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A generalized solution of the differential equation 1 in the class is any generalized function in satisfying equation 1 in , that is, for any test function , the equation must be satisfied. Here is the operator adjoint to in the sense of Lagrange: A generalized solution of a boundary value problem should satisfy the boundary condition in the appropriate generalized sense in or , etc.
Generalized-function solutions of differential and functional differential equations
References  S. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Gel'fand, G. Komatsu ed.
Generalized solution - Encyclopedia of Mathematics
Katata, , Lect. Vladimirov, "Equations of mathematical physics" , M. Dekker Translated from Russian  V.
Euler, "Institutionum calculi integralis" G. Kowalewski ed. Abolina and A. MathSciNet Google Scholar.
Generalized solutions of functional differential equations
Myshkis and A. Anwendungen , 16 , No. Pura Appl. Milano , 52 , — Van, M.
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Abstract and Applied Analysis
Kamenskij and A. The notion of weak solution based on distributions is sometimes inadequate. In the case of hyperbolic systems , the notion of weak solution based on distributions does not guarantee uniqueness, and it is necessary to supplement it with entropy conditions or some other selection criterion. In fully nonlinear PDE such as the Hamilton—Jacobi equation , there is a very different definition of weak solution called viscosity solution.
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