The Rayleigh quotient and dynamic programming by Richard Bellman

“…The purpose of this paper is to derive a nonlinear partial differential equation for which λ given by (1.3), is one value of the solution. …”
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Application of PDE Constrained Optimization in Internal Combustion Engine Pollution Control by Hongliang Guan

“…Aiming at the problem of secondary pollution of waters due to the difficulty of controlling the dosage of purifiers in the treatment of internal combustion engine pollution, a partial differential equation (referred to as PDE) constrained optimization algorithm based on l1-norm is proposed. …”
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Computational Modelling of Thermal Stability in a Reactive Slab with Reactant Consumption by O. D. Makinde

“…The nonlinear partial differential equation governing the transient reaction-diffusion problem is solved numerically using a semidiscretization finite difference technique. …”
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Synchronization of the Coupled Distributed Parameter System with Time Delay via Proportional-Spatial Derivative Control by Kun Yuan, Abdulaziz Alofi, Jinde Cao, Abdullah Al-Mazrooei, Ahmed Elaiw

“…By combining parabolic partial differential equation (PDE) theory with Lyapunov technique, the synchronization is studied for a class of coupled distributed parameter systems (DPS) described by PDEs. …”
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Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme by Dagnachew Mengstie Tefera, Awoke Andargie Tiruneh, Getachew Adamu Derese

“…This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. …”
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On the Korteweg-de Vries equation: an associated equation by Eugene P. Schlereth, Ervin Y. Rodin

“…The purpose of this paper is to describe a relationship between the Korteweg-de Vries (KdV) equation ut−6uux+uxxx=0and another nonlinear partial differential equation of the form zt+zxxx−3zxzxxz=H(t)z.The second equation will be called the Associated Equation (AE) and the connection between the two will be explained. …”
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Computer-Aided Numerical Inversion of Laplace Transform by Umesh Kumar

“…This technique is applicable to the solution of partially differentiable equations and certain classes of linear systems with time varying components.…”
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Dynamic Analysis for a Kaldor–Kalecki Model of Business Cycle with Time Delay and Diffusion Effect by Wenjie Hu, Hua Zhao, Tao Dong

“…Moreover, the normal form of Hopf bifurcations is obtained by using the center manifold theorem and normal form theory of the partial differential equation, which can determine the bifurcation direction and the stability of the periodic solutions. …”
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Analytical Solution of Heat Conduction for Hollow Cylinders with Time-Dependent Boundary Condition and Time-Dependent Heat Transfer Coefficient by Te-Wen Tu, Sen-Yung Lee

“…By dividing the Biot function into a constant plus a function and introducing two specially chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions only. …”
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An Adaptive Time-Stepping Algorithm for the Allen–Cahn Equation by Chaeyoung Lee, Jintae Park, Soobin Kwak, Sangkwon Kim, Yongho Choi, Seokjun Ham, Junseok Kim

“…The AC equation is a nonlinear partial differential equation, which was first proposed by Allen and Cahn for antiphase boundary motion and antiphase domain coarsening. …”
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Investigating wave solutions in coupled nonlinear Schrödinger equation: insights into bifurcation, chaos, and sensitivity by Adil Jhangeer, Abdallah M. Talafha, Ariana Abdul Rahimzai, Lubomír Říha

“…The transformation of the coupled partial differential equation into ordinary differential equation is achieved by utilizing a complex wave variable. …”
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Dynamic Mean-Variance Model with Borrowing Constraint under the Constant Elasticity of Variance Process by Hao Chang, Xi-min Rong

“…Secondly, we use dynamic programming principle to get the Hamilton-Jacobi-Bellman (HJB) equation for the value function, which is a more sophisticated nonlinear second-order partial differential equation. Furthermore, we use Legendre transform and dual theory to transform the HJB equation into its dual one. …”
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Per‐unit transformation for the combined analysis of multi‐energy integration by Zhang Zhang, Fei Wang, Hui Fan

“…Abstract To handle numerical issues caused by dimensional differences among heterogeneous states in integrated energy systems (IES), this paper proposes a novel per‐unit transformation method and explores its applications to the partial differential equation‐governed models. Initially, a unified IES model is developed, which encompasses both fast and slow dynamics in a generalized form. …”
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Shaping Single Offset Reflector Antennas Using Local Axis-Displaced Confocal Quadrics by Rafael A. Penchel, José R. Bergmann, Fernando J. S. Moreira

“…This work investigates a novel numerical procedure for the solution of an exact formulation for the Geometrical Optics synthesis of a single reflector antenna by simultaneously imposing Snell’s Law and Conservation of Energy in a tube of rays, yielding a second-order nonlinear partial differential equation of Monge-Ampère type, which can be solved as a boundary value problem. …”
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Asymptotic Behavior of an Optimal Investment-Reinsurance Problem with General Utility Functions by Yan Zhang, Peibiao Zhao

“…Moreover, applying the dual method along with the partial differential equation method, we verify the asymptotic behavior of the optimal control strategy, which means that the optimal investment-reinsurance strategy with the general utility functions converges to that with the power utility function at any level of the initial wealth.…”
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Numerical Solutions to Optimal Portfolio Selection and Consumption Strategies under Stochastic Volatility by Lei Ge, Qiang Zhang

“…Based on the method of dynamic programming, this paper uses analysis methods governed by the nonlinear and inhomogeneous partial differential equation to study modern portfolio management problems with stochastic volatility, incomplete markets, limited investment scope, and constant relative risk aversion (CRRA). …”
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Well-Posed Inhomogeneous Nonlinear Diffusion Scheme for Digital Image Denoising by V. B. Surya Prasath, Arindama Singh

“…We study an inhomogeneous partial differential equation which includes a separate edge detection part to control smoothing in and around possible discontinuities, under the framework of anisotropic diffusion. …”
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An Improved C-V Model and Application to the Coal Rock Mesocrack Images by Yulong Chen, Hongwei Zhang

“…In order to accurately and comprehensively obtain information about coal rock mesocrack images, image processing technique based on partial differential equation (PDE) is introduced in order to expound on the active contour model without edges and overcome the deficiency of the C-V model. …”
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Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem by Stéphane Malek

“…The equation under study represents a q−analog of a singular partial differential equation, recently investigated by the author, which comprises Fuchsian operators and entails a forcing term that combines polynomial and logarithmic type functions in time. …”
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Simulation and Domain Identification of Sea Ice Thermodynamic System by Bing Tan, Peng Lu, Zhijun Li, Enmin Feng, Juan Wang

“…Based on the measured data and characteristics of sea ice temperature distribution in space and time, this study is intended to consider a parabolic partial differential equation of the thermodynamic field of sea ice (coupled by snow, ice, and sea water layers) with a time-dependent domain and its parameter identification problem. …”
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