In the present work, a new direct computational method for solving definite integrals based on Haar wavelets is introduced. The definite integral of the functions from Holder classes is replaced with the approximation of the function by Haar wavelets and the the calculation of definite integrals is reduced to the problem of solving algebraic equation formed by the Fourier coefficients in terms of Haar wavelets. Based on the properties of the Haar wavelets it is shown that the such approximations much better approximate the value of the integrals for the functions from Holder classes. The Error analysis of the approximation method are worked out in the classes of Holder to show the efficiency of the new method and connection of the module of difference with smoothness of the function is established. Finally, some numerical examples of the implementation the method for the functions from Holder classes are presented.

The mathematical models of the heat and mass transfer processes on the ball type solids can be solved using the theory of convergence of Fourier-Laplace series on unit sphere. Many interesting models have divergent Fourier-Laplace series, which can be made convergent by introducing Riesz and Cesaro means of the series. Partial sums of the Fourier-Laplace series summed by Riesz method are integral operators with the kernel known as Riesz means of the spectral function. In order to obtain the convergence results for the partial sums by Riesz means we need to know an asymptotic behavior of the latter kernel. In this work the estimations for Riesz means of spectral function of Laplace-Beltrami operator which guarantees the convergence of the Fourier-Laplace series by Riesz method are obtained.

Previous numerical methods for solving systems of higher order ordinary differential equations (ODEs) directly require calculating the integration coefficients at every step. This research provides a block multi step method for solving orbital problems with periodic solutions in the form of higher order ODEs directly. The advantage of the proposed method is, it requires calculating the integration coefficients only once at the beginning of the integration is presented. The derived formulae is then validated by running simulations with known higher order orbital equations. To provide further efficiency, a relationship between integration coefficients of various order is obtained.

In the present work, a new direct computational method for solving definite integrals based on linear Legendre multi wavelets is introduced. This approach is an improvement of previous methods which are based on Haar wavelets functions. An algorithm using properties of the linear Legendre multi wavelets is developed in order to find numerical approximations for double, triple and improper integrals. The main advantage of this method is its efficiency and simple applicability. To validate the algorithm, numerical experiments are conducted to illustrate the accuracy of the method.

The study of chaotic motion in periodic self-excited oscillators are an area of interest in science and engineering. In the current research, a numerical solution hi backward difference form is proposed for solving these chaotic motions in periodic-self excited oscillators. Study conducted in this article focuses on chaotic motions in the form of Duffing-Van Der Pol Oscillators. A backward difference formulation in predictor-corrector (PeCe) mode is introduced for solving these Duffing-Van Der Pol directly. Numerical simulations provided will show the accuracy of the PeCe backward difference formulation.

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken).

A two-point block backward difference technique is established for solving nonlinear Riccati differential equations directly. The proposed method is coded using a variable order step size (VOS) algorithm. The advantage of the two-point block method is its programmability to implement parallel programming techniques. Combination of the block method and VOS algorithm allows for a significant reduction of computation cost in comparison to conventional methods. With an added advantage of the recursive relationship between integration coefficients of different orders, the proposed two-point block method provides efficient computation without loss of accuracy.

Nonlinear phenomena in science and engineering such as a periodically forced oscillator with nonlinear elasticity are often modeled by the Duffing oscillator (Duffing equation). The Duffling oscillator is a type of nonlinear higher order differential equation. In this research, a numerical approximation for solving the Duffing oscillator directly is introduced using a variable order stepsize (VOS) algorithm coupled with a backward difference formulation. By selecting the appropriate restrictions, the VOS algorithm provides a cost efficient computational code without affecting its accuracy. Numerical results have demonstrated the advantages of a variable order stepsize algorithm over conventional methods in terms of total steps and accuracy.