An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

The material in this thesis is the result of a year's experience in the solution of problems on the Caltech Electric Analog Computer. Although much work has been done elsewhere, the solution of partial differential equations is a relatively new field for the Caltech Computer. It is natural that such an undertaking should initiate points of view and techniques that differ from those of other investigators., This thesis contains the development of certain ideas that have been useful to the author in the solution of problems on the computer. In Part I, finite difference methods are treated with reference to problems with one space variable. Techniques are developed for the representation of differential operators by means of electrical networks and the question of unequal lumping is discussed. The solutions of the fourth order differential equations of a beam are treated in Part II. Such practical considerations as the effects of parasitic impedances and cell size are investigated. Solutions are presented for the normal modes of a cantilever beam, the transient vibration of a cantilever beam, and the coupled modes of vibration of an airplane wing. Problems involving the scalar Laplacian operator are treated in Part III. A general asymmetric network is developed that is useful for problems with irregular boundaries and for problems where it is desired to have variable cell size. These techniques are illustrated with respect to a cavity resonator problem and an electromagnetic radiation problem. Elastic plate problems are treated in Part IV. The analogy for the elastic plate is an extension of the beam analogy to two dimensions. Here the difficult problems are those relating to the representation of boundary conditions, particularly of boundary conditions along an irregular edge. Some conclusions regarding the construction of a network analyzer designed specifically for the solution of partial differential equations are given in Part V. The chief conclusion is that such a computer must contain a much larger number of electrical elements than are at present available in the Caltech Computer.

In this article, we present a numerical simulation of one-dimensional problem of quasi-static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ2 + h2), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

A comparison of differential quadrature methods for the solution of partial differential equations

Several research papers experimentally validated the derived theoretical solution of the partial differential equation (PDE) of Transient Land Drainage (TLD) using laboratory sand tanks, Hele-Shaw models, resistance networks, and field data. In this research, the main objective is to use an Arduino microcontroller linked with the LabVIEW software to derive experimentally-by using an RC analog circuit connected to the Arduino microcontroller - a solution to Kirchhoff’s partial differential equation describing the unsteady electrical flow through the RC circuit. The RC circuit electrically simulates a non-dimensional virtual aquifer for which an experimental Fourier sine series solution in terms of the analog components is to be derived and used for comparison with the theoretical Fourier sine series solution of the transient partial differential equation of land drainage. The derived Fourier sine series solution of the partial differential equation governing the unsteady electrical flow through the RC analog is written in a non-dimensional form to enable its general application to the case study of this research. Isochrones of the water table drawdown based on those derived from the analog solution are similar in trend and behavior to those of the theoretical solution, indicating that both Fourier solutions are physically and mathematically identical.

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in C 1 C^1 wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of C 1 C^1 quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct C 1 C^1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.

In this paper, the strategies and tests that guarantee a reliable solution of partial differential equations (PDE) solved with the finite difference method are proposed. The elements to be considered are presented so that the solution of equations in partial derivatives is the closest to reality. The experience of the proponents and previous work play an important role in validating simulations in science and engineering problems. A series of tests are proposed that must be carried out to validate and make the simulation results reliable, due to the fact that the selection of the finite difference scheme, the initial data, the boundary conditions and the restrictions in the time step, the stability and convergence of the solution cannot be guaranteed a priori. There are situations in which the solution is stable but converges to values ??that have no physical meaning. To illustrate the strategies and tests inherent in a simulation, such that the solutions can be trusted, the 3D wave equation (three spatial dimensions and one temporal dimension) is solved numerically, by implementing a code in FORTRAN 90, implementing a scheme finite difference to second order approximation. The stability and convergence of the solution are studied according to the schemes proposed in this research, in such a way that the results are reliable, and guarantee the possible practical implementation of the solutions. Finally, the evolution of the scalar field for different times is shown, validated with the principle of conservation of energy. The strategies implemented in this work to guarantee the reliability of the simulations are also valid when solving EDP problems with neural networks, genetic algorithms and other intelligent computing techniques.

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.

On the numerical solution of partial differential equations with two spatial scales

Numerical convective schemes based on accurate computation of space derivatives

The problem of checking existence of infinitely differentiable solutions of linear partial differential equations with zero boundary conditions is considered. The coefficients of the equations are assumed to be polynomials over Z in independent variables. It is proved that this problem is algorithmically undecidable. This result extends results of our earlier studies of analytic solutions. The proof relies on the result obtained by Denef and Lipshitz concerning the relationship between a certain subset of nonhomogeneous differential equations of the considered form (but without boundary conditions) and Diophantine equations.

L iSS — An environment for the parallel multigrid solution of partial differential equations on general 2D domains

Solution of the partial differential equation of the Hodgkin-Huxley model using differential quadrature

A curvelet method for numerical solution of partial differential equations