This note is concerned with a semi-analytical method for the solution of 2-D Helmholtz equation in unit square. The method uses orthogonal functions to project the problem down to finite dimensional space. After the projection, the problem simplifies to that of obtaining solutions for second order constant coefficient differential equations which can be done analytically. Numerical results indi...

We present a method to obtain approximate polynomial solutions for a class of second order nonlinear differential equations arising in heat transfer problems. We apply this method to find a approximate polynomial solution for the nonlinear temperature distribution equation of a thick rectangular fin. Key-Words: Nonlinear differential equation; boundary value problem; ε -approximate analytical a...

Analyses of dynamic responses are significantly important for the design, maintenance and rehabilitation of asphalt pavement. In order to evaluate the dynamic responses of asphalt pavement under moving loads, a specific computational program, SAFEM, was developed based on a semi-analytical finite element method. This method is three-dimensional and only requires a two-dimensional FE discretizat...

We use an approximation method to study the solution to a nonlinear heat conduction equation in a semi-infinite domain. By expanding an energy density function defined as the internal energy per unit volume as a Taylor polynomial in a spatial domain, we reduce the partial differential equation to a set of first-order ordinary differential equations in time. We describe a systematic approach to ...

It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. ...

LetM denote the maximal function along the polynomial curve (γ1t, . . . , γdt ): M( f )(x) = sup r>0 1 2r ∫ |t|≤r | f (x1 − γ1t, . . . , xd − γdt )|dt. We show that the L norm of this operator grows atmost logarithmically with the parameter d: ‖M f ‖L2(Rd) ≤ c log d ‖ f ‖L2(Rd), where c > 0 is an absolute constant. The proof depends on the explicit construction of a “parabolic” semi-group of op...