Construction of solutions to parabolic and hyperbolic initial–boundary value problems

Abstract

Assume that, in a parabolic or hyperbolic equation, the right-hand side is analytic in time and the coefficients are analytic in time at each fixed point of the space. We show that the infinitely differentiable solution to this equation is also analytic in time at each fixed point of the space. This solution is given in the form of the Taylor expansion with respect to time t with coefficients depending on x. The coefficients of the expansion are defined by recursion relations, which are obtained from the condition of compatibility of order $k=\mathrm{\infty }$. The value of the solution on the boundary is defined by the right-hand side and initial data, so that it is not prescribed. We show that exact regular and weak solutions to the initial–boundary value problems for parabolic and hyperbolic equations can be determined as the sum of a function that satisfies the boundary conditions and the limit of the infinitely differentiable solutions for smooth approximations of the data of the corresponding problem with zero boundary conditions. These solutions are represented in the form of the Taylor expansion with respect to t. The suggested method can be considered as an alternative to numerical methods of solution of parabolic and hyperbolic equations.

Keywords:

• Parabolic equation
• hyperbolic equation
• smooth solution
• regular solution
• Taylor expansion

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35C10
• 35K20
• 35K51
• 35K55
• 35L20
• 35L53
• 35L75

Disclosure statement

No potential conflict of interest was reported by the authors.