These equations are called Lagrange–Charpit equations. In interpreting these equations, it is convenient to allow zero denominators. For example, if Fp = 0, these equations require that dx = 0; that is, the denominator being zero just means that the numerator is also zero. Assume that from equations (1) and (2) one can derive a new equation (3) φ(x, y, u, p, q) = a, ∗Written for the course Math...

An extension of the contour integral method (CIM) for the computation of stress intensity factors for loaded cracks is presented. Numerical solutions of the two-dimensional linear elasticity equations are computed with the p-version of the generalized finite element method. Polynomial enrichment functions as well as enrichment with near crack tip asymptotic expansion are used. The robustness of...

We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukers’ double integral. The two-dimensional version of the saddle method, which we call C2-saddle method, is applied. 0. Introduction F. Beukers [1] has introduced the following double and triple integrals: ∫∫ S L(x)(1 − y) 1− xy dx dy and ∫∫∫ B L(x)L(y) 1− u(1− xy) du dx dy, (0.1) givi...

In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integral equation is obtained from the weak form of elastoplasticity based on Green-Naghdi’s theory over a local sub-domain, and the moving least-squares approximation is used for meshl...

Preface In our course we will consider the volume integral equations in the following form) () () () () () (x f dy y u y b y x y x K x u x a Q m = − − + ∫ , 3 ≤ m. Many important classes of the wave scattering problems can be described by equations of this form; for example, this is the case for problems of electromagnetic and acoustic scattering on 3D transparent bodies. The corresponding inte...

Implementations of the heat balance integral method are discussed in which exponential functions are used in place of the familiar polynomial approximants. The rationale is based upon that of least-squares in that the use of ‘appropriate’ basis functions can enhance solution accuracy. Whilst this is true in principle it is shown that considerable skill must be exercised when deviating from poly...

one of the main steps within the geoid computation methodology without applying the stokes formula is downward continuation of the harmonic residual observables from the surface of the earth down to the surface of the reference ellipsoid. this downward continuation is done via the abel-poisson integral and its derivatives. this integral in which the unknowns, i.e. harmonic residual potential va...

the aim of this work is to investigate the variational discretization and mixed finite element methods for optimal control problem governed by semi linear parabolic equations with integral constraint. the state and co-state are approximated by the lowest order raviart-thomas mixed finite element spaces and the control is not discreted. optimal error estimates in l2 are established for the state...

in this paper, we propose a successive approximation method based on fuzzy wavelet like operator to approximate the solution of linear fuzzy fredholm integral equations of the second kind with arbitrary kernels. we give the convergence conditions and an error estimate. also, we investigate the numerical stability of the computed values with respect to small perturbations in the first iteration....