives are analytically derived, which are used to verify the CVDM. The paper is organized as follows. In Section, review and modification of the direct method in [] are provided. In Section 3, numerical examples are given to show the
2 Mathematical Problems in Engineering effectiveness and the accuracy of the present method. Finally, conclusions are drawn from computational results and discussions. x x p Q ρ P. Development of the Direct Method for Evaluating Arbitrary High Order Singular Curved Boundary Integrals ξ Q r P.. Review of Singular Integrals []. Singular integrals over an element can be classified into the following form: I e (x p )= Γ e f(x p, x) r β (x p, x) dγ (x) () in which x p and x represent the source and field points, respectively; Γ e is the boundary element under integration, which is a curved line or a curved surface for D or 3D problems, respectively; and r is the distance between the source and field points. The term f(x p, x) in () is a regular function and β is the order of the singularity. In D problems, the following type of singular line integrals is also frequently encountered: J e (x p )= Γ e f(x p, x) ln [r (x p, x)] dγ (x). () r β (x p, x) x x Figure : The curved line boundary and the projection line. In D problems, the boundary element is a curved line as shown in Figure. The local orthogonal coordinate system is denoted by (x,x ) and the projection line is along the axis x. There is only one independent variable for a line, and all other geometry quantities can be expressed by x.similarto the treatment in [], the coordinate x at the field point over thecurvedlinecanbeexpressedbyx using Taylor series. Different from the treatment in [], the third-order derivative is considered in the present work: It is assumed that the integrals in () and () always exist: that is, each integration has a finite value... Review of Expansions of Geometry Quantities on the Projection Line/Plane []. In [], a projection line for D or a projection plane for 3D problems, which is the tangential line/plane of the element to the origin of the intrinsic coordinate system, was introduced. A local orthogonal coordinate system was established on the projection line/plane. The coordinate transformation between the local and global systems can be performed using the following relationships: x i =L ij (x j x o j ), (3) x i =x o i +L ji x j, (4) L ij is the direction cosine of the local coordinate axes with respect to the global one. The repeated subscripts represent summation; x o i is the global coordinates of the origin of the local coordinate system. The original curved element can be projected onto the projection line or plane to form a straight or a flat projection element by (3), and then all geometry quantities can be expressed in terms of variables defined on the projection line/plane. x = x x x + x x x x x 3 x 3 =a (x )x +a (x )x +a 3 (x )x 3, a (x )= x x = x x, a (x )= x x = x a 3 (x )= 3 x 6 x 3 = 6 3 x 3 + x 3 ξ. x 3 = 6 x x x x x x ( x ) x +3 x + x x ξ x ξ, x (5) (6).3. High Order Expansion of Geometry Quantities in terms of Variables on the Projection Line for D Problems. For D and 3D problems, derivations for the modification are different, which are separately described in detail. It can be seen that the expressions are more complex. For D problem, the corresponding derivatives are analytically derived, and the detailed derivation process is omitted here.
3 Mathematical Problems in Engineering 3 In (5) and (6), x Nnode = N α α= x α, (7a) p(x )=a x p +a (x )x p x p (4) +a 3 (x )x p x p x p =x p, x N node = α= 3 x N node 3 = α= N α x α, (7b) 3 N n 3 x α, (7c) x = m, (8a) ξ x = m 4 ( x x + x x ), (8b) 3 ξ x 3 = x ( ξ x )= m B 4C. (8c) m 7 In (7a) (8c), m = ( x ) +( x ), (9) B=( x ) +( x ) + x 3 x 3 + x 3 x 3, C=( x ) ( x ) + x x x x +( x ) ( x ). Define a local distance: (0a) (0b) ρ= x x p, () x p is the local coordinate of point P which is the projection point of the source point P and x is the local coordinate of point Q which is the projection point of the field point Q. The derivative of ρ with respect to local coordinate x is In (a), ρ, =or. Then, we can obtain ρ, = ρ x = x x p. (a) ρ x =x p +ρ,ρ. (b) By substituting (a) into (5), we can obtain x =p(x )+q(x,ρ)ρ, (3) In (5), q (x ) q(x,ρ)=q (x )+q (x )ρ+q 3 (x )ρ. (5) =[a (x )+a (x )x p +3a 3 (x )x p ]ρ,, (6a) q (x )=a (x )ρ, +3a 3 (x )x p (6b) =a (x )+3a 3 (x )x p, q 3 (x,ρ)=a 3 (x )ρ,. (6c) Similarly, we can express the intrinsic coordinate in terms of the local distance as ξ=p(ξ) +q(ξ, ρ) ρ, (7) p(ξ) and q(ξ, ρ) can be determined by replacing x with ξ in (3) (6c). Substituting (b) and (3) into (4), we can obtain x i =x o i +L ji x j =xo i +L i x +L ix =x o i +L i (x p +ρ,ρ) + L i [x p +q(x,ρ)ρ] =x p i +L i ρ, ρ+l i q(x,ρ)ρ. Then, substituting (5) into (8), we can obtain (8) x i =x p i +(b i +c i ρ+d i ρ )ρ, (9) b i =L i ρ, +L i q (y ), c i =L i q (y ), d i =L i q 3 (y ). (0a) (0b) (0c) Referring to Figure, the global distance r canbeeasily expressed as r= (x i x p i )(x i x p i ) = g (ρ) ρ, () g(ρ)= (b i +c i ρ+d i ρ )(b i +c i ρ+d i ρ ) = G 0 +G ρ+g ρ +G 3 ρ 3 +G 4 ρ 4. () In (), G 0 =b i b i, G =b i c i, G =c i c i +b i d i, G 3 =c i d i,and G 4 =d i d i. Itcanbeeasilyderivedthat r,i = r = x i x p i x i r = b i +c i ρ+d i ρ. (3) g(ρ)
4 4 Mathematical Problems in Engineering.4. The Complex-Variable-Differentiation Method (CVDM). From the above derivations, it can be seen that the corresponding expressions have been changed, by considering the third-order derivatives when expanding the geometry quantities at the field point as Taylor series. The third-order derivatives should be first determined. In the present work, the complex-variable-differentiation method (CVDM) [0] is introduced to determine the high order derivatives. For D analysis, the third-order derivatives are determined explicitly by (7c), (8c), (9), (0a), and (0b), which are also calculated by the CVDM. In the CVDM, the variable x of a real function f(x) is replaced by a complex one x+ih, with the imaginary part h being very small (usually h =0 0 ). The function f(x + ih) can be expanded into Taylor s series as x C C B B x x 3 o n 0 A A D D f (x+ih) =f(x) +ihf (x) h f (x) +o(h 3 ). (4) Since h issmallenough,(5)canbeobtained: f (x) = Im (f (x+ih)), (5) h Im denotes the imaginary part. The CVDM is a very promising method, since the derivatives only require direct evaluations []. The CVDM has been validated to be an efficient and accurate approach for determination of derivatives [ 4]. For example, considering 3 x / x 3 = ( / x )( x / x ), we can add the very small imaginary part h to x ; then the real value of 3 x / x 3 yields Im( x / x )/h. In addition, the value of x / x can be easily obtained using the analytical expressions in the previous work []. The CVDM and the derived expressions in (7c) (8c) can be validatedbyeachother..5. Expansion of Geometry Quantities in terms of Variables ontheprojectionplanefor3dproblems. In 3D problems, the boundary element is a curved surface marked as ABCD in Figure. A local orthogonal coordinate system (x,x,x 3 )is established on the projection plane with its origin being at the point (ξ =0, ξ =0), in which axes x and x are located within the plane. The axis x is along ξ direction and axis x 3 is along the outward normal direction to the element. By (4), wecanprojecttheoriginalcurvedsurfaceelementontothe projection plane, to form a flat projection element marked as A B C D in Figure. In a plane, there are only two independent variables. We choose x and x as the independent variables, and x 3 over the curved surface can be expressed in terms of x and x. Compared with the treatment in [], x 3 is expanded as Taylor series by x and x, by considering the third-order derivatives: x 3 =a I (x 3 )x I +a IJ (x 3 )x I x J +a IJK (x 3 )x I x J x K, (6) Figure : The curved surface boundary and the projection plane. I, J,andK are integers varying from to, and a I (x 3 )= x 3 x I a IJ (x 3 )= x 3 xi x J = x 3 K, (7a) K x I = x 3 K L K L xi xj + x 3 K a IJK (x 3 )= 3 x 3 6 xi = a IJ (x 3 ) x J x K 3 xk = a IJ (x 3 ) L. 3 L x K ξ K xi, x J (7b) (7c) The treatment is the same as that in [], to determine a I (x 3 ) in (7a) and a IJ(x 3 ) in (7b). For a IJK(x 3 ) in (7c), we use the CVDM, to avoid rather complicated analytical derivations. The local distance ρ projected from the global distance r ontotheprojectionplaneanditsderivativeρ,i are introduced as follows: as ρ= (x x p ) +(x x p ), ρ,i = ρ x I = x I x p I. ρ (8) Thelocalcoordinatesatthefieldpointcanbeexpressed x I =x p I +ρ,i ρ. (9) Similar to the D analysis, substituting (9) into (6) yields x 3 =p(x 3 )+q(x 3,ρ)ρ, (30)
5 Mathematical Problems in Engineering 5 In (3), p(x 3 )=a I (x 3 )x p I +a IJ (x 3 )x p I x p J +a IJK (x 3 )x p I x p J x p K =x p 3, (3) q(x 3,ρ)=q (x 3 )+q (x 3 )ρ+q 3 (x 3 )ρ. (3) q (x 3 )=a I (x 3 )ρ,i +a IJ (x 3 )x p I ρ,j +a IJ (x 3 )x p J ρ,i +a IJK (x 3 )x p I x p K ρ,j +a IJK (x 3 )x p J x p K ρ,i (33a) In (39), G 0 =b i b i, G =b i c i, G =c i c i +b i d i, G 3 =c i d i,and G 4 =d i d i. Itcanbeeasilyderivedthatr,i has the same expression as that in D analysis. Considering that I=J=K=and ρ, =or for D problems, it can be seen that unified expressions are obtained for both two-dimensional and three-dimensional problems. Then, we can handle the singular boundary integrals in () and (), with the above modifications. The D problem is taken as an example to describe the procedure to determine the integral in (). The differential relationship between the localdistanceandtherealcurvedlinecanbewrittenas dρ = dγ cos φ=dγn 0 i n i, (40) +a IJK (x 3 )x p I x p J ρ,k, q (x 3 )=a IJ (x 3 )ρ,iρ,j +a IJK (x 3 )x p K ρ,iρ,j +a IJK (x 3 )x p I ρ,jρ,k +a IJK (x 3 )x p J ρ,iρ,k, q 3 (x 3 )=a IJK (x 3 )ρ,iρ,j ρ,k. (33b) (33c) n 0 i and n i are the outward normals to the tangential linesofpassingtheoriginofthelocalcoordinatesystemand the field point, respectively, and φ is the angle between the two outward normals. Substituting (40) and (38) into () leads to I e (x p )=lim ρ E ε 0 ε F(ρ) dρ, (4) ρβ Similar to the D analysis, intrinsic coordinates can be expressed as ξ K =p(ξ K )+q(ξ K,ρ)ρ, (34) p(ξ K ) and q(ξ K,ρ)can be determined by replacing x 3 with ξ K in (3) (33c). Substituting (9) and (30) into (4) yields x i =x o i +L Ji x J +L 3ix 3 =x o i +L Ji (x p J =x p i +L Ji ρ,j ρ+l 3i q(x 3,ρ)ρ. +ρ,jρ) + L 3i [x p 3 +q(x 3,ρ)ρ] Substituting (3) into (35), we can obtain (35) x i =x p i +(b i +c i ρ+d i ρ )ρ, (36) b i =L Ji ρ,j +L 3i q (x 3 ), c i =L 3i q (x 3 ), d i =L 3i q 3 (x 3 ). The expression of r is as follows: (37a) (37b) (37c) r= (x i x p i )(x i x p i ) = g (ρ) ρ, (38) g(ρ)= (b i +c i ρ+d i ρ )(b i +c i ρ+d i ρ ) = G 0 +G ρ+g ρ +G 3 ρ 3 +G 4 ρ 4. (39) ρ E isthedistancefromtheprojectedsourcepointp to one of the ends of the projected line element and F(ρ) is the regular part: that is, F(ρ)= f(xp, x (ρ)) g β (ρ) n 0 i n i (ρ). (4) In order to integrate (4), the nonsingular part F is expanded as a power series in ρ,suchthat F(ρ)= N n=0 B (n) ρ n (43) in which N is the order of the power series, usually taking a value between and 7 depending on the size of ρ E,andB (n) are constants which are determined by collocating N+points over the integration region (0, ρ E ).Inthepaper,N+equally spaced points are used: that is, (0, ρ,...,ρ N ).Thecoefficient for the first point (n = 0) is B (0) = F(0) and other coefficients can be solved using the following equation set: [R] B = Y, (44) [R] is a square matrix with the order of N: ρ ρ N ρ ρ N [R] = [... (45). ] [ ρ N ρ N N ]
6 6 Mathematical Problems in Engineering B and Yare vectors as follows: B = B B. B N, [F(ρ ) B (0) ] ρ [F(ρ ) B (0) ] Y = ρ... [F(ρ N ) B (0) ] ρ N (46) Solving (44) for coefficient vector B and then substituting (43) into (4) yield E n I e (x p )= N n=0 ρ E ε 0 ε B (n) lim ρ n β dρ = N n=0 B (n) E n, (47) n β+ ( ρ lim β n ρ = E ε 0ρ ) (n =β ) β n ε ln ρ E lim ln ρ ε (n = β ). ρε 0 Then, singularities can be analytically removed. 3. Numerical Examples (48) BasedontheFortrancodeforsingularintegralsin[],anew Fortrancodehasbeendevelopedwiththeabovemodification presented in this paper. It should be emphasized that the functions of the old Fortran code are fully included in the new Fortran code. If all the third-order derivatives are set to bezero,thefunctionsarethesamewiththoseofthemethod in []. Besides, the new code has its unique characteristics. For D analysis, the third-order derivatives are computed by both CVDM and analytical expression. For 3D analysis, the high order derivatives are computed by the CVDM, because analytical derivations are very difficult and complicatedtobeobtained. In this section, the first example is given to show the advantageofcvdmoveradifferencemethod.then,two singular integrals are calculated, to show the effectiveness and the accuracy of the present method. 3.. The Advantage of the CVDM. A singular function is selected: f (x, x p ) = (x x p ). (49) Table : The first-order derivatives with different h for f(x, x p )= /(x x p ). h CVDM Difference method.00e E E E E E E E E E E E E E E E E E E E E E E E E E E + 00 Then, the exact derivative with respect to the variable x p can be obtained: f x p = (x x p ) 3. (50) For comparison, the central difference formulation is adopted: that is, f x p = f(x,xp +h) f(x,x p h). (5) h When x =.00, x p =, the exact first-order derivative is 0 9. The dependence of the derivatives on h for the CVDM and the difference method is examined, and the results are listed in Table. It can be seen that the difference method can give satisfactory results only for h between 0 5 and 0 0.For smaller h, its result becomes unacceptable. However, the result of the CVDM tends to be independent of h as the step size becomes sufficiently small. This clearly verifies the advantage of the CVDM. 3.. A Weakly Singular Line Integral over a Cubed Curve. In this example, a singular line integral for a D BEM problem is chosen: J= ln (r) dγ. (5) Γ The integration line is a cubed curve as shown in Figure 3. The four typical points are (, ), (, ), 3 ( /3, /7), and 4(/3, /7). The computational point (sourcepoint)isat.thecurveistakenasoneboundary element. It should be emphasized here that the direct method in [] has been invalid, which cannot be used for the singular line integration in (5) over the cubed curve in Figure 4, with only one element, because only quadratic terms were truncated when expanding the geometry quantities at the field point as Taylor series. Firstly, the accuracy of the CVDM for the proposed technique is validated. We use the derived analytical expressions in (7c) (8c) to compute the third-order derivatives. Then, the CVDM is employed to determine the same derivatives.
7 Mathematical Problems in Engineering 7 Table : Results for two points a and b (β =3). Point a Point b Method in [4] Present method z Figure 3: The cubed curve for integration. a b The boundary integral is as follows: I(x p )= Γ 4πr [3r r β,3 n n 3]dΓ. (54) Two source points, a and b, are located at positions with intrinsic coordinates of (0, 0) and (0.66, 0). Guiggiani et al. [4] computed this integral for the case of β=3. Table lists the results using both methods. For the present method, the Gauss order is 4. From Table, it can be seen that he results using the present method agree well with the Guiggiani et al. s results, which indicate the high accuracy of the present method for singular integrals over a highly curved surface. Figure 4: The cylindrical surface. Using the derived analytical expressions in this paper, the third-order derivatives can be obtained: y 3 x x 3 = , 3 ξ x 3 = x (53) TheresultsoftheCVDMareexactlythesamewiththat in (53). It can be seen that the CVDM and the analytical expressions in (7c) (8c) are validated by each other. Secondly, the singular integral is evaluated using the present method. The result is , with the Gauss order being 6. This integral is also evaluated using the Gauss quadrature rule for logarithmic function []. The result is J = It can be seen that the present method is effective and with high accuracy, for higher order curved boundary A Supersingular Integral over a Curved Surface. In this example, the accuracy of the present method in handling singular integrals over a highly curved surface is examined. The boundary element is a cylindrical surface with radius and the rotational angle is 90. The geometrical dimensions are shown in Figure Conclusion and Future Work Inthepresentwork,highorderprojectionplanemethod forevaluationofsupersingularcurvedboundaryintegrals in BEM has been developed, by considering the thirdorder derivatives when expanding the geometry quantities at the field point as Taylor series. Corresponding analytical expressions are derived, and unified expressions are obtained for both two-dimensional and three-dimensional problems. The present method extends the application range of the method in []. That is, the singular integrals over higher order elements, such as cubed curve, can be directly evaluated. The complex-variable-differentiation method is introduced to determine the third-order derivatives, which effectively avoids rather complicated analytical derivations for the three-dimensional analysis. For the two-dimensional boundary integral, the third-order derivatives are analytically derived and calculated, which are used to verify the complexvariable-differentiation method. The results show that the present method is effective and with high accuracy for high order singular curved boundary integrals. Future work is to apply the mathematical derivations and equations to evaluate the singularities involved in radial integration boundary element method in solving nonhomogeneous thermoelastic problems. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments The authors gratefully acknowledge the National Natural Science Foundation of China ( and 57075) and
8 8 Mathematical Problems in Engineering the Fundamental Research Funds for the Central Universities (DUT4LK03). References [] C. F. Rowlatt and T. N. Phillips, A spectral element formulation of the immersed boundary method for Newtonian fluids, Computer Methods in Applied Mechanics and Engineering, vol. 98, pp. 9 57, 06. []H.Qi,L.M.Ruan,H.C.Zhang,Y.M.Wang,andH.P.Tan, Inverse radiation analysis of a one-dimensional participating slab by stochastic particle swarm optimizer algorithm, International Thermal Sciences,vol.46,no.7,pp , 007. [3] S.Zhang,Y.Zhu,B.Qiao,andZ.Zhang, Systemmodelofheat andmasstransferprocessformobilesolventvaporphasedrying equipment, Mathematical Problems in Engineering, vol.04, Article ID 6776, pages, 04. [4] T. Lu, W. W. Han, P. X. Jiang, Y. H. Zhu, J. Wu, and C. L. Liu, A two-dimensional inverse heat conduction problem for simultaneous estimation of heat convection coefficient, fluid temperature and wall temperature on the inner wall of a pipeline, Progress in Nuclear Energy,vol.8,pp.6 68,05. [5] H.Qi,Y.B.Qiao,S.C.Sun,Y.Yao,andL.Ruan, Imagereconstruction of two-dimensional highly scattering inhomogeneous medium using MAP-based estimation, Mathematical Problems in Engineering, vol. 05, Article ID 435, 9 pages, 05. [6] J. Sladek, V. Sladek, and S. N. Atluri, Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties, Computational Mechanics,vol.4,no.6,pp ,000. [7] J. T. Chen and H.-K. Hong, Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series, Applied Mechanics Reviews, vol. 5, no., pp. 7 33, 999. [8]Ch.Zhang,M.Cui,J.Wang,X.W.Gao,J.Sladek,andV. Sladek, 3D crack analysis in functionally graded materials, Engineering Fracture Mechanics,vol.78,no.3,pp ,0. [9] Y. Ochiai, Three-dimensional thermo-elastoplastic analysis by tripe-reciprocity boundary element method, Engineering Analysis with Boundary Elements,vol.35,pp ,0. [0] J. Bremer and Z. Gimbutas, On the numerical evaluation of the singular integrals of scattering theory, Computational Physics,vol.5,pp ,03. [] X.-W. Gao, W.-Z. Feng, K. Yang, and M. Cui, Projection plane method for evaluation of arbitrary high order singular boundary integrals, Engineering Analysis with Boundary Elements, vol. 50, pp , 05. [] J. L. Tsalamengas, Quadrature rules for weakly singular, strongly singular, and hypersingular integrals in boundary integral equation methods, Computational Physics, vol. 303, pp , 05. [3] V.SladekandJ.Sladek,Singular Integrals in Boundary Element Methods, WIT Press, Southampton, UK, 998. [4] M. Guiggiani, G. Krishnasamy, T. J. Rudolphi, and F. J. Rizzo, A general algorithm for the numerical solution of hypersingular boundary integral equations, Applied Mechanics,vol. 59,no.3,pp ,99. [5] P. Bettini, M. F. Palumbo, and R. Specogna, A boundary element method for eddy-current problems in fusion devices, Fusion Engineering and Design, vol , pp , 05. [6] A. Frangi and M. Guiggiani, Direct approach for boundary integral equations with high-order singularities, International Journal for Numerical Methods in Engineering,vol.49,no.7,pp , 000. [7] S. Mukherjee and Y. X. Mukherjee, The hypersingular boundary contour method for three-dimensional linear elasticity, Applied Mechanics, vol. 65, no., pp , 998. [8] W. Chen, J.-Y. Zhang, and Z.-J. Fu, Singular boundary method for modified Helmholtz equations, Engineering Analysis with Boundary Elements,vol.44,pp. 9,04. [9] M. H. Bazyar and A. Talebi, Scaled boundary finite-element method for solving non-homogeneous anisotropic heat conduction problems, Applied Mathematical Modelling, vol. 39, no. 3-4, pp , 05. [0] J. N. Lyness and C. B. Moler, Numerical differentiation of analytic functions, SIAMJournalonNumericalAnalysis,vol. 4, pp. 0 0, 967. [] M. Cui, X. Gao, and H. Chen, Inverse radiation analysis in an absorbing, emitting and non-gray participating medium, International Thermal Sciences,vol.50,no.6,pp , 0. [] M. Cui, X. W. Gao, and H. G. Chen, A new approach for the equivalent gray radiative property of a non-gray medium using a modified zonal method and the complex-variabledifferentiation method, Quantitative Spectroscopy and Radiative Transfer,vol.,no.8,pp ,0. [3] M. Cui, Q. H. Zhu, and X. W. Gao, A modified conjugate gradient method for transient nonlinear inverse heat conduction problems: a case study for identifying temperature-dependent thermal conductivities, ASME Heat Transfer,vol.36, no.9,articleid0930,04. [4] M. Cui, W.-W. Duan, and X.-W. Gao, A new inverse analysis method based on a relaxation factor optimization technique for solving transient nonlinear inverse heat conduction problems, International Heat and Mass Transfer,vol.90,pp , 05.
9 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization
Research Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components
Applied Mathematics Volume 202, Article ID 689820, 3 pages doi:0.55/202/689820 Research Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components
More informationResearch Article Bessel Equation in the Semiunbounded Interval x [x 0, ]: Solving in the Neighbourhood of an Irregular Singular Point
International Mathematics and Mathematical Sciences Volume 2016, Article ID 6826482, 7 pages http://dx.doi.org/10.1155/2016/6826482 Research Article Bessel Equation in the Semiunbounded Interval x [x 0,
More informationResearch Article Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation
International Scholarly Research Network ISRN Mathematical Analysis Volume 2012 Article ID 384906 10 pages doi:10.5402/2012/384906 Research Article Two Different Classes of Wronskian Conditions to a 3
More informationDepartment of Structural, Faculty of Civil Engineering, Architecture and Urban Design, State University of Campinas, Brazil
Blucher Mechanical Engineering Proceedings May 2014, vol. 1, num. 1 www.proceedings.blucher.com.br/evento/10wccm A SIMPLIFIED FORMULATION FOR STRESS AND TRACTION BOUNDARY IN- TEGRAL EQUATIONS USING THE
More informationResearch Article A New Global Optimization Algorithm for Solving Generalized Geometric Programming
Mathematical Problems in Engineering Volume 2010, Article ID 346965, 12 pages doi:10.1155/2010/346965 Research Article A New Global Optimization Algorithm for Solving Generalized Geometric Programming
More informationResearch Article The Characteristic Solutions to the V-Notch Plane Problem of Anisotropy and the Associated Finite Element Method
Mathematical Problems in Engineering Volume 2013, Article ID 593640, 11 pages http://dx.doi.org/10.1155/2013/593640 Research Article The Characteristic Solutions to the V-Notch Plane Problem of Anisotropy
More informationCalculation of Potential Second Derivatives by Means of a Regularized Indirect Algorithm in the Boundary Element Method
Copyright 2010 Tech Science Press CMES, vol.69, no.1, pp.19-42, 2010 Calculation of Potential Second Derivatives by Means of a Regularized Indirect Algorithm in the Boundary Element Method H.B. Chen 1
More informationResearch Article New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients
Abstract and Applied Analysis Volume 2010, Article ID 564068, 11 pages doi:10.1155/2010/564068 Research Article New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive
More informationResearch Article An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem
Abstract and Applied Analysis, Article ID 60813, 5 pages http://dx.doi.org/10.1155/014/60813 Research Article An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem Luoyi
More informationResearch Article Adaptive Control of Chaos in Chua s Circuit
Mathematical Problems in Engineering Volume 2011, Article ID 620946, 14 pages doi:10.1155/2011/620946 Research Article Adaptive Control of Chaos in Chua s Circuit Weiping Guo and Diantong Liu Institute
More informationResearch Article New Examples of Einstein Metrics in Dimension Four
International Mathematics and Mathematical Sciences Volume 2010, Article ID 716035, 9 pages doi:10.1155/2010/716035 Research Article New Examples of Einstein Metrics in Dimension Four Ryad Ghanam Department
More informationResearch Article Existence of Periodic Positive Solutions for Abstract Difference Equations
Discrete Dynamics in Nature and Society Volume 2011, Article ID 870164, 7 pages doi:10.1155/2011/870164 Research Article Existence of Periodic Positive Solutions for Abstract Difference Equations Shugui
More informationTransactions on Modelling and Simulation vol 8, 1994 WIT Press, ISSN X
Analysis of singular and near-singular integrals by the continuation approach D. Rosen & D.E. Cormack Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto M5S 1A4, Canada
More informationResearch Article Soliton Solutions for the Wick-Type Stochastic KP Equation
Abstract and Applied Analysis Volume 212, Article ID 327682, 9 pages doi:1.1155/212/327682 Research Article Soliton Solutions for the Wick-Type Stochastic KP Equation Y. F. Guo, 1, 2 L. M. Ling, 2 and
More informationResearch Article A PLS-Based Weighted Artificial Neural Network Approach for Alpha Radioactivity Prediction inside Contaminated Pipes
Mathematical Problems in Engineering, Article ID 517605, 5 pages http://dxdoiorg/101155/2014/517605 Research Article A PLS-Based Weighted Artificial Neural Network Approach for Alpha Radioactivity Prediction
More informationKingSaudBinAbdulazizUniversityforHealthScience,Riyadh11481,SaudiArabia. Correspondence should be addressed to Raghib Abu-Saris;
Chaos Volume 26, Article ID 49252, 7 pages http://dx.doi.org/.55/26/49252 Research Article On Matrix Projective Synchronization and Inverse Matrix Projective Synchronization for Different and Identical
More informationResearch Article L-Stable Derivative-Free Error-Corrected Trapezoidal Rule for Burgers Equation with Inconsistent Initial and Boundary Conditions
International Mathematics and Mathematical Sciences Volume 212, Article ID 82197, 13 pages doi:1.1155/212/82197 Research Article L-Stable Derivative-Free Error-Corrected Trapezoidal Rule for Burgers Equation
More informationResearch Article Dynamic Carrying Capacity Analysis of Double-Row Four-Point Contact Ball Slewing Bearing
Mathematical Problems in Engineering Volume 215, Article ID 8598, 7 pages http://dx.doi.org/1.1155/215/8598 Research Article Dynamic Carrying Capacity Analysis of Double-Row Four-Point Contact Ball Slewing
More informationResearch Article Modified T-F Function Method for Finding Global Minimizer on Unconstrained Optimization
Mathematical Problems in Engineering Volume 2010, Article ID 602831, 11 pages doi:10.1155/2010/602831 Research Article Modified T-F Function Method for Finding Global Minimizer on Unconstrained Optimization
More informationResearch Article Uniqueness Theorems on Difference Monomials of Entire Functions
Abstract and Applied Analysis Volume 202, Article ID 40735, 8 pages doi:0.55/202/40735 Research Article Uniqueness Theorems on Difference Monomials of Entire Functions Gang Wang, Deng-li Han, 2 and Zhi-Tao
More informationTHE TRACTION BOUNDARY CONTOUR METHOD FOR LINEAR ELASTICITY
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 46, 1883}1895 (1999) THE TRACTION BOUNDARY CONTOUR METHOD FOR LINEAR ELASTICITY ZHOU SHENJIE*, CAO ZHIYUAN AND SUN
More informationResearch Article The Dirichlet Problem on the Upper Half-Space
Abstract and Applied Analysis Volume 2012, Article ID 203096, 5 pages doi:10.1155/2012/203096 Research Article The Dirichlet Problem on the Upper Half-Space Jinjin Huang 1 and Lei Qiao 2 1 Department of
More informationResearch Article Circle-Uniqueness of Pythagorean Orthogonality in Normed Linear Spaces
Function Spaces, Article ID 634842, 4 pages http://dx.doi.org/10.1155/2014/634842 Research Article Circle-Uniqueness of Pythagorean Orthogonality in Normed Linear Spaces Senlin Wu, Xinjian Dong, and Dan
More informationResearch Article Normal and Osculating Planes of Δ-Regular Curves
Abstract and Applied Analysis Volume 2010, Article ID 923916, 8 pages doi:10.1155/2010/923916 Research Article Normal and Osculating Planes of Δ-Regular Curves Sibel Paşalı Atmaca Matematik Bölümü, Fen-Edebiyat
More informationResearch Article Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients
Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 606149, 15 pages doi:10.1155/2010/606149 Research Article Frequent Oscillatory Behavior of Delay Partial Difference
More informationResearch Article Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation
Applied Mathematics Volume 22, Article ID 39876, 9 pages doi:.55/22/39876 Research Article Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation Xiuming Li
More informationResearch Article Solving the Matrix Nearness Problem in the Maximum Norm by Applying a Projection and Contraction Method
Advances in Operations Research Volume 01, Article ID 357954, 15 pages doi:10.1155/01/357954 Research Article Solving the Matrix Nearness Problem in the Maximum Norm by Applying a Projection and Contraction
More informationResearch Article An Optimized Grey GM(2,1) Model and Forecasting of Highway Subgrade Settlement
Mathematical Problems in Engineering Volume 015, Article ID 606707, 6 pages http://dx.doi.org/10.1155/015/606707 Research Article An Optimized Grey GM(,1) Model and Forecasting of Highway Subgrade Settlement
More informationResearch Article Constrained Solutions of a System of Matrix Equations
Journal of Applied Mathematics Volume 2012, Article ID 471573, 19 pages doi:10.1155/2012/471573 Research Article Constrained Solutions of a System of Matrix Equations Qing-Wen Wang 1 and Juan Yu 1, 2 1
More informationResearch Article Some Generalizations and Modifications of Iterative Methods for Solving Large Sparse Symmetric Indefinite Linear Systems
Abstract and Applied Analysis Article ID 237808 pages http://dxdoiorg/055/204/237808 Research Article Some Generalizations and Modifications of Iterative Methods for Solving Large Sparse Symmetric Indefinite
More informationResearch Article Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem
Abstract and Applied Analysis Volume 01, Article ID 958040, 11 pages doi:10.1155/01/958040 Research Article Self-Adaptive and Relaxed Self-Adaptive Proection Methods for Solving the Multiple-Set Split
More informationResearch Article Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance
Advances in Decision Sciences Volume, Article ID 893497, 6 pages doi:.55//893497 Research Article Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance Junichi Hirukawa and
More informationResearch Article A Generalization of a Class of Matrices: Analytic Inverse and Determinant
Advances in Numerical Analysis Volume 2011, Article ID 593548, 6 pages doi:10.1155/2011/593548 Research Article A Generalization of a Class of Matrices: Analytic Inverse and Determinant F. N. Valvi Department
More informationResearch Article On New Wilker-Type Inequalities
International Scholarly Research Network ISRN Mathematical Analysis Volume 2011, Article ID 681702, 7 pages doi:10.5402/2011/681702 Research Article On New Wilker-Type Inequalities Zhengjie Sun and Ling
More informationThe iterated sinh transformation
The iterated sinh transformation Author Elliott, David, Johnston, Peter Published 2008 Journal Title International Journal for Numerical Methods in Engineering DOI https://doi.org/10.1002/nme.2244 Copyright
More informationResearch Article Propagation Characteristics of Oblique Incident Terahertz Wave in Nonuniform Dusty Plasma
Antennas and Propagation Volume 216, Article ID 945473, 6 pages http://dx.doi.org/1.1155/216/945473 Research Article Propagation Characteristics of Oblique Incident Terahert Wave in Nonuniform Dusty Plasma
More informationResearch Article A New Roper-Suffridge Extension Operator on a Reinhardt Domain
Abstract and Applied Analysis Volume 2011, Article ID 865496, 14 pages doi:10.1155/2011/865496 Research Article A New Roper-Suffridge Extension Operator on a Reinhardt Domain Jianfei Wang and Cailing Gao
More informationResearch Article A New Fractional Integral Inequality with Singularity and Its Application
Abstract and Applied Analysis Volume 212, Article ID 93798, 12 pages doi:1.1155/212/93798 Research Article A New Fractional Integral Inequality with Singularity and Its Application Qiong-Xiang Kong 1 and
More informationResearch Article An Inverse Eigenvalue Problem for Jacobi Matrices
Mathematical Problems in Engineering Volume 2011 Article ID 571781 11 pages doi:10.1155/2011/571781 Research Article An Inverse Eigenvalue Problem for Jacobi Matrices Zhengsheng Wang 1 and Baoiang Zhong
More informationResearch Article Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming
Discrete Dynamics in Nature and Society Volume 20, Article ID 7697, pages doi:0.55/20/7697 Research Article Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming Wei-Xiang
More informationResearch Article A Note about the General Meromorphic Solutions of the Fisher Equation
Mathematical Problems in Engineering, Article ID 793834, 4 pages http://dx.doi.org/0.55/204/793834 Research Article A Note about the General Meromorphic Solutions of the Fisher Equation Jian-ming Qi, Qiu-hui
More informationResearch Article A Novel Filled Function Method for Nonlinear Equations
Applied Mathematics, Article ID 24759, 7 pages http://dx.doi.org/0.55/204/24759 Research Article A Novel Filled Function Method for Nonlinear Equations Liuyang Yuan and Qiuhua Tang 2 College of Sciences,
More informationResearch Article A Note on the Discrete Spectrum of Gaussian Wells (I): The Ground State Energy in One Dimension
Mathematical Physics Volume 016, Article ID 15769, 4 pages http://dx.doi.org/10.1155/016/15769 Research Article A Note on the Discrete Spectrum of Gaussian Wells (I: The Ground State Energy in One Dimension
More informationResearch Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation
International Differential Equations Volume 2010, Article ID 764738, 8 pages doi:10.1155/2010/764738 Research Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation
More informationResearch Article Data-Driven Fault Diagnosis Method for Power Transformers Using Modified Kriging Model
Hindawi Mathematical Problems in Engineering Volume 2017, Article ID 3068548, 5 pages https://doi.org/10.1155/2017/3068548 Research Article Data-Driven Fault Diagnosis Method for Power Transformers Using
More informationResearch Article Unicity of Entire Functions concerning Shifts and Difference Operators
Abstract and Applied Analysis Volume 204, Article ID 38090, 5 pages http://dx.doi.org/0.55/204/38090 Research Article Unicity of Entire Functions concerning Shifts and Difference Operators Dan Liu, Degui
More informationResearch Article Variational Iteration Method for the Magnetohydrodynamic Flow over a Nonlinear Stretching Sheet
Abstract and Applied Analysis Volume 213, Article ID 573782, 5 pages http://dx.doi.org/1.1155/213/573782 Research Article Variational Iteration Method for the Magnetohydrodynamic Flow over a Nonlinear
More informationResearch Article Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler Foundation Using Homotopy Perturbation Method
Mathematical Problems in Engineering Volume 213, Article ID 721294, 9 pages http://dx.doi.org/1.1155/213/721294 Research Article Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler
More informationResearch Article Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices
Hindawi Advances in Mathematical Physics Volume 07, Article ID 680, 7 pages https://doi.org/0.55/07/680 Research Article Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices
More informationResearch Article The Approximate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels
Applied Mathematics, Article ID 72327, 7 pages http://ddoiorg/055/204/72327 Research Article The Approimate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels Qinghua Wu School
More informationResearch Article Strong Convergence of a Projected Gradient Method
Applied Mathematics Volume 2012, Article ID 410137, 10 pages doi:10.1155/2012/410137 Research Article Strong Convergence of a Projected Gradient Method Shunhou Fan and Yonghong Yao Department of Mathematics,
More informationResearch Article SPH Simulation of Acoustic Waves: Effects of Frequency, Sound Pressure, and Particle Spacing
Mathematical Problems in Engineering Volume 215, Article ID 348314, 7 pages http://dx.doi.org/.1155/215/348314 Research Article SPH Simulation of Acoustic Waves: Effects of Frequency, Sound Pressure, and
More informationResearch Article Quasilinearization Technique for Φ-Laplacian Type Equations
International Mathematics and Mathematical Sciences Volume 0, Article ID 975760, pages doi:0.55/0/975760 Research Article Quasilinearization Technique for Φ-Laplacian Type Equations Inara Yermachenko and
More informationResearch Article An Analysis of the Quality of Repeated Plate Load Tests Using the Harmony Search Algorithm
Applied Mathematics, Article ID 486, 5 pages http://dxdoiorg/55/4/486 Research Article An Analysis of the Quality of Repeated Plate Load Tests Using the Harmony Search Algorithm Kook-Hwan Cho andsunghomun
More informationResearch Article Partial Pole Placement in LMI Region
Control Science and Engineering Article ID 84128 5 pages http://dxdoiorg/11155/214/84128 Research Article Partial Pole Placement in LMI Region Liuli Ou 1 Shaobo Han 2 Yongji Wang 1 Shuai Dong 1 and Lei
More informationThe Evaluation Of early Singular Integrals In The Direct Regularized Boundary Element Method
The Evaluation Of early Singular Integrals In The Direct Regularized Boundary Element Method * School of Science Shandong university of technology Zhangzhou Road #, Zibo, Shandong, 55049 China Zhengbin60@6.com
More information2 Basic Equations in Generalized Plane Strain
Boundary integral equations for plane orthotropic bodies and exterior regions G. Szeidl and J. Dudra University of Miskolc, Department of Mechanics 3515 Miskolc-Egyetemváros, Hungary Abstract Assuming
More informationResearch Article On the System of Diophantine Equations
e Scientific World Journal, Article ID 63617, 4 pages http://dx.doi.org/1.1155/14/63617 Research Article On the System of Diophantine Equations x 6y = 5and x=az b Silan Zhang, 1, Jianhua Chen, 1 and Hao
More informationResearch Article Assessment of Haar Wavelet-Quasilinearization Technique in Heat Convection-Radiation Equations
Applied Computational Intelligence and So Computing, Article ID 45423, 5 pages http://.doi.org/.55/24/45423 Research Article Assessment of Haar Wavelet-Quasilinearization Technique in Heat Convection-Radiation
More informationResearch Article A Matrix Method Based on the Fibonacci Polynomials to the Generalized Pantograph Equations with Functional Arguments
Advances in Mathematical Physics, Article ID 694580, 5 pages http://dx.doi.org/10.1155/2014/694580 Research Article A Matrix Method Based on the Fibonacci Polynomials to the Generalized Pantograph Equations
More informationResearch Article Extended Precise Large Deviations of Random Sums in the Presence of END Structure and Consistent Variation
Applied Mathematics Volume 2012, Article ID 436531, 12 pages doi:10.1155/2012/436531 Research Article Extended Precise Large Deviations of Random Sums in the Presence of END Structure and Consistent Variation
More informationSolving Poisson Equation within Local Fractional Derivative Operators
vol. (207), Article ID 0253, 2 pages doi:0.3/207/0253 AgiAl Publishing House http://www.agialpress.com/ Research Article Solving Poisson Equation within Local Fractional Derivative Operators Hassan Kamil
More informationJournal of Computational Science and Technology
Science and Technology Explicit Evaluation of Hypersingular Boundary Integral Equation for 3-D Helmholtz Equation Discretized with Constant Triangular Element Toshiro MATSUMOTO, Changjun ZHENG,, Shinya
More informationResearch Article The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem
Journal of Applied Mathematics Volume 2012, Article ID 219478, 15 pages doi:10.1155/2012/219478 Research Article The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity
More informationResearch Article Nonlinear Conjugate Gradient Methods with Wolfe Type Line Search
Abstract and Applied Analysis Volume 013, Article ID 74815, 5 pages http://dx.doi.org/10.1155/013/74815 Research Article Nonlinear Conjugate Gradient Methods with Wolfe Type Line Search Yuan-Yuan Chen
More informationResearch Article Solvability of a Class of Integral Inclusions
Abstract and Applied Analysis Volume 212, Article ID 21327, 12 pages doi:1.1155/212/21327 Research Article Solvability of a Class of Integral Inclusions Ying Chen and Shihuang Hong Institute of Applied
More information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More informationResearch Article Existence and Duality of Generalized ε-vector Equilibrium Problems
Applied Mathematics Volume 2012, Article ID 674512, 13 pages doi:10.1155/2012/674512 Research Article Existence and Duality of Generalized ε-vector Equilibrium Problems Hong-Yong Fu, Bin Dan, and Xiang-Yu
More informationA note on the unique solution of linear complementarity problem
COMPUTATIONAL SCIENCE SHORT COMMUNICATION A note on the unique solution of linear complementarity problem Cui-Xia Li 1 and Shi-Liang Wu 1 * Received: 13 June 2016 Accepted: 14 November 2016 First Published:
More informationResearch Article A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
Mathematical Problems in Engineering Volume 1, Article ID 693453, 1 pages doi:11155/1/693453 Research Article A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
More informationResearch Article Multiple-Decision Procedures for Testing the Homogeneity of Mean for k Exponential Distributions
Discrete Dynamics in Nature and Society, Article ID 70074, 5 pages http://dx.doi.org/0.55/204/70074 Research Article Multiple-Decision Procedures for Testing the Homogeneity of Mean for Exponential Distributions
More informationResearch Article Remarks on Asymptotic Centers and Fixed Points
Abstract and Applied Analysis Volume 2010, Article ID 247402, 5 pages doi:10.1155/2010/247402 Research Article Remarks on Asymptotic Centers and Fixed Points A. Kaewkhao 1 and K. Sokhuma 2 1 Department
More informationResearch Article Completing a 2 2Block Matrix of Real Quaternions with a Partial Specified Inverse
Applied Mathematics Volume 0, Article ID 7978, 5 pages http://dx.doi.org/0.55/0/7978 Research Article Completing a Block Matrix of Real Quaternions with a Partial Specified Inverse Yong Lin, and Qing-Wen
More informationResearch Article Some Results on Characterizations of Matrix Partial Orderings
Applied Mathematics, Article ID 408457, 6 pages http://dx.doi.org/10.1155/2014/408457 Research Article Some Results on Characterizations of Matrix Partial Orderings Hongxing Wang and Jin Xu Department
More informationA SIMILARITY SOLUTION OF FIN EQUATION WITH VARIABLE THERMAL CONDUCTIVITY AND HEAT TRANSFER COEFFICIENT
Mathematical and Computational Applications, Vol. 11, No. 1, pp. 25-30, 2006. Association for Scientific Research A SIMILARITY SOLUTION OF FIN EQUATION WITH VARIABLE THERMAL CONDUCTIVITY AND HEAT TRANSFER
More informationResearch Article Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent
International Differential Equations Volume 2015, Article ID 494907, 4 pages http://dx.doi.org/10.1155/2015/494907 Research Article Existence for Elliptic Equation Involving Decaying Cylindrical Potentials
More informationResearch Article On a Quasi-Neutral Approximation to the Incompressible Euler Equations
Applied Mathematics Volume 2012, Article ID 957185, 8 pages doi:10.1155/2012/957185 Research Article On a Quasi-Neutral Approximation to the Incompressible Euler Equations Jianwei Yang and Zhitao Zhuang
More informationTransactions on Modelling and Simulation vol 12, 1996 WIT Press, ISSN X
Simplifying integration for logarithmic singularities R.N.L. Smith Department ofapplied Mathematics & OR, Cranfield University, RMCS, Shrivenham, Swindon, Wiltshire SN6 SLA, UK Introduction Any implementation
More informationResearch Article Semi-Online Scheduling on Two Machines with GoS Levels and Partial Information of Processing Time
e Scientific World Journal Volume 04, Article ID 57634, 6 pages http://dx.doi.org/0.55/04/57634 Research Article Semi-Online Scheduling on Two Machines with GoS Levels and Partial Information of Processing
More informationMoving screw dislocations in piezoelectric bimaterials
phys stat sol (b) 38 No 1 10 16 (003) / DOI 10100/pssb00301805 Moving screw dislocations in piezoelectric bimaterials Xiang-Fa Wu *1 Yuris A Dzenis 1 and Wen-Sheng Zou 1 Department of Engineering Mechanics
More informationResearch Article A Necessary Characteristic Equation of Diffusion Processes Having Gaussian Marginals
Abstract and Applied Analysis Volume 01, Article ID 598590, 9 pages doi:10.1155/01/598590 Research Article A Necessary Characteristic Equation of Diffusion Processes Having Gaussian Marginals Syeda Rabab
More informationResearch Article Opinion Impact Models and Opinion Consensus Methods in Ad Hoc Tactical Social Networks
Discrete Dynamics in ature and Society Volume 203, Article ID 758079, 6 pages http://dx.doi.org/0.55/203/758079 Research Article Opinion Impact Models and Opinion Consensus Methods in Ad Hoc Tactical Social
More informationResearch Article Some New Fixed-Point Theorems for a (ψ, φ)-pair Meir-Keeler-Type Set-Valued Contraction Map in Complete Metric Spaces
Applied Mathematics Volume 2011, Article ID 327941, 11 pages doi:10.1155/2011/327941 Research Article Some New Fixed-Point Theorems for a (ψ, φ)-pair Meir-Keeler-Type Set-Valued Contraction Map in Complete
More informationResearch Article Fixed Point Theorems of Quasicontractions on Cone Metric Spaces with Banach Algebras
Abstract and Applied Analysis Volume 2013, Article ID 187348, 5 pages http://dx.doi.org/10.1155/2013/187348 Research Article Fixed Point Theorems of Quasicontractions on Cone Metric Spaces with Banach
More informationResearch Article Statistical Tests for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Probability and Statistics Volume 2015, Article ID 723924, 5 pages http://dx.doi.org/10.1155/2015/723924 Research Article Statistical Tests for the Reciprocal of a Normal Mean with a Known Coefficient
More informationResearch Article Translative Packing of Unit Squares into Squares
International Mathematics and Mathematical Sciences Volume 01, Article ID 61301, 7 pages doi:10.1155/01/61301 Research Article Translative Packing of Unit Squares into Squares Janusz Januszewski Institute
More informationResearch Article Hermite Wavelet Method for Fractional Delay Differential Equations
Difference Equations, Article ID 359093, 8 pages http://dx.doi.org/0.55/04/359093 Research Article Hermite Wavelet Method for Fractional Delay Differential Equations Umer Saeed and Mujeeb ur Rehman School
More informationResearch Article The Numerical Solution of Problems in Calculus of Variation Using B-Spline Collocation Method
Applied Mathematics Volume 2012, Article ID 605741, 10 pages doi:10.1155/2012/605741 Research Article The Numerical Solution of Problems in Calculus of Variation Using B-Spline Collocation Method M. Zarebnia
More informationResearch Article Approximation of Analytic Functions by Bessel s Functions of Fractional Order
Abstract and Applied Analysis Volume 20, Article ID 923269, 3 pages doi:0.55/20/923269 Research Article Approximation of Analytic Functions by Bessel s Functions of Fractional Order Soon-Mo Jung Mathematics
More informationResearch Article An Unbiased Two-Parameter Estimation with Prior Information in Linear Regression Model
e Scientific World Journal, Article ID 206943, 8 pages http://dx.doi.org/10.1155/2014/206943 Research Article An Unbiased Two-Parameter Estimation with Prior Information in Linear Regression Model Jibo
More informationResearch Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations
Abstract and Applied Analysis Volume 212, Article ID 391918, 11 pages doi:1.1155/212/391918 Research Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations Chuanjun Chen
More informationAnalysis of thin acoustic barriers over an
Analysis of thin acoustic barriers over an impedance plane using a dual BEM formulation L.A. de Lacerda*\ L.C. Wrobel*, WJ. Mansur^ "Wessex Institute of Technology, Ashurst, Southampton, UK ^COPPE/URRJ,
More informationResearch Article Dynamic Time Warping Distance Method for Similarity Test of Multipoint Ground Motion Field
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2, Article ID 74957, 2 pages doi:.55/2/74957 Research Article Dynamic Time Warping Distance Method for Similarity Test of Multipoint
More informationResearch Article Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to F p, q, s Space on the Unit Ball
Abstract and Applied Analysis Volume 2011, Article ID 152635, 9 pages doi:10.1155/2011/152635 Research Article Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to F p,
More informationResearch Article Design of PDC Controllers by Matrix Reversibility for Synchronization of Yin and Yang Chaotic Takagi-Sugeno Fuzzy Henon Maps
Abstract and Applied Analysis Volume 212, Article ID 35821, 11 pages doi:1.1155/212/35821 Research Article Design of PDC Controllers by Matrix Reversibility for Synchronization of Yin and Yang Chaotic
More informationResearch Article Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential Equations
Applied Mathematics Volume 2012, Article ID 615303, 13 pages doi:10.1155/2012/615303 Research Article Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential
More informationResearch Article The Characterization of the Variational Minimizers for Spatial Restricted N+1-Body Problems
Abstract and Applied Analysis Volume 23, Article ID 845795, 5 pages http://dx.doi.org/.55/23/845795 Research Article he Characterization of the Variational Minimizers for Spatial Restricted +-Body Problems
More informationResearch Article Chaos Control on a Duopoly Game with Homogeneous Strategy
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 16, Article ID 74185, 7 pages http://dx.doi.org/1.1155/16/74185 Publication Year 16 Research Article Chaos Control on a Duopoly
More informationResearch Article Local Fractional Variational Iteration Method for Inhomogeneous Helmholtz Equation within Local Fractional Derivative Operator
Mathematical Problems in Engineering, Article ID 9322, 7 pages http://d.doi.org/.55/24/9322 Research Article Local Fractional Variational Iteration Method for Inhomogeneous Helmholtz Equation within Local
More informationResearch Article Taylor s Expansion Revisited: A General Formula for the Remainder
International Mathematics and Mathematical Sciences Volume 2012, Article ID 645736, 5 pages doi:10.1155/2012/645736 Research Article Taylor s Expansion Revisited: A General Formula for the Remainder José
More information