Sentence examples for boundary value problems including from inspiring English sources
The classical theory of Maxwell's equations; electrostatics, magnetostatics, boundary value problems including numerical solutions, currents and their interactions, and force and energy relations.
Boundary value problems, including the ϕ-Laplacian operator, have received a lot of attention with respect to the existence and multiplicity of solutions.
Fourth-order boundary value problems, including those with the p-Laplacian operator, have their origin in beam theory [1, 2], ice formation [3, 4], fluids on lungs [5], brain warping [6, 7], designing special curves on surfaces [6, 8], etc.
One of the main advantages of the presented algorithms is their availability for application on both linear and nonlinear second-order boundary value problems including some important singular perturbed equations and also a Bratu-type equation.
However, multi-point boundary value problems included the most recent works [1 4, 6 9] and boundary value problems with integral boundary conditions for ordinary differential equations have been studied by many authors; one may refer to [5, 10 12].
The advancement in the study of fractional-order boundary value problems includes different kinds of boundary conditions such as two-point, multi-point, nonlocal, periodic/anti-periodic, and integral conditions.
Since the nonlocal boundary value problems include the multi-point boundary value problem (A is a step function) and the Riemann integral boundary value problem (A has a continuous derivative), it has become a more general case where we study the boundary value problem with integral boundary conditions of Riemann-Stieltjes type.
In continuous case, since integral boundary value problems include two-point, three-point,..., n-point boundary value problems, such boundary value problems for continuous systems have received more and more attention and many results have worked out during the past ten years, see Refs. [21 27] for more details.
By means of the Lagrange multiplier in the calculus of variations and using the formula for fractional integration by parts, the Euler-Lagrange equations are derived in terms of a two-point fractional boundary value problem including an advance term as well as the delay argument.
Simulations of a Maxwell fluid as three-dimensional free boundary value problem include the die swell effect [10, 11], but fail for large Weissenberg numbers, see large We-limit in [12].
