In the last decade, Australia has experienced an overall decline in red cell demand, but there has been an increased need for phenotyped matched red cells. Lifeblood and mathematicians from Queensland universities have developed a probabilistic model to determine the percentage of the donor panel that would need extended antigen typing to meet this increasing demand, and an estimated timeline to achieve the optimum required phenotyped (genotyped) panel. Mathematical modelling, based on Multinomial distributions, was used to provide guidance on the percentage of typed donor panel needed, based on recent historical blood request data and the current donor panel size. Only antigen combinations determined to be uncommon, but not rare, were considered. Simulations were run to attain at least 95% success percentage. Modelling predicted a target of 38% of the donor panel, or 205,000 donors, would need to be genotyped to meet the current demand. If 5% of weekly returning donors were genotyped, this target would be reached within 12 years. For phenotyping, 35% or 188,000 donors would need to be phenotyped to meet Lifeblood's demand. With the current level of testing, this would take eight years but could be performed within three years if testing was increased to 9% of weekly returning donors. An additional 26,140 returning donors need to be phenotyped annually to maintain this panel. This mathematical model will inform business decisions and assist Lifeblood in determining the level of investment required to meet the desired timeline to achieve the optimum donor panel size.

In this paper, we forecast cumulative production for stimulated gas wells using a combination of fast-to-implement modeling methodologies, including polynomial chaos expansion (PCE) and Gaussian processes (GP) proxy models coupled with populations of phenomenological models (POMs). These modeling techniques allow for a reduction in forecast uncertainty and are shown to be effective techniques for extrapolating early time data for stimulated well production from a field of wells in the Surat Basin, Queensland, Australia. The proposed techniques strategically capture and capitalize on production trends across an entire gas field, even in the presence of early production transients. We demonstrate that learning cycles can be shortened, leading to reasonable forecasts, as well as meaningful and actionable insights.

The petroleum industry uses high level dynamic simulations applied to geocellular models to guide forecasts of oil, gas and water production. Uncertainty in model choice and input variable selection is often addressed through large numbers of computationally slow Monte Carlo simulations designed around physics based models. Here, an alternate approach is proposed, which uses a relatively small amount of data and a reduced number of simulations of the high level physics model to train a fast (to evaluate) proxy or surrogate model based on a Polynomial Chaos Expansion. We give details of the theory and incorporated techniques, which significantly increase flexibility. Input variables (e.g. cell-by-cell variations in porosity and permeability) are sampled from unknown probability distributions and sensitivity analysis is based on low level proxy models. The theory is tested by developing proxy models to predict total gas production from a five-spot well configuration in the Hermitage area that taps into the Walloon Coal Measures of the Surat Basin in Queensland. Synthetic training data is simulated using commercial dynamic simulation software based on a high level physics model.

This paper investigated the impact of geological and engineering factors on coal seam gas production in horizontal wells. The results were then used to compare the performance of proxy models based on linear and quadratic response surfaces and Universal Kriging (UK) to models based on Polynomial Chaos Expansion (PCE). A simple reservoir model was created using a commercial reservoir modelling software package which includes the capability to construct proxy models. The simple model was used for the dynamic modelling of cumulative gas production and peak gas rate under uncertainty in the input variables, e.g. the reservoir principal permeability and permeability in orthogonal directions varied as ratios of the principal permeability, porosity, gas content, coal saturation and the angle between the horizontal well and the principal permeability direction. The simulation results for cumulative gas production and peak gas rate were used to generate training data for proxy models, which were then used to predict the simulated output for other combinations of input parameters. Error analysis was conducted for each proxy model and used to compare and contrast the different modelling techniques. In addition, we investigated the sensitivity of the model to changes in the input variables. The results indicate that, for the given study, proxy models based on linear regression were not good estimators for cumulative gas production and peak gas rate. However, when enough training points were utilized, all other techniques provided good estimates. The best performing proxy model was a cubic PCE. In addition, the cubic PCE proxy model provided direct access to the sensitivity of the gas production to changes in the values of the input variables. Considering the main effects, the changes in the principle permeability and the porosity were predominant factors for the cumulative gas production and peak gas rate, respectively. As for the pairwise interactions, the combined effect of the principle permeability and the coal saturation had the most impact on the cumulative gas production, followed by the combined effect of the drilling angle and the ratio of the permeabilities in the y-direction and the x-direction, indicating that gas production can be improved by optimizing the orientation of the horizontal well. In addition, the combined effect of the coal saturation and the porosity had the most impact on the peak gas rate, an interesting result that warrants further investigation in future studies.

In this paper, we deal with the existence of solutions to the f requency problem of a perturbed system, x′ − A(t)x = ε f (x,sinwt,coswt,ε) with three-point boundary condition. The topological tech nique is used to obtain existence theorem. Two examples are given to illustrate our results.

A surrogate model approximates a computationally expensive solver. Polynomial Chaos is a method to construct surrogate models by summing combinations of carefully chosen polynomials. The polynomials are chosen to respect the probability distributions of the uncertain input variables (parameters); this allows for both uncertainty quantification and global sensitivity analysis. In this paper we apply these techniques to a commercial solver for the estimation of peak gas rate and cumulative gas extraction from a coal seam gas well. The polynomial expansion is shown to honour the underlying geophysics with low error when compared to a much more complex and computationally slower commercial solver. We make use of advanced numerical integration techniques to achieve this accuracy using relatively small amounts of training data.

In this paper we use counting arguments to prove that the expected percentage coverage of a d dimensional parameter space of size n when performing k trials with either Latin Hypercube sampling or Orthogonal Array-based Latin Hypercube sampling is the same. We then extend these results to an experimental design setting by projecting onto a t < d dimensional subspace. These results are confirmed by simulations. The theory presented has both theoretical and practical significance in modelling and simulation science when sampling over high dimensional spaces. (C) 2017 Elsevier Inc. All rights reserved.

In this paper we use counting arguments to prove that the expected percentage coverage of a d dimensional parameter space of size n when performing k trials with either Latin Hypercube sampling or Orthogonal sampling (when n=p^d) is the same. We then extend these results to an experimental design setting by projecting onto a 2 dimensional subspace. In this case the coverage is equivalent to the Orthogonal sampling setting when the dimension of the parameter space is two. These results are confirmed by simulations. The ideas presented here have particular relevance when attempting to perform uncertainty quantification or when building populations of models.

We introduce the \(p\)-Mawhin–Ureña–Nagumo and \(p\)-Hartman–Nagumo conditions and apply them to prove the Dirichlet problem for the vector ordinary \(p\)-Laplacian, \((\varPhi _p(x'))'=f(t,x,x')\), for \(t\in [0,1],\) has a solution \(x\) with \((t,x(t))\in \varOmega \subset [0,1]\times \mathbb {R}^n\) where \((\varOmega ,v,p)\) is a \(p\)-admissible bounding set. For \(1<p<2\), we turn the vector ordinary \(p\)-Laplacian into an equivalent system of second-order ordinary differential equations to prove existence. For \(p>2\) we approximate the \(p\)-Laplacian and then turn the approximation into a system, proving existence as a limit of solutions to the approximating problems.

We introduce the p -Mawhin–Ureña–Nagumo and p -Hartman–Nagumo conditions and apply them to prove the Dirichlet problem for the vector ordinary p -Laplacian, ( _p(x'))'=f(t,x,x') , for t∈ [0,1], has a solution x with (t,x(t))∈⊂ [0,1]×ℝ^n where ( ,v,p) is a p -admissible bounding set. For 12 we approximate the p -Laplacian and then turn the approximation into a system, proving existence as a limit of solutions to the approximating problems.

We show that systems of second-order ordinary differential equations, , subject to compatible nonlinear boundary conditions and impulses, have a solution x such that lies in an admissible bounding subset of when f satisfies a Hartman-Nagumo growth bound with respect to . We reformulate the problem as a system of nonlinear equations and apply Leray-Schauder degree theory. We compute the degree by homotopying to a new system of nonlinear equations based on the simpler system of ordinary differential equations, , subject to Picard boundary conditions and impulses and using the Leray index theorem. Our proof is simpler than earlier existence proofs involving nonlinear boundary conditions without impulses and requires weak assumptions on f . MSC: 34A37, 34A34, 34B15.

The main purpose of this paper is to investigate the existence of solutions of BVPs for a very general case in which both the system of ordinary differential equations and the boundary conditions are nonlinear. By employing the implicit function theorem, sufficient conditions for the existence of three-point boundary value problems are established.

This paper deals with existence of solutions to three-point BVPs in perturbed systems of first-order ordinary differential equations at resonance. An existence theorem is established by using the Theorem of Borsuk and some examples are given to illustrate it. A result for computing the local degree of polynomials whose terms of highest order have no common real linear factors is also presented.

We consider three-point boundary value problems for a system of first-order equations in perturbed systems of ordinary differential equations at resonance. We obtain new results for the above boundary value problems with nonlinear boundary conditions. The existence of solutions is established by applying a version of Brouwer's Fixed Point Theorem which is due to Miranda.

We establish stability and nonstability results for a domain-wall brane model arising in classical field theory. In particular, we show the nonexistence of nontrivial bounded solutions on the real line for a coupled pair of parameter dependent linear second order ordinary differential equations for an open set of those parameters. Moreover, we establish the existence of nontrivial solutions for a hypersurface of the parameters. We use Fredholm theory for compact linear operators combined with the Lyapunov–Schmidt method to prove our results. The model is stable, respectively unstable, for those parameters for which the coupled system does not, respectively does, have nontrivial solutions.

In this paper we present existence results for solutions of nonlinear second-order boundary value problems with impulses. Our impulses are applied at p points in the interval and given implicitly by nonlinear functions of the solution. Moreover we allow functional dependence on the solution. Our existence results follow from the existence of a pair of well ordered lower and upper solutions. We generalize earlier results of Cabada and Tomec˘ek, allowing more general compatible boundary conditions, impulses and φ-Laplacian equations.

We apply the strong maximum principle to obtain a priori bounds and uniqueness of solutions for some initial value and boundary value problems as well as to establish oscillation results for second-order dynamic equations on time scales. Our comparison, uniqueness, and oscillation results are new and are extensions of results for ordinary differential equations to the times scale setting.

We are concerned with determining values of λ, for which there exist nodal solutions of the boundary value problemsu″+ra(t)f(u)=0,0

We show the existence of forced periodic solutions to certain symmetric ordinary differential equations. First and second order systems of ordinary differential equations are investigated with and without damping with periodic and symmetric forcings. We study both resonance and nonresonance cases.

We establish existence results for multiple solutions to boundary value problems for nonlinear, second order, ordinary differential equations subject to nonlinear boundary conditions involving two points. We apply our theory to a problem from chemical reactor theory. Our results are extended to systems of equations.