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The homotopy analysis Rangaig transform method for nonlinear partial differential equations

Ziane Djelloul, Cherif Mountassir Hamdi

Journal of Applied Mathematics and Computational Mechanics

2022

Vol. 21, nr 2

111--122

The idea suggested in this article is to combine the Rangaig transform with the homotopy analysis method in order to facilitate the solution of nonlinear partial differentia equations. This method may be called the homotopy analysis Rangaig transform method (HARTM). The proposed example results showed that HARTM is an effective method for solving nonlinear partial differential equations.

A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis

Cherniha R., Stachowska-Piętka J., Waniewski J.

International Journal of Applied Mathematics and Computer Science

2014

Vol. 24, no. 4

837--851

A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a three-component nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.

Generalized solutions to a characteristic Cauchy problem

Allaud E, Devoue V

Journal of Applied Analysis

2013

Vol. 19, nr 1

1--29

We give a meaning to the nonlinear characteristic Cauchy problem for the wave equation in base form by replacing it by a family of non-characteristic ones. This leads to a well-formulated problem in an appropriate algebra of generalized functions. We prove existence of a solution and we precise how it depends on the choice made. We also check that in the classical case (non-characteristic) our new solution coincides with the classical one.

Generalized solutions to a non Lipschitz- Cauchy problem

Devoue V.

Journal of Applied Analysis

2009

Vol. 15, nr 1

1-32

In this article we investigate solutions to a semilinear partial differential equation with non Lipschitz nonlinearity by using recent theories of generalized functions. To give a meaning to a non Lipschitz characteristic Cauchy problem with irregular data, we replace it by a three parameter family of problems. The first parameter turns the problem into a family of Lipschitz problems, the second one converts the given problem to a non-characteristic one, whereas the third one regularizes the data. Finally, the family of problems is solved in an appropriate three parametric (C,Ε,P) algebra.