Abstract:In this paper, we study a one-dimensional random motion with switching process having an Erlang-distribution for the sojourn times. We obtain the solution of the associated hyperbolic PDE of sixth Note :Red parts indicate major changes. Please check them carefully.order through algebraic techniques.

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In this paper some systems of differential equations with partial derivatives are studied by using the properties of Gâteaux differentiable functions on commutative algebras. The connection between solutions of systems of differential equations in partial derivatives and components of monogenic functions on corresponding commutative algebras is shown. We also give some examples of systems of differential equations with partial derivatives and find their solutions.

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

In this paper we expand the concept of a really significant probabilistic measure in the case when the measure takes values in the algebra of bihyperbolic numbers. The basic properties of bihyperbolic numbers are given, in particular idempotents, main ideals generated by idempotents, Pierce's decompo\-sition and the set of zero divisors of the algebra of bihyperbolic numbers are determined. We entered the relation of partial order on the set of bihyperbolic numbers, by means of which the bihyperbolic significant modulus is defined and its basic properties are proved. In addition, some bihyperbolic modules can be endowed with a bihyperbolic significant norms that take values in a set of non-negative bihyperbolic numbers. We define $\sigma$-additive functions of sets in a measurable space that take appropriately normalized bihyperbolic values, which we call a bihyperbolic significant probability. It is proved that such a bihyperbolic probability satisfies the basic properties of the classical probability. A representation of the bihyperbolic probability measure is given and its main properties are proved. A bihyperbolically significant random variable is defined on a bihyperbolic probability space, and this variable is a bihyperbolic measurable function in the same space. We proved the criterion of measurability of a function with values in the algebra of bihyperbolic numbers, and the basic properties of bihyperbolic random variables are formulated and proved. Special cases have been studied in which the bihyperbolic probability and the bihyperbolic random variable take values that are zero divisors of bihyperbolic algebra. Although bihyperbolic numbers are less popular than hyperbolic numbers, bicomplex numbers, or quaternions, they have a number of important properties that can be useful, particularly in the study of partial differential equations also in mathematical statistics for testing complex hypotheses, in thermodynamics and statistical physics.