Homogeneous difference schemes for hyperbolic equations

The original problem. In a common setting it is required to find in the rectangle 

Homogeneous difference schemes. In preparation for designing a homogeneous weighted scheme associated with problem (l)-(3), let 

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Before going further, we must append to (7) the initial and boundary conditions. These depend on the range of variables thus, the boundary conditions and the first initial condition are specified exactly  

There are two ways of approximating the second initial condition 5u/5f ( o = Uoi ) OHS of which generates an approximation of order 2 in r  

The second one is connected with the difference equation for determination of j/(t)  

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Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

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