modern research physics

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Some boundary value problems for the Cauchy-Riemann equation with non-local boundary conditions in several regions of plane have been investigated and solved by authors. In this paper, by making use of fundamental solutions of Cauchy-Riemann equations and by presenting analytic solutions to the above-mentioned boundary value problems, we try to present an analytic expression for the solution of Cauchy-Riemann equation in the first semi-quarter.

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Boundary value problems (BVPs) are one of the most important fields in engineering and mathematical physics. In self-adjoint case of these problems, there are some facilities to solve them, such as
eigenvalues of adjoint equations are real numbers and associated eigenfunctions make an orthogonal
basis system.
In this paper a new method for investigation of self-adjoint B.V.Ps including ordinary differential
equations (O.D.Es) is introduced. Based on this method, at first, some necessary conditions
are obtained by making use of fundamental solutions of adjoint equations. Then an algebraic system is made by this necessary conditions and boundary conditions of given boundary value problem.
Finally, by making use of Lagrangian identity and boundary values of unknown function, sufficient conditions for having a self-adjoint problem are presented.

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In this papear, we produce the method for formation and recognizing boundary layers in singular perturbation problems. This method involves four step for localization of non-local boundary conditions to local case.For the given problem some sufficient and necessary conditions are given for formation and non formation of boundary layers. Since the existence of boundary layers and their places has a direct relation with the structure of approximate solutions and uniform solutions, therefore the main purpose of this paper is recognition and formation of boundary layers in singular perturbation problems with non-local boundary conditions. This process will be done by using fundamental solution of adjoint given differential equation and necessary conditions.In fact by using these necessary conditions and given boundary conditions, we make an algebraic system.By solving this algebraic system by Cramer rule we obtain boundary values of unknown function.These values of unknown function are local boundary conditions.The mathematical model for this kind of problem usually is in the form
of either ordinary differential equations (O.D.E) or partial differential equations (P.D.E) in which the highest derivative is multiplied by some powers of as a positive small parameter.