Conserved quantities for a class of (1 + n)-dimensional linear evolution equation

A special type of (1 + n)-dimensional linear evolution equation is considered. A class of the equations generated by the Fokker–Planck equation becomes the subcase of the considered equation. Conserved vectors using the partial Lagrangian approach is derived in terms of the coefficients of the discussed equation. Derived results are used for the different models from different sciences. We also discuss the conservation laws of the heat equation on curved manifolds and in different coordinate systems. Potential systems are also obtained for some models. At last conclusion is given.