Methodology of Syntheses of Knowledge: Overcoming Incorrectness of the Problems of Mathematical Modeling

J. Hadamard's ideas of correct formulation of problems of mathematical physics as well as related Banach's theorem on the inverse operator are analyzed. Modern techniques of numerical simulations are shown to be in drastic contradiction to the concepts of J. Hadamard, S. Banach and a number of other outstanding scientists in the sense that the priority is given to the realization of inefficient algorithms, based on a belief that ill-posed problems are adequate to real phenomena. A new method of the solution of problems, traditionally associated with Fredholm integral equations of the first kind, is developed. Its key aspect is a constructive use of possibilities of the functional space $l_2$ to ensure the conditions of correctness. A well-known phenomenon of smoothing of information is taken into account by means of a special composition that explicitly involves the sought function and is infinitesimal in the space $L_2$. By relatively simple transformations, the outlined class of problems is reduced to the solution of Fredholm integral equations of the second kind with properties most favorable for the numerical realization. We demonstrate a reduction to Fredholm integral equations of the first kind and, correspondingly, a possibility to extend the suggested approach to wide classes of linear and nonlinear boundary-value and initial-boundary-value problems. We put forward arguments that the determination of causal relationships, based on the formulation restricted to a primitive renaming of known and unknown functions of the corresponding direct problem, is essentially illegitimate.Comment: 172 page

Solution to the Volterra Operator Equations of the 1st kind with Piecewise Continuous Kernels

The sufficient conditions for existence and uniqueness of continuous solutions of the Volterra operator equations of the first kind with piecewise continuous kernel are derived. The asymptotic approximation of the parametric family of solutions are constructed in case of non-unique solution. The algorithm for the solution's improvement is proposed using the successive approximations method.Comment: 17 page

The convergence of operational Tau method for solving a class of nonlinear Fredholm fractional integro-differential equations on Legendre basis

In this paper, we investigate approximate solutions for nonlinear Fredholm integro-differential equations of fractional order. We present an operational Tau method by obtaining the Tau matrix representation. We solve a special class of nonlinear Fredholm integro-differential equations based on Legendre-Tau method. By using the Sobolev inequality and some of Banach algebra properties, we prove that our proposed method converges to the exact solution in L^2-norm.Comment: 16 pages, 6 figures,2 table

The Problem of Modelling of Economic Dynamics in Differential Form

Traditional models of macroeconomic dynamics are fundamentally incorrect. The reason lies in a misunderstanding of peculiarities of the analysis of infinitesimal quantities. However, even those types of solutions that are envisaged by the above-mentioned models are nonrepresentative in the sense of the reflection of realities. It became obvious that the techniques of the theory of linear differential equations were insufficient here. Accordingly, the scientists' attention switched to the theory of nonlinear differential equations. At the same time, balance and, accordingly, the model with matrix properties are objectively inherent in the economic system. For the reduction of this model to a differential form, there exist rather elementary means that proved to be unclaimed. Macroeconomic rhetoric - the power of the accelerator, a lag on the part of demand, etc. - accompanied by the use of a lot of abstract coefficients prevailed. However, there is no organic interrelation between matrix and nonlinear differential equations. On the contrary, it can be said that linear theory of integral equations originated in matrix analysis. The Fredholm linear integral equation of the second kind with a parameter-dependent kernel proves to be rather representative with regard to the class of possible solutions. It seems that it can be used for the description of any zigzags of the economy. The price one has to pay for this is the nontriviality of existing theory.Comment: 36 pages; made minor textual correction

Approximation of Weakly Singular Integral Equations by Sinc Projection Methods

In this paper, two numerical schemes for a nonlinear integral equation of Fredholm type with weakly singular kernel are proposed. These numerical methods combine sinc-collocation and sinc-convolution approximations with Newton and steepest descent iterative methods that involve solving a nonlinear system of equations. The convergence rate of the approximation schemes is also analyzed. Numerical experiments have been performed to illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation. The comparison between the schemes indicates that sinc-convolution method is more effective.Comment: 19 pages, 4 figure

Analytical method and its convergence analysis based on homotopy analysis for the integral form of doubly singular boundary value problems

In this paper, we consider the nonlinear doubly singular boundary value problems $(p(x)y'(x))'+ q(x)f(x,y(x))=0,~0<x<1$ with Dirichlet/Neumann boundary conditions at $x=0$ and Robin type boundary conditions at $x=1$. Due to the presence of singularity at $x=0$ as well as discontinuity of $q(x)$ at $x=0$, these problems pose difficulties in obtaining their solutions. In this paper, a new formulation of the singular boundary value problems is presented. To overcome the singular behavior at the origin, with the help of Green's function theory the problem is transformed into an equivalent Fredholm integral equation. Then the optimal homotopy analysis method is applied to solve integral form of problem. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. For speed up the calculations, the discrete averaged residual error is used to obtain optimal value of the adjustable parameter $c_0$ to control the convergence of solution. The proposed method \textbf{(a)} avoids solving a sequence of transcendental equations for the undetermined coefficients \textbf{(b)} it is a general method \textbf{(c)} contains a parameter $c_0$ to control the convergence of solution. Convergence analysis and error estimate of the proposed method are discussed. Accuracy, applicability and generality of the present method is examined by solving five singular problems.Comment: 2

Riemann--Hilbert Problems

These lectures introduce the method of nonlinear steepest descent for Riemann-Hilbert problems. This method finds use in studying asymptotics associated to a variety of special functions such as the Painlev\'{e} equations and orthogonal polynomials, in solving the inverse scattering problem for certain integrable systems, and in proving universality for certain classes of random matrix ensembles. These lectures highlight a few such applications

A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator's banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form $K(x,y)=K(x-y)$ but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments on problems with or without known analytic solutions and comparisons with other methods.Comment: 24 pages, 8 figure

Integral equations and applications

The goal of this Section is to formulate some of the basic results on the theory of integral equations and mention some of its applications. The literature of this subject is very large. Proofs are not given due to the space restriction. The results are taken from the works mentioned in the references

Numerical Solution of the Neural Field Equation in the Two-dimensional Case

We are concerned with the numerical solution of a class integro-differential equations, known as Neural Field Equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many applications in Neuroscience and Robotics. We describe a numerical method for the approximation of solutions in the two-dimensional case, including a space-dependent delay in the integrand function. Compared with known algorithms for this type of equation we propose a scheme with higher accuracy in the time discretisation. Since computational efficiency is a key issue in this type of calculations, we use a new method for reducing the complexity of the algorithm. The convergence issues are discussed in detail and a number of numerical examples is presented, which illustrate the performance of the method.Comment: 25 pages, 1 figur