Let [mathematical expression not reproducible] take expectations for [j.sub.t] on both sides of above equation, and cancel items of d[t.sup.2]; then we have a

parabolic partial differential equation as follows:

Fifth-degree B-spline solution for a fourth-order

parabolic partial differential equation. Appl.

In this study, we will deal with a

parabolic partial differential equation in one space variable, where t and x are the time and space coordinates respectively, and the quantities h and k are the mesh sizes in the space and time directions.

This book focuses on optimal control problems in which the state equation is an elliptic or

parabolic partial differential equation. Topics include the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions, and main principles of selected numerical techniques.

In this study, we will deal with the simplest one dimensional

parabolic partial differential equation of second order, where t and x are the time and space coordinates respectively, and the quantities h and k are the mesh sizes in the space and time directions.

Gu, Second order

parabolic partial differential equations, The publishing company of Xiamen University, China, 2004.

Wazwaz, "Analytic treatment for variable coefficient fourth-order

parabolic partial differential equations," Applied Mathematics and Computation, vol.

An H 1-Galerkin mixed finite element method for

parabolic partial differential equations. SIAM Journal on Numerical Analysis, 35(2):712-727, 1998.

Finally, a hierarchy of

parabolic partial differential equations is shown in the thermodynamic theory of heat conduction with current multipliers.

Russel: A unified boundary controllability theory for hyperbolic and

parabolic partial differential equations, Studies in Appl.

Lin presents a comprehensive examination of initial-boundary value problems for

parabolic partial differential equations of the second order.

Manafian, "The solution of the variable coefficients fourth-order

parabolic partial differential equations by homotopy perturbation method," Zeitschrift fur Naturforschung, vol.