A numerical technique based on hybrid of radial Basis functions including Guassians (GAs) and Multiquadrics (MQs) is proposed to obtain the solution of nonlinear Fredholm integral equations. Zeros of the shifted Legendre polynomials are used as the collocation points. The integral involved in the formulation of the problems are approximated based on Legendre-Gauss-Lobatto integration rule. Some numerical examples illustrate the accuracy and validity of the proposed method.

In this paper a meshless method using exponential Basis functions is developed for fluid-structure interaction in liquid tanks undergoing non-linear sloshing. The formulation in the fluid part is based on the use of Navier-Stokes equations, presented in Lagrangian description as Laplacian of the pressure, for inviscid incompressible fluids. The use of exponential Basis functions satisfying the Laplace equation leads to a strong form of volume preservation which has a direct effect on the accuracy of the pressure field. In a boundary node style, the bases are used to incrementally solve the fluid part in space and time. The elastic structure is discretized by the finite elements and analyzed by the Newmark method. The direct use of the pressure, as the 䐀 䐀 ential of the acceleration, helps to find the loads acting on the structure in a straight-forward manner. The interaction equations are derived and used in the analysis of a tank with elastic walls. The overall formulation may be implemented simply. To demonstrate the efficiency of the solution, the obtained results are compared with those obtained from a finite elements solution using Lagrangian description. The results show that while the wave height and the oscillations of elastic walls of the two analyses are in good agreement with each other; the use of the proposed meshless analysis not only leads to accurate hydrodynamic pressure but also reduces the computational time to one-eighth of the time needed for the finite e☮ leKmeeynwtso radnsa

The assessment of Tsunami propagation in the sea and ocean is the primary aim of this research. Tsunami propagation is generally simulated by solitary waves. In this research, for the simulation of solitary wave propagation in fluid with free surface, an exponential Basis functions meshless method is introduced. The formulation of this method is based on Lagrangian form of Navier-stokes equations for non-viscous fluids based in pressure. In this regard, the Laplace equation of pressure is solved at each time step. Then, the geometry is updated based on the Lagrangian formulation of the motion through an implicit algorithm. Considering the changes of geometry in simulation time, the introduced meshless method is efficient and very quick in calculations. The results for water surface profile before breaking are in good agreement with experimental data.

Due to the high approximation power and simplicity of computation of smooth radial Basis functions (RBFs), in recent decades they have received much attention for function approximation. These RBFs contain a shape parameter that regulates their approximation power and stability but its optimal selection is challenging. To avoid this difficulty, this paper follows a novel and computationally efficient strategy to propose a space of radial polynomials with even degree that well approximates flat RBFs. The proposed space, $\mathcal{H}_n$, is the shifted radial polynomials of degree $2n$. By obtaining the dimension of $\mathcal{H}_n$ and introducing a Basis set, it is shown that $\mathcal{H}_n$ is considerably smaller than $\mathcal{P}_{2n}$ while the distances from RBFs to both $\mathcal{H}_n$ and $\mathcal{P}_{2n}$ are nearly equal. For computation, by introducing new Basis functions, two computationally efficient approaches are proposed. Finally, the presented theoretical studies are verified by the numerical results.

According to the standards of the International Telecommunication Union, shaping a reflector antenna to cover a specific area of the Earth from a satellite in geostationary orbit (GEO) is essential. In this paper, we propose using radial Basis functions to shape the reflector antenna surface. The key feature of the proposed method is that the distortion created on the surface of the antenna using radial Basis functions is smoother than that created by other Basis functions introduced so far. As a result, this method simplifies the antenna manufacturing process. Additionally, by employing radial Basis functions, there is no need for constrained optimization to control the distortion of the antenna surface. To demonstrate the effectiveness of radial Basis functions in achieving a smooth shape for the reflector antenna, we simulate coverage of Australia by a reflector antenna mounted on a satellite in GEO orbit and compare the results with those obtained using other Basis functions, such as Zernike and B-spline Basis functions. This comparison shows that radial Basis functions produce a smoother antenna surface, facilitating the realization of the antenna design.

In Shamir threshold scheme one that is called dealer, chooses the key and then shares some partial information about it, called among the participants, secretly. In this paper, we use some numerical methods with piecewise constant Basis functions in Shamir threshold scheme. We first introduce operational matrix of this functions and then show how dealer multiplies this matrix by vector of shares to obtain a new vector and distributes it.

In this paper, a direct method for solving Volterra-Fredholm integral equations with time delay by using orthogonal functions and their stochastic operational matrix of integration is proposed. Stochastic integral equations can be reduced to a sparse system which can be directly solved. Numerical examples show that the proposed scheme has a suitable degree of accuracy.