Kolmogorov-Petrovskii-Piskunov equation can be regarded as a generalized form of the Fitzhugh-Nagumo, Fisher and Huxley equations which have many applications in physics, chemistry biology. In this paper, two fractional ex-tended versions non-linear are solved by analytical methods. Firstly, new more general derivative is defined some properties it given. Secondly, solution operator representat...

In Section 1 we present a general principle for associating nonlinear equations of evolutions with linear operators so that the eigenvalues of the linear operator are integrals of the nonlinear equation. A striking instance of such a procedure is the discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation. In Secti...

In this paper an observability analysis is performed for an axial dispersion tubular bioreactor that involves one nonlinear growth reaction. The process is described by a semilinear parabolic Partial Differential Equation (PDE). More precisely, the analysis is performed on a tangent linearized model, that is described by a linear PDE with a spatial-dependent coefficient. It is reported that the...

is a solvable linear equation in a Hilbert space H , A is a linear, closed, densely defined, unbounded operator in H , which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the closure of the operator (AA + α I )−1A∗, with the domain D(A), where α > 0 is a constant, is a linear bounded everywhere defined operator with norm ≤ 1 2 √ α . This result is applied to the va...

This paper considers the Riemann–Liouville fractional operator as a tool to reduce linear ordinary equations with variable coefficients to simpler problems, avoiding the singularities of the original equation. The main result is that this technique allow us to obtain an extension of the classical integral representation of the special functions related with the original differential equations. ...

The stability in L norm is considered for the Ritz Volterra projection and some applications are presented in this paper As a result point wise error estimates are established for the nite ele ment approximation for the parabolic integro di erential equation Sobolev equations and a di usion equation with non local boundary value problem This work is supported in part by NSERC CANADA Journal of ...

In this paper we will consider a partial reduction for nonhomogeneous linear systems of the operator equations with the system matrix in the companion form and with different operators. As a result of this method we will get an equivalent system consisting of the linear operator equations having only one or two variables. Homogeneous part of the equation in one unknown is obtained using general...

Some existence and uniqueness theorems are established for weakly singular Volterra and Fredholm-Volterra integral equations in C[a, b]. Our method is based on fixed point theorems which are applied to the iterated operator and we apply the fiber Picard operator theorem to establish differentiability with respect to parameter. This method can be applied only for linear equations because otherwi...