By V. I. Agoshkov

The ebook examines the vintage and customarily approved tools for fixing mathematical physics difficulties (method of the capability concept, the eigenfunction process, indispensable transformation tools, discretisation characterisation equipment, splitting methods). A separate bankruptcy is dedicated to equipment for fixing nonlinear equations. The booklet bargains a good number of examples of ways those tools are utilized to the answer of particular mathematical physics difficulties, utilized within the components of technological know-how and social actions, akin to power, environmental defense, hydrodynamics, idea of elasticity, and so forth.

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E. its canonic form contains n–1 positive or n–1 negative coefficients. If any of the conditions a),b),c) is valid at all x ∈ Ω, where Ω is the domain R n , then we talk about the ellipticity, hyperbolicity or parabolicity in domain Ω. , equations having different type at different points of the examined domain. For example, the Tricomi equation yu xx +u yy = 0 (55) examined in R 2 is elliptical at y > 0, hyperbolic at y < 0 and parabolic at the straight line y = 0. This equation forms when describing the motion of a solid in a gas with the velocity close to supersonic velocity: the hyperbolic domain y < 0 corresponds to motion with subsonic velocity, and the elliptical domain y > 0 to motion with supersonic velocity.

The problem, satisfying these requirements, is referred to as a correctly (well) posed problem (according to Hadamard) and the set of function X 1 ∩X 2 is the class of validity. The problem which does not satisfy at least one of the conditions a)– b) is referred to as an ill-posed problem. The ill-posed problems often result from inverse problems of mathematical physics: use information on the solution of the direct problem to restore some unknown physical quantities (sources, boundary-value conditions, the coefficients of an equation, etc) determining this problem.

C, ρ and k are constant, equation (40) has the form: ρ ∂u = a 2 ∆u + f , ∂t (41) where k F , f = . cρ cρ Equation (41) is the heat conductivity equation. x n in this equation may have any value. As in the case of the oscillation equation, for complete description of the sources of heat propagation it is necessary to specify the initial distribution of temperature u in medium (in initial condition) and the conditions at the boundary of this medium (boundary condition). a2 = Examples of boundary conditions a) if the given temperature distribution u 0 is maintained at boundary ∂Ω, then u ∂Ω = u0 .

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