PDE in Modern Science and Engineering: Cross-Cutting Practices
and Challenges
Jincan Li
a
School of Mathematics, University of Leeds, Leeds, U.K.
Keywords: Partial Differential Equations, Cross-Cutting Practices, Black-Scholes Equation.
Abstract: Partial differential equations (PDEs), as a core tool for mathematical modelling, have demonstrated
remarkable universality in science and engineering by describing the spatio-temporal evolution laws of
multivariate dynamical systems. From fluid motion (Navier-Stokes equations) and heat conduction (Fourier
equations) in classical mechanics, to pricing of financial derivatives (Black-Scholes model), and prediction
of tumor growth (reaction-diffusion equations) in biomedicine, PDEs provide a unifying theoretical
framework for interdisciplinary complex problems. However, their applications face two core challenges: first,
high-dimensional PDEs (e.g., the quantum many-body problem) lead to ‘dimensional catastrophe’, where the
computational complexity of traditional numerical methods grows exponentially with the dimensionality;
second, the deviation of the idealised physical assumptions (e.g., homogeneous medium, linear eigenstructure
relationship) from the actual scenarios leads to the limitation of the accuracy of the model predictions. The
study shows that interdisciplinary collaboration and algorithmic innovation are the keys to breaking through
the existing limitations, and future research needs to find a balance between theoretical rigor, computational
efficiency and engineering applicability, in order to promote the paradigm change of PDEs in the era of
artificial intelligence and quantum.
1 INTRODUCTION
Partial Differential Equations (PDEs), as a
mathematical tool for describing the spatio-temporal
evolution of a continuous medium, have always been
the central bridge connecting physical phenomena
and mathematical theories since Fourier proposed the
heat conduction equation in the 18th century. The
parabolic equations established by Fourier in The
Analytic Theory of Heat:
It is not only reveals the mathematical nature of
thermal diffusion, but also creates a modelling
paradigm for coupling spatial and temporal variables
with differential operators. In the following two
centuries, the application of PDE has been extended
from classical mechanics to quantum mechanics,
financial engineering and biomedicine, etc., and it has
become a ‘universal language’ for describing multi-
scale dynamical systems. Their mathematical forms
can be classified as elliptic (e.g., Poisson equation for
electrostatic field), parabolic (e.g., heat conduction
equation), and hyperbolic (e.g., fluctuation equation),
a
https://orcid.org/ 0009-0000-1775-0551
which can be distinguished by discriminant B2-4AC,
corresponding to steady state equilibrium, diffusion-
dissipation, and fluctuation-propagation,
respectively. Unlike ordinary differential equations
(ODEs), the solution space of PDEs is an infinite
dimensional function space, which needs to be
combined with the initial margin conditions to
construct the adapted problem, a property that enables
it to express the non-local interactions of complex
systems.
In engineering and science, the universality of
PDE is reflected in interdisciplinary scenarios. For
example, the Navier-Stokes equation can directly
help to optimise aircraft performance by balancing
viscous and inertial terms (Chau & Zingg, 2021); the
Black-Scholes equation transforms financial
derivative pricing into a stochastic differential
equation problem, and its extended models (e.g.,
jump-diffusion processes) significantly improve
prediction accuracy in the presence of extreme market
risks (Marengo et al., 2021). Among emerging
applications, reaction-diffusion equations quantify
Li, J.
PDE in Modern Science and Engineering: Cross-Cutting Practices and Challenges.
DOI: 10.5220/0013688800004670
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 2nd International Conference on Data Science and Engineering (ICDSE 2025), pages 317-323
ISBN: 978-989-758-765-8
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.
317

the spatial dynamics of ecological invasions and
outbreaks spread by coupling species competition or
viral transmission mechanisms.
However, there are two major challenges in
solving PDEs: firstly, high-dimensional problems
(e.g., quantum many-body systems) lead to a
‘dimensional catastrophe’, where the computational
complexity of traditional numerical methods (finite
element, Monte Carlo) increases exponentially with
the dimensionality; and secondly, the deviation of
idealised physical assumptions (e.g., homogeneous
media, linear eigenstructure relations) from the actual
scenarios causes the model prediction errors.
This paper systematically sort out the key
applications of PDEs in modern science and
engineering, compare the differences in the solution
paradigms between classical methods and emerging
technologies (quantum computing, data-driven), and
aim to reveal their core strengths and common
challenges, so as to provide theoretical frameworks
and methodological references for cross-disciplinary
research.
2 TYPICAL AREAS OF
APPLICATION OF PARTIAL
DIFFERENTIAL EQUATIONS
2.1 Black-Scholes Equation in
American Option Pricing
The Black-Scholes equation is a core partial
differential equation (parabolic) used in financial
mathematics for option pricing, proposed by Fischer
Black, Myron Scholes and Robert Merton in 1973,
which laid the theoretical foundations for derivatives
pricing and for which he was awarded the 1997 Nobel
Prize in Economics. The Black-Scholes equation can
provide an efficient numerical framework for pricing
American options through an extended application of
the finite difference method.
2.1.1 Theoretical Framework and
Mathematical Modeling
The Black-Scholes equation, the classical parabolic
partial differential equation (PDE) for option pricing,
Its standard form is shown as follows.
𝜕𝑉
𝜕𝑡
+
1
2
𝜎

𝑆

𝜕

𝑉
𝜕𝑆

+𝑟𝑆
𝜕𝑉
𝜕𝑆
−𝑟𝑉=0
(
1
)
V
(
S,t
)
≥max
(
K−S,0
)
,∀S>0,t∈
[
0,T
]
(
2
)
Where V(S,t) is the option price, S is the
underlying asset price,
σ
is the volatility, and r is the
risk-free rate. For the American put option, since the
holder is allowed to exercise the option at any
moment before expiration, the pricing problem is
transformed into a free boundary problem, which
needs to satisfy the following variational inequality:
if the PDE is valid:
𝜕𝑉
𝜕𝑡
+𝐿,𝑉≤0
(
3
)
complement each other:
(
V−max
(
K−S,0
))

𝜕𝑉
𝜕𝑡
+LV=0
(
4
)
Here L is a Black-Scholes differential operator
and the free boundary 𝑆
∗
(𝑡) is defined as a critical
price satisfying V
(
𝑆
∗
(
𝑡
)
,𝑡
)
=𝐾−𝑆
∗
(𝑡).
2.1.2 Numerical Implementation of the
Finite Difference Method
To solve the above free boundary problem, a
discretisation in Crank-Nicolson format is used:
Introducing the log-price transformation x=lnS,
the equation is rewritten as:
𝜕𝑉
𝜕𝑡
+
𝜎

2
𝜕

𝑉
𝜕𝑥

+𝑟−
𝜎

2

𝜕𝑉
𝜕𝑥
−𝑟𝑉
(
5
)
Truncate the computational domain to x∈
[𝑥

,𝑥

] and divide it into a uniform grid:
𝑥

=𝑥

+𝑖∆𝑥,𝑡

=𝑛∆𝑡,
∆𝑥=
𝑥

−𝑥

𝑀
,∆𝑡=
𝑇
𝑁
(
6
)
The Crank-Nicolson format combines implicit
and explicit time integration with discrete equations:
𝑉


−𝑉


∆𝑡
=
1
2
(
𝐿

𝑉


+𝐿

𝑉


)(
7
)
Where the spatial discrete operator L
h
is defined
as:
𝐿

𝑉

=
𝜎

2
𝑉

−2𝑉

+𝑉

(
∆𝑥
)

+
𝑟 −
𝜎

2

𝑉

−𝑉

2∆𝑥
−r𝑉

(
8
)
Dirichlet Boundary:
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318

x→−∞
(
S→0
)
:V
(
0,t
)
=K𝑒

(

)
(
9
)
X→+∞
(
S→∞
)
:V
(
S,t
)
→0
(
10
)
Each time step is forced to satisfy by the
projection method: 𝑉


=max (𝑉


,𝐾−𝑒


).
The discrete equations can be constructed as a
tridiagonal linear system A𝑉

=B𝑉

+b, which
is solved iteratively using the projected successive
over-relaxation method (PSOR) to ensure that the
solution satisfies the advance exercise constraint
(Marengo et al., 2021).
This study constructs a variational inequality
model for American option pricing based on the
Black-Scholes equation, and reveals the essential
features of the free boundary through mathematical
analysis. In the continuous-time financial framework,
the American option exercise strategy is transformed
into a coupled problem of parabolic partial
differential equations with complementary
conditions, the logarithmic price transformation is
introduced to eliminate the singularity of the
equations, and the uniqueness of the existence of the
weak solution is proved by using Sobolev space
theory. The local Lipschitz continuity of the critical
price function S*(t) and its asymptotic behaviour of
convergence to the strike price are rigorously derived
for the non-smooth property of the free boundary. A
finite difference method in Crank-Nicolson format is
proposed, which is combined with a projection
relaxation algorithm to deal with the early strike
constraints, to establish the convergence theory of the
numerical solution in terms of operator splitting. The
model is further extended to fractional order
derivatives to portray market memory effects (Chen
et al., 2024), and sparse tensor product spaces are
constructed to solve the high-dimensional problem
dimensionality catastrophe. This study provides a
rigorous mathematical framework for American
derivatives pricing and deepens the theoretical
knowledge of the free boundary dynamics
mechanism.
2.2 Turbulence Modelling and
Aerodynamic Efficiency
Optimization
The mathematical description of fluid motion is the
intersection of classical mechanics and engineering
science. the Navier-Stokes system of equations, as the
controlling equations of viscous fluid dynamics, has
nonlinear characteristics originating from the
coupling of inertial and viscous terms, which
profoundly reflects the energy transfer and
dissipation mechanisms of fluid motion. The system
of equations is derived from the laws of conservation
of mass and momentum:
ρ
∂u
∂t
+u∇u=−∇p+μ∇

𝑢+𝑓
(
∇𝑢=0
)(
11
)
he coupling relationship between the velocity
field u and the pressure field p dominates complex
phenomena such as flow separation and vortex
evolution. In the field of aeronautical engineering, the
solution of the equations is directly related to the
optimisation of the aerodynamic performance of the
aircraft, whose core objective is to reduce wind
resistance by suppressing turbulence dissipation, and
thus to improve fuel efficiency (Chau & Zingg,
2021).
Theoretical studies have shown that the
distribution of pressure gradient ∇p on the airfoil
surface is a key regulating parameter for aerodynamic
efficiency (Deng et al., 2022). The region of inverse
pressure gradient is prone to trigger boundary layer
separation, leading to a surge in differential pressure
drag. Adjusting the airfoil curvature by constructing
a shape function can optimise the mathematical
properties of the pressure distribution function p(x)
and shift the separation point back. For example,
increasing the radius of curvature of the leading edge
delays the flow instability, while controlling the
trailing edge curvature attenuates the intensity of
trailing vortex shedding. This type of optimisation is
essentially a problem of solving a generalised
extremum problem under the constraints of the N-S
equations, which is mathematically expressed as:
𝑚𝑖𝑛

𝐶

(
𝛤
)(
12
)
Such that
𝑁
(
𝑢,𝑃;𝛤
)
=0
(
13
)
where Γ is the airfoil geometry parameter, CD is
the drag coefficient, and N is the N-S equation
operator.
Current theoretical challenges focus on the
construction of closed models for high Reynolds
number turbulence (Zhang et al., 2023). The classical
RANS method simplifies the pulsation correlation
term by introducing the turbulent viscosity μt, but its
prediction of anisotropic turbulence suffers from
systematic bias. The data assimilation technique
embeds the flow field observation data into the PDE
constrained optimisation framework, which provides
a new idea to improve the generality of the model.
The theoretical progress of the N-S equations
PDE in Modern Science and Engineering: Cross-Cutting Practices and Challenges
319

continues to promote the development of green
aviation technology, and its nonlinear nature has
become a bridge between mathematical analysis and
engineering innovation.
2.3 Fourier Equations in Heat Transfer
The mathematical description of heat transfer as a
fundamental physical process of energy transfer
began with the formulation of Fourier's law. This law
establishes a linear ontological relationship between
the density of heat flow and the temperature gradient:
q=−k∇t
(
14
)
Where k is the thermal conductivity of the
material and ∇T characterises the spatial temperature
inhomogeneity. Combined with the law of
conservation of energy, the classical parabolic partial
differential equation, the Fourier heat conduction
equation, can be derived:
ρc

𝜕𝑇
𝜕𝑡
=∇
(
𝑘∇𝑇
)
+𝑄
(
15
)
Where ρ is the density, Cp is the specific heat
capacity, and Q is the endothermic term. The equation
has a missing second-order derivative term in time,
which is essentially a parabolic PDE, and the spatial
and temporal evolutionary properties of its solution
T(x,t) reflect the nonequilibrium nature of the thermal
diffusion process with memory effects (Narasimhan,
1999).
In engineering thermal design, this equation needs
to be combined with mixed boundary conditions to
form a suitable problem. As an example, a typical
boundary condition for heat dissipation on metal
substrates of electronic devices contains:
Dirichlet condition: fixed temperature boundary
(e.g. heat sink contact surface 𝑇
|


=𝑇

);
Robin condition: convective heat transfer
boundary −k(∂T ∕ ∂n)
|

=h(T−T
∞
), where h is
the convective heat transfer coefficient and T∞ is the
ambient temperature. The analytical solutions of such
marginal problems are usually difficult to obtain, and
the uniqueness of the existence of their weak
solutions can be proved in Sobolev space by the
energy estimation method, which lays the theoretical
foundation for the finite element numerical methods.
The nonlinear expansion of the heat transfer
equation is particularly important in phase change
materials and anisotropic media. For example, in
phase change energy storage systems, the enthalpy
function H(T) is introduced to reconstruct the
governing equations:
𝜕𝐻
𝜕𝑡
=∇
(
𝑘
(
𝑇
)
∇𝑇
)(
16
)
In this case, the thermal conductivity k(T) exhibits
strong nonlinear characteristics in the solid-liquid
phase transition interval, which leads to degradation
of the smoothness of the equation solution
(Simoncelli et al., 2019). The mathematical analysis
of such problems requires the use of regularisation
methods with monotone operator theory to reveal the
dynamics of the phase interface movement.
Modern engineering challenges focus on multi-
physics field coupling effects. For example,
thermoelectric coupling problems in microelectronic
packages require solving the heat conduction
equation in conjunction with the current continuity
equation:
∇∙
(
k∇T
)
+J∙E=0
(
17
)
∇∙
(
σ∇∅
)
=0
(
18
)
Where σ is the conductivity, ϕ is the potential
field, and J is the current density. The suitable
qualitative analysis of such coupled systems involves
the interaction mechanism of the elliptic-parabolic
system of equations, and its numerical stability
conditions are significantly stricter than that of the
single physical field case (Bar-Kohany & Jain, 2024).
The theoretical extension and coupled modeling of
Fourier equations continue to drive the thermal
management technology innovation of new energy
systems and high-end equipment.
2.4 Modeling Practices for Partial
Differential Equations in Ecology
and Biomedicine
The universality of partial differential equations
(PDEs) has made them a central tool in the modeling
of complex systems across disciplines, with
applications ranging from the prediction of
epidemiological transmission to the dynamics of
tumor growth demonstrating profound scientific
value.
2.4.1 Ecology and Epidemiology:
Reaction-Diffusion Equations
In the COVID-19 pandemic, the reaction-diffusion
equation quantifies the spatio-temporal heterogeneity
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320

of virus transmission by coupling the mechanisms of
spatial diffusion and population interaction (Ahmed
et al., 2021). Its standard form is:
∂u
∂t
=𝐷∇

𝑢+𝛽𝑈

1−
𝑢
𝐾

−𝛾𝑢
(
19
)
Where u(x,t) denotes the regional infection density, D
is the diffusion coefficient (positively correlated with
the intensity of population mobility), β is the contact
transmission rate, γ is the recovery rate, and K is the
environmental carrying capacity (e.g., healthcare
resource constraints). By integrating mobile phone
signaling data to construct a spatial dynamic function
of D(x,t), the cross-city transmission path can be
accurately simulated. The DIMON framework
(Diffeomorphic Mapping Operator Learning)
significantly improves the efficiency of multi-area
propagation models by geometrically dependent PDE
solving, reducing the computation time from hours to
seconds (Yin et al., 2024). Theoretical analysis shows
that when the basic regeneration number 𝑅

=𝛽/𝛾>
1, the solution of the equation presents the traveling
wave front (Traveling Wave) characteristic, which
corresponds to the pattern of the epidemic spreading
from the core city to the periphery. Based on this
model, the optimisation of the quarantine policy can
be transformed into a control problem under the PDE
constraints: limiting population movement by
adjusting the diffusion term coefficient D delays the
peak of infection and reduces the overall scale of
transmission.
2.4.2 Biomedicine: Tumor Growth and
Therapeutic Response
The process of tumor invasion can be modeled by an
improved reaction-diffusion equation:
𝜕𝑐
𝜕𝑡
=∇
(
𝐷

∇𝑐
)
+𝜌𝑐𝑙𝑛
𝐶

𝑐
−𝜆𝑐
(
20
)
Where c(x,t) is the tumor cell density, Dc
characterises the cell migration ability, ρ is the
proliferation rate, and λ is the killing coefficient of
chemotherapy or radiotherapy. This model reveals the
phenomenon of ‘infiltration fronts’ at the edge of the
tumor: highly migratory cells ( 𝐷

↑ ) develop a
diffusion advantage, leading to local failure of
conventional treatments. Based on this, combination
therapies (e.g., targeted migration inhibitors with
immune activation have been proposed to achieve
synergistic therapeutic effects by simultaneously
modulating Dc and λ(Kohli et al., 2022).In addition,
angiogenesis models:
∂v
∂t
=𝐷

∇

+Κ
𝑐

𝑐

+𝑐


−𝜇𝑣
(
21
)
It describes the process of tumour-induced
vascular neovascularisation (v is the vessel density)
and provides a theoretical basis for dose optimisation
of antivascular drugs.
2.4.3 Environmental Science: Atmospheric
Pollution Dispersion
Pollutant transport follows the convection-diffusion
equation:
∂C
∂t
+𝑢∙∇𝐶=∇∙
(
K∇C
)
+S
(
x,t
)(
22
)
Where C is the pollutant concentration, u is the
wind velocity field, K is the turbulent diffusion
coefficient, and S is the pollution source term.
Coupling meteorological data to solve this equation
can predict the spatial and temporal distribution of
PM2.5 and guide the dynamic regulation of industrial
emissions. The DIMON framework further optimises
the solution efficiency of 3D pollution dispersion
models through parametric domain and geometric
mapping to support real-time environmental
decision-making(Yin et al., 2024).
From the traveling wave dynamics of virus
propagation to the diffusion front of tumor
infiltration, partial differential equations reveal the
essential laws of multidisciplinary dynamic systems
through a rigorous mathematical framework. Its
successful applications in isolation policy
optimisation, combination therapy design and
environmental governance highlight the
irreplaceability of mathematical tools in solving real-
life complex problems. With the development of data
assimilation and multi-scale modeling techniques,
PDE will continue to promote the deep integration of
scientific frontiers and engineering practices.
3 LIMITATIONS OF PARTIAL
DIFFERENTIAL EQUATIONS
AND FUYURE DIRECTIONS
Partial differential equations (PDEs) are widely used
in science and engineering, but their theoretical
framework and computational methods still face core
challenges. This section analyses the current
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321

limitations from a mathematical and computational
point of view and looks at future breakthroughs.
3.1 Limitations
lthough the interdisciplinary applications of partial
differential equations (PDEs) are wide-ranging, their
theoretical and computational methods still face core
challenges.
3.1.1 The ‘Dimensional Disaster’ of
High-Dimensional PDEs
Higher dimensional parabolic type equations (such as
Schrödinger's equation for the quantum many-body
problem) take the form:
𝜕𝑢
𝜕𝑡
+
1
2
𝑇𝑟
(
𝜎𝜎

𝐻𝑒𝑠𝑠

𝑢
)
+∇𝑢∙𝜇+
𝑓
(
𝑡,𝑥,𝑢,𝜎

∇𝑢
)
=0
(
23
)
The spatial dimension d often reaches hundreds
(e.g., the number of associated assets in the pricing of
financial derivatives), and the computational
complexity of conventional numerical methods (e.g.,
finite element, Monte Carlo) grows exponentially
with d (Kohli et al., 2022; Hafiz et al., 2024). For
example, the Black-Scholes equation requires the
introduction of non-linear terms when considering
default risk, but the memory requirements after high-
dimensional discretisation far exceed the classical
computer limits (Brunton & Kutz, 2024).
3.1.2 Physical Assumptions and Actual
Deviations
PDE modelling often relies on idealised assumptions
such as a homogeneous medium or linear ontological
relationships. Taking the heat transfer equation as an
example, Fourier's law assumes instantaneous
equilibrium of the heat flow with the temperature
gradient, but in micro- and nanoscale or ultra-fast heat
transfer, the non-locality and relaxation time effects
are significant, leading to deviation of model
predictions from experimental observations.
Similarly, the turbulence closure model of the Navier-
Stokes equations suffers from a universality
deficiency, making it difficult to characterise
complex boundary layer separation phenomena
(Hafiz et al., 2024).
3.2 Future Directions
Quantum computing provides a new paradigm for
solving high-dimensional PDEs. Based on the
Schrödingerisation technique, linear PDEs can be
transformed into quantum simulatable Schrödinger
equations, which can be solved in parallel via
quantum superposition states with complexity
reduced to poly(d,log (


)). For example, quantum
finite-difference algorithms for the Poisson equation
have been realised to solve with high accuracy in the
spatial dimension d = 103 , and the computation time
has been reduced by two orders of magnitude
compared to the classical methods (Hafiz et al., 2024;
Brunton & Kutz, 2024). In addition, hybrid quantum-
classical algorithms (e.g., adaptive grids combined
with homogenisation) for multiscale PDEs can
effectively reduce the CFL condition limitations and
are suitable for the simulation of heat transfer in
composites and groundwater flow (Hafiz et al., 2024).
Innovations in quantum computing provide
leapfrog solutions to high-dimensional, nonlinear
problems. However, the deep integration of physical
mechanisms and data-driven, and synergistic
optimisation of quantum-classical computing
architectures still require interdisciplinary
collaboration. These advances will deepen the
knowledge of complex systems and drive disruptive
technological breakthroughs in areas such as financial
risk modeling and new energy material design (Hafiz
et al., 2024; Brunton & Kutz, 2024).
4 CONCLUSIONS
PDEs have become a central framework for modeling
complex systems in modern science and engineering
due to their mathematical universality and physical
interpretability. From classical fluid dynamics and
heat transfer to financial derivatives pricing,
biomedical and environmental sciences, PDEs reveal
the intrinsic laws of multi-scale dynamic systems
through a unified theoretical language. However, the
‘dimensional catastrophe’ of high-dimensional
problems and the simplicity of physical assumptions
are still the main bottlenecks that restrict their wide
application. Emerging technologies such as Physical
Information Neural Networks (PINNs) and quantum
algorithms have provided new paths to address these
challenges: the former fuses data-driven and physical
constraints to achieve efficient solutions to high-
dimensional nonlinear problems, while the latter
utilises quantum parallelism to achieve exponential
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computational acceleration. Future research needs to
further deepen the synergy between mathematical
theory and engineering practice, and promote
innovations in multi-scale modeling, uncertainty
quantification and interdisciplinary algorithm design.
By balancing model accuracy, computational
efficiency and engineering applicability, PDEs will
continue to lead the paradigm change in the cognition
and regulation of complex systems in the era of
artificial intelligence and quantum computing.
REFERENCES
Ahmed, N., Elsonbaty, A., Raza, A., Rafiq, M., & Adel, W.
(2021). Numerical simulation and stability analysis of a
novel reaction–diffusion COVID-19 model. Nonlinear
Dynamics, 106, 1293-1310.
Bar-Kohany, T., & Jain, A. (2024). Dissipation of boundary
effects in multilayer heat conduction problems.
International Journal of Heat and Mass Transfer, 223,
125207.
Brunton, S.L., & Kutz, J.N. (2024). Promising directions of
machine learning for partial differential equations. Nat
Comput Sci, 4, 483–494.
https://doi.org/10.1038/s43588-024-00643-2
Chau, T., & Zingg, D. W. (2021). Aerodynamic
Optimization of a Transonic Strut-Braced-Wing
Regional Aircraft Based on the Reynolds-Averaged
Navier-Stokes Equations. In AIAA AVIATION 2021
FORUM (p. 2526).
Chen, J., Li, X., & Shao, Y. (2024). Numerical analysis of
fractional order Black–Scholes option pricing model
with band equation method. Journal of Computational
and Applied Mathematics, 451, 115998.
Deng, F., Xue, C., & Qin, N. (2022). Parameterizing airfoil
shape using aerodynamic performance parameters.
AIAA Journal, 60(7), 4399-4412.
Hafiz, A.M., Faiq, I., & Hassaballah, M. (2024). Solving
partial differential equations using large-data models: a
literature review. Artif Intell Rev, 57, 152.
https://doi.org/10.1007/s10462-024-10784-5
Kohli, K., Pillarisetty, V. G., & Kim, T. S. (2022). Key
chemokines direct migration of immune cells in solid
tumors. Cancer Gene Therapy, 29(1), 10-21.
Marengo, A., Patton, A., Negri, M., Perego, U., & Reali, A.
(2021). A rigorous and efficient explicit algorithm for
irreversibility enforcement in phase-field finite element
modeling of brittle crack propagation. Computer
Methods in Applied Mechanics and Engineering, 387,
114137.
Narasimhan, T.N. (1999). Fourier’s heat conduction
equation: History, influence, and connections. Proc.
Indian Acad. Sci. (Earth Planet Sci.), 108, 117–148.
https://doi.org/10.1007/BF02842327
Simoncelli, M., et al. (2019). Unified theory of thermal
transport in crystals and glasses. Nature Physics,
15(8):1. DOI:10.1038/s41567-019-0520-x
Yin, M., Charon, N., Brody, R. et al. (2024). A scalable
framework for learning the geometry-dependent
solution operators of partial differential equations. Nat
Comput Sci, 4, 928–940.
https://doi.org/10.1038/s43588-024-00732-2
Zhang, Y., Zhang, D., & Jiang, H. (2023). Review of
challenges and opportunities in turbulence modeling: A
comparative analysis of data-driven machine learning
approaches. Journal of Marine Science and
Engineering, 11(7), 1440.
PDE in Modern Science and Engineering: Cross-Cutting Practices and Challenges
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