Semiorthogonal B-spline Wavelet for Solving 2D- Nonlinear Fredholm-Hammerstein Integral Equations

This work is concerned with the study of the second order (linear) semiorthogonal B-spline wavelet method to solve one-dimensional nonlinear Fredholm-Hammerstein integral equations of the second kind. Proof of the existence and uniqueness solution for the two-dimensional Fredholm-Hammerstein nonlinear integral equations of the second kind was introduced. Moreover, generalization the second order (linear) semiorthogonal B-spline wavelet method was achieved and then using it to solve two-dimensional nonlinear Fredholm-Hammerstein integral equations of the second kind. This method transform the one-dimensional and two-dimensional nonlinear Fredholm-Hammerstein integral equations of the second kind to a system of algebraic equations by expanding the unknown function as second order (linear) semiorthogonal B-spline wavelet with unknown coefficients. The properties of these wavelets functions are then utilized to evaluate the unknown coefficients. Also some of illustrative examples which show that the second order (linear) semiorthogonalB-spline wavelet method give good agreement with the exact solutions

A New Numerical Approach for the Solutions of Partial Differential Equations in Three-Dimensional Space

This paper deals with the numerical computation of the solutions of nonlinear partial differential equations in threedimensional space subjected to boundary and initial conditions. Specifically, the modified cubic B-spline differential quadrature method is proposed where the cubic B-splines are employed as a set of basis functions in the differential quadrature method. The method transforms the three-dimensional nonlinear partial differential equation into a system of ordinary differential equations which is solved by considering an optimal five stage and fourth-order strong stability preserving Runge-Kutta scheme. The stability region of the numerical method is investigated and the accuracy and efficiency of the method are shown by means of three test problems: the threedimensional space telegraph equation, the Van der Pol nonlinear wave equation and the dissipative wave equation. The results show that the numerical solution is in good agreement with the exact solution. Finally the comparison with the numerical solution obtained with some numerical methods proposed in the pertinent literature is performed

A fractional B-spline collocation method for the numerical solution of fractional predator-prey models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost