Reflection and transmission coefficients have been defined as integrals over the circumference of a single rod, in terms of the velocity potential and auxiliary velocity terms on the surface of the rod.
Thermography has potential as an auxiliary tool for assessment of the TMJ region because of its higher specificity for TMD, although its sensitivity is low [29].
Unlike spike-layered solutions, whose concentration occurs around minimum points of V, the solutions we are going to study concentrate around minimum points of an auxiliary potential.
We define the ground-energy function (mathcal{E}: mathbb{R}^ tomathbb{R}^) by mathcal{E}(a) = inf_{mathcal{N}_{a}}I_{a}. Finally, we define the auxiliary potential (mathcal{M}:mathbb {R}^{N} to 0,+infty]) by mathcal{M}(x) = biglvert x bigrvert ^{k} mathcal{E}bigl(V x bigr), where (x = (x',x")), (x'inmathcal{H}) and (x inmathcal {H}^{perp}).
On the auxiliary potential (mathcal{M}), we impose the following condition: ((mathcal{M}_{1})) : There exists an open bounded set (Omega subsetmathbb{R}^{N}) such that if ((x',x") inOmega) then ((x',y") inOmega) for all (y inmathcal{H}^{perp}), (vert x vert =vert y vert ).
The complex potentials for the problem of an inclusion in an infinite body are given as two boundary integrals, and the boundary integral equation governing the complex potentials for the auxiliary problem is provided.
Introducing some auxiliary and potential functions, an exact analytical procedure is applied on the governing equations to decouple the displacement variables.
New high-pressure pumps are designed with special focus on critical areas such as rotordynamics, performance, potential excitations, safety, reliability, auxiliary systems, fabrication processes, and seals.
Based on the present solution, five governing complicated partial differential equations of motion were exactly solved by introducing the auxiliary and potential functions and using the method of separation of variables.
