[7] **viXra:1705.0410 [pdf]**
*replaced on 2017-05-30 20:59:45*

**Authors:** Hong Lai Zhu

**Comments:** 71 Pages.

This is the first part of the total paper. Since the theory of partial differential equations (PDEs) has been established nearly 300 years, there are many important problems have not been resolved, such as what are the general solutions of Laplace equation, acoustic wave equation, Helmholtz equation, heat conduction equation, Schrodinger equation and other important equations? How to solve the problems of definite solutions which have universal significance for these equations? What are the laws of general solution of the mth-order linear PDEs with n variables (n,m≥2)? Is there any general rule for the solution of a PDE in arbitrary orthogonal coordinate systems? Can we obtain the general solution of vector PDEs? Are there very simple methods to quickly and efficiently solve the exact solutions of nonlinear PDEs? And even general solution? Etc. These problems are all effectively solved in this paper. Substituting the results into the original equations, we have verified that they are all correct.

**Category:** Functions and Analysis

[6] **viXra:1705.0399 [pdf]**
*submitted on 2017-05-28 00:50:44*

**Authors:** Andrzej Peczkowski

**Comments:** 15 Pages.

This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Hi 1 is the OY axis that crosses the OX axis at any angle.

**Category:** Functions and Analysis

[5] **viXra:1705.0398 [pdf]**
*submitted on 2017-05-28 00:55:17*

**Authors:** Andrzej Peczkowski

**Comments:** 14 Pages.

This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Hi 1 is the OX axis that crosses the OY axis at any angle.

**Category:** Functions and Analysis

[4] **viXra:1705.0397 [pdf]**
*submitted on 2017-05-28 01:04:24*

**Authors:** Andrzej Peczkowski

**Comments:** 17 Pages.

This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Part 3. Axes OX and OY intersect at any angle

**Category:** Functions and Analysis

[3] **viXra:1705.0249 [pdf]**
*submitted on 2017-05-16 08:26:20*

**Authors:** Andrej Liptaj

**Comments:** 6 Pages.

A set of functions which allows easy derivative-matching is proposed. Several examples of approximations are shown.

**Category:** Functions and Analysis

[2] **viXra:1705.0165 [pdf]**
*replaced on 2018-02-14 22:45:03*

**Authors:** Nicholas R. Wright

**Comments:** 6 Pages. Replaced "Pareto" with "Pareto Improvement"

We prove the Navier-Stokes equations, by means of the Metabolic Theory of Ecology and the Rule of 72. Macroecological theories are proof to the Navier-Stokes equations. A solution could be found using Kleiber’s Law. Measurement is possible through the heat calorie. A Pareto Improvement exists within the Navier-Stokes equations. This is done by superposing dust solutions onto fluid solutions. In summary, the Navier-Stokes equations require a theoretical solution. The Metabolic Theory of Ecology, along with Kleiber’s Law, form a theory by such standards.

**Category:** Functions and Analysis

[1] **viXra:1705.0028 [pdf]**
*submitted on 2017-05-02 15:33:07*

**Authors:** Morad Ahmad, Shaher Momani, Omar Abu Arqub, Mohammed Al-Smadi, Ahmed Alsaedi

**Comments:** 13 Pages.

In this paper, a powerful computational algorithm is developed for the solution of classes of singular second-order, three-point Volterra integrodifferential equations in favorable reproducing kernel Hilbert spaces. The solutions is represented in the form of series in the Hilbert space W₂³[0,1] with easily computable components. In finding the computational solutions, we use generating the orthogonal basis from the obtained kernel functions such that the orthonormal basis is constructing in order to formulate and utilize the solutions. Numerical experiments are carried where two smooth reproducing kernel functions are used throughout the evolution of the algorithm to obtain the required nodal values of the unknown variables. Error estimates are proven that it converge to zero in the sense of the space norm. Several computational simulation experiments are given to show the good performance of the proposed procedure. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to multipoint singular boundary value problems restricted by Volterra operator.

**Category:** Functions and Analysis