E. A . Abdel-rehim - Academia.edu (original) (raw)

Papers by E. A . Abdel-rehim

Research paper thumbnail of The classification of lung cancer using deep learning Techniques

Advances in Environmental and Life Sciences, Mar 3, 2024

Research paper thumbnail of The Classification of the Bladder Cancer based on Vision Transformers (ViT)

Bladder cancer is a prevalent malignancy with diverse subtypes, including invasive, non-invasive,... more Bladder cancer is a prevalent malignancy with diverse subtypes, including invasive, non-invasive, and normal tissue. Accurate classification of these subtypes is crucial for personalized treatment and prognosis. In this paper, we present a comprehensive study on the classification of bladder cancer into three classes using various deep-learning models. We utilized a dataset containing histopathological images of bladder tissue samples, split into a training set (70%), a validation set (15%), and a test set (15%). Four different deep-learning architectures were evaluated for their performance in classifying bladder cancer, EfficientNetB2, InceptionResNetV2, InceptionV3, and ResNet50V2. Additionally, we explored the potential of Vision Transformers with two different configurations, ViT_B32 and ViT_B16, for this classification task. Our experimental results revealed significant variations in the models' accuracies for classifying bladder cancer. The highest accuracy was achieved u...

Research paper thumbnail of The Classification of the Prostate Cancer based on Transfer Learning Techniques

The most common cause of mortality worldwide and the most common male cancer is prostate cancer. ... more The most common cause of mortality worldwide and the most common male cancer is prostate cancer. According to the American Cancer Society. In the United States, there were 164,690 new instances of prostate cancer and at least 29,430 deaths from the disease in 2018, making up 9.5% of all new cancer cases. This will have a significant socioeconomic impact. Having the ability to determine the aggressiveness risk of confirmed prostate cancer could enhance the choice of proper treatment for individuals. This could lead to better outcomes, especially in terms of prostate cancer specific mortality. Deep learning-based significant prostate cancer classification has attracted a lot of attention because it may one day be used to support therapeutic decision-making. In this research we propose four models for classification the prostate cancer based on transfer learning algorithms (EfficentNet, DenseNet and Xception). We used two datasets for diagnosing prostate cancer. One of them is the stan...

Research paper thumbnail of The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process

Electronic Research Archive

The Feller exponential population growth is the continuous analogues of the classical branching p... more The Feller exponential population growth is the continuous analogues of the classical branching process with fixed number of individuals. In this paper, I begin by proving that the discrete birth-death process, $ M/M/1 $ queue, could be mathematically modelled by the same Feller exponential growth equation via the Kolmogorov forward equation. This equation mathematically formulates the classical Markov chain process. The non-classical linear birth-death growth equation is studied by extending the first-order time derivative by the Caputo time fractional operator, to study the effect of the memory on this stochastic process. The approximate solutions of the models are numerically studied by implementing the finite difference method and the fourth order compact finite difference method. The stability of the difference schemes are studied by using the Matrix method. The time evolution of these approximate solutions are compared for different values of the time fractional orders. The ap...

Research paper thumbnail of The Reversible Diffusion in Genetic

Biostatistics and Biometrics Open Access Journal, 2017

Research paper thumbnail of The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

Journal of Physics: Conference Series, 2021

The fractional calculus gains wide applications nowadays in all fields. The implementation of the... more The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov ...

Research paper thumbnail of From the space–time fractional integral of the continuous time random walk to the space–time fractional diffusion equations, a short proof and simulation

Physica A: Statistical Mechanics and its Applications, 2019

Research paper thumbnail of Quadratic Spline Function for the Approximate Solution of an Intermediate Space-Fractional Advection Diffusion Equation

Journal of Applied Nonlinear Dynamics, 2017

Research paper thumbnail of Fundamental solutions of the fractional diffusion and the fractional Fokker–Planck equations

Journal of the Egyptian Mathematical Society, 2016

Research paper thumbnail of Implicit Difference Scheme of the Space-Time Fractional Advection Diffusion Equation

Fractional Calculus and Applied Analysis, 2015

The space-time fractional advection diffusion equations are linear partial pseudo-differential eq... more The space-time fractional advection diffusion equations are linear partial pseudo-differential equation with spatial fractional derivatives in time and in space and are used to model transport at the earth surface. We use the implicit difference scheme, the theta-method, to find the approximation solution of these equations in the long run. The proofs of stability of the difference scheme of each models are given. We compare the numerical results of these models for different values of the space and the time fractional orders and for different values of theta.

Research paper thumbnail of Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations

Applied Mathematics, 2013

Research paper thumbnail of Simulation of the Space–Time-Fractional Ultrasound Waves with Attenuation in Fractal Media

Springer Proceedings in Mathematics & Statistics, 2019

In this paper, we are interested in studying the propagation of the over diagnostic ultrasound wa... more In this paper, we are interested in studying the propagation of the over diagnostic ultrasound waves through complex biological vascular networks such as the tumor tissue. Evidence shows that the over diagnostic wave propagates through complex media with power law of non-integer order \(t^{-\nu }, \, 1< \nu <2\). Evidence shows also that the vascular morphology of the tumor is non-smooth and is a complex media that means it is a fractal media. The wave propagates through this fractal media which exhibits with extremely long jumps whose length is distributed according to the Levy long tail \(\sim |x|^{-1-\alpha }\), \(0<\alpha <2\). Therefore, the space–time-fractional forced wave equation with attenuation, or the so-called multi-term wave equation, mathematically models this medicine problem. This equation mathematically models many other physical, biological, chemical, and environmental problems. We get the approximate solution of this model to study the time evolution ...

Research paper thumbnail of On simulating the short and long memory of ergodic Markov and Non-Markov genetic diffusion processes on the long run

Chaos, Solitons & Fractals, 2021

Patient: A 66-year-old woman, who had a bilateral free-end edentulous mandible and no experience ... more Patient: A 66-year-old woman, who had a bilateral free-end edentulous mandible and no experience with dentures, was examined for the chief complaint of masticatory dysfunction on left side of dental arch. A unilateral distal extension removable partial denture (RPD) replacing lower-left molars was selected. Tomographic images were obtained using Fluorine-18 NaF positron emission computerized tomography (NaF-PET)/ computed tomography (CT) before the RPD use and at 1, 6, and 13 weeks after the RPD use to observe the metabolic changes in residual bone caused by the RPD use. PET standardized uptake values (SUVs) and CT values were calculated for lower-left edentulous site (test side) and lowerright edentulous site (control side). As a result, SUVs on the control side remained static after the RPD use, whereas those on the test side increased at 1 and 6 weeks after the RPD use and then decreased. However, CT images showed no obvious changes in the bone shape and structure beneath RPD, and CT values both on the control and test sides did not change either. Discussion: This report shows that NaF-PET could detect bone metabolic changes soon after the RPD use, which cannot be detected by clinical Xrays. The SUV changes may be a mechanobiological reaction to the pressure due to the RPD use, and wearing of the RPD may increase the bone turnover beneath denture. Conclusion: This report demonstrates that wearing of an RPD increases bone turnover beneath denture immediately after the RPD use without clinically detectable changes in bone structure or volume.

Research paper thumbnail of Time evolution of the approximate and stationary solutions of the Time-Fractional Forced-Damped-Wave equation

Tbilisi Mathematical Journal, 2017

In this paper, the simulation of the time-fractional-forced-damped-wave equation (the diffusion a... more In this paper, the simulation of the time-fractional-forced-damped-wave equation (the diffusion advection forced wave) is given for different parameters. The common finite difference rules beside the backward Grünwald-Letnikov scheme are used to find the approximation solution of this model. The paper discusses also the effects of the memory, the internal force (resistance) and the external force on the travelling wave. We follow the waves till they reach their stationary waves. The Von-Neumann stability condition is also considered and discussed. Besides the simulation of the time evolution of the approximation solution of the classical and time-fractional model, the stationary solutions are also simulated. All the numerical results are compared for different values of time.

Research paper thumbnail of Simulation of the approximate solutions of the time-fractional multi-term wave equations

Computers & Mathematics with Applications, 2017

In this paper, simulations of the approximation solutions of time-fractional wave, forced wave (s... more In this paper, simulations of the approximation solutions of time-fractional wave, forced wave (shear wave), and damped wave equations are given. The common finite difference rules besides the backward Grünwald-Letnikov scheme are used to find the approximation solution of these models. The paper discusses also the effects of the memory, the internal force (resistance) and the external force on the travelling wave. The Von-Neumann stability conditions are also considered and discussed for these models. Besides the simulations of the time evolutions of the approximation solutions, the stationary solutions are also simulated. The numerical results are obtained by the Mathematica software.

Research paper thumbnail of The need for the fractional operators

Journal of the Egyptian Mathematical Society, 2021

In this review paper, I focus on presenting the reasons of extending the partial differential equ... more In this review paper, I focus on presenting the reasons of extending the partial differential equations to space-time fractional differential equations. I believe that extending any partial differential equations or any system of equations to fractional systems without giving concrete reasons has no sense. The experiments agrees with the theoretical studies on extending the first order-time derivative to time-fractional derivative. The simulations of some processes also agrees with the theory of continuous time random walks for extending the second-order space fractional derivative to the Riesz–Feller fractional operators. For this aim, I give a condense review the theory of Brownian motion, Langevin equations, diffusion processes and the continuous time random walk. Some partial differential equations that are successfully extended to space-time-fractional differential equations are also presented.

Research paper thumbnail of Review of: "Time evolution and convergence of simple migration models

Research paper thumbnail of Modelling and Simulating of Classical and Non-Classical Diffusion Processes by Random Walks

In this thesis we discuss the equation of one-dimensional space-time fractional diffusion with dr... more In this thesis we discuss the equation of one-dimensional space-time fractional diffusion with drift. In Chapter 3, discrete approximations to time-fractional diffusion processes, (alpha = 2) with drift towards the origin are obtained as explicit and implicit difference schemes and as a random walk models. We have simulated these random walk models and given numerical results for the discrete approximations. Then we discuss the convergence of the discrete solutions to the stationary solutions of the model. Numerical solutions are displayed for central linear drift and for cubic central drift. Furthermore we discuss in detail the relations to the classical Ehrenfest model which is described carefully in Chapter 2. In Chapter 4, we give a survey of the theory of continuous time random walk. We show how the above space-time-time-fractional diffusion equation, with F(x)=0, can be obtained from the integral equation for a continuous time random walk or from that describing a cumulative r...

Research paper thumbnail of The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker–Planck Operator and Applications

Axioms, 2021

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Plan... more In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function G3(t) of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function 1F1, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The e...

Research paper thumbnail of A study on two-person zero-sum rough interval continuous differential games

OPSEARCH, 2019

In this paper, we concentrate on solving the zero-sum two-person continuous differential games us... more In this paper, we concentrate on solving the zero-sum two-person continuous differential games using rough programming approach. A new class defined as rough continuous differential games is resulted from the combination of rough programming and continuous differential games. An effective and simple technique is given for solving such problem. In addition, the trust measure and the expected value operator of rough interval are used to find the α-trust and expected equilibrium strategies for the rough zero-sum two-person continuous differential games. Moreover, sufficient and necessary conditions for an open loop saddle point solution of rough continuous differential games are also derived. Finally, a numerical example is given to confirm the theoretical results.

Research paper thumbnail of The classification of lung cancer using deep learning Techniques

Advances in Environmental and Life Sciences, Mar 3, 2024

Research paper thumbnail of The Classification of the Bladder Cancer based on Vision Transformers (ViT)

Bladder cancer is a prevalent malignancy with diverse subtypes, including invasive, non-invasive,... more Bladder cancer is a prevalent malignancy with diverse subtypes, including invasive, non-invasive, and normal tissue. Accurate classification of these subtypes is crucial for personalized treatment and prognosis. In this paper, we present a comprehensive study on the classification of bladder cancer into three classes using various deep-learning models. We utilized a dataset containing histopathological images of bladder tissue samples, split into a training set (70%), a validation set (15%), and a test set (15%). Four different deep-learning architectures were evaluated for their performance in classifying bladder cancer, EfficientNetB2, InceptionResNetV2, InceptionV3, and ResNet50V2. Additionally, we explored the potential of Vision Transformers with two different configurations, ViT_B32 and ViT_B16, for this classification task. Our experimental results revealed significant variations in the models' accuracies for classifying bladder cancer. The highest accuracy was achieved u...

Research paper thumbnail of The Classification of the Prostate Cancer based on Transfer Learning Techniques

The most common cause of mortality worldwide and the most common male cancer is prostate cancer. ... more The most common cause of mortality worldwide and the most common male cancer is prostate cancer. According to the American Cancer Society. In the United States, there were 164,690 new instances of prostate cancer and at least 29,430 deaths from the disease in 2018, making up 9.5% of all new cancer cases. This will have a significant socioeconomic impact. Having the ability to determine the aggressiveness risk of confirmed prostate cancer could enhance the choice of proper treatment for individuals. This could lead to better outcomes, especially in terms of prostate cancer specific mortality. Deep learning-based significant prostate cancer classification has attracted a lot of attention because it may one day be used to support therapeutic decision-making. In this research we propose four models for classification the prostate cancer based on transfer learning algorithms (EfficentNet, DenseNet and Xception). We used two datasets for diagnosing prostate cancer. One of them is the stan...

Research paper thumbnail of The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process

Electronic Research Archive

The Feller exponential population growth is the continuous analogues of the classical branching p... more The Feller exponential population growth is the continuous analogues of the classical branching process with fixed number of individuals. In this paper, I begin by proving that the discrete birth-death process, $ M/M/1 $ queue, could be mathematically modelled by the same Feller exponential growth equation via the Kolmogorov forward equation. This equation mathematically formulates the classical Markov chain process. The non-classical linear birth-death growth equation is studied by extending the first-order time derivative by the Caputo time fractional operator, to study the effect of the memory on this stochastic process. The approximate solutions of the models are numerically studied by implementing the finite difference method and the fourth order compact finite difference method. The stability of the difference schemes are studied by using the Matrix method. The time evolution of these approximate solutions are compared for different values of the time fractional orders. The ap...

Research paper thumbnail of The Reversible Diffusion in Genetic

Biostatistics and Biometrics Open Access Journal, 2017

Research paper thumbnail of The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

Journal of Physics: Conference Series, 2021

The fractional calculus gains wide applications nowadays in all fields. The implementation of the... more The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov ...

Research paper thumbnail of From the space–time fractional integral of the continuous time random walk to the space–time fractional diffusion equations, a short proof and simulation

Physica A: Statistical Mechanics and its Applications, 2019

Research paper thumbnail of Quadratic Spline Function for the Approximate Solution of an Intermediate Space-Fractional Advection Diffusion Equation

Journal of Applied Nonlinear Dynamics, 2017

Research paper thumbnail of Fundamental solutions of the fractional diffusion and the fractional Fokker–Planck equations

Journal of the Egyptian Mathematical Society, 2016

Research paper thumbnail of Implicit Difference Scheme of the Space-Time Fractional Advection Diffusion Equation

Fractional Calculus and Applied Analysis, 2015

The space-time fractional advection diffusion equations are linear partial pseudo-differential eq... more The space-time fractional advection diffusion equations are linear partial pseudo-differential equation with spatial fractional derivatives in time and in space and are used to model transport at the earth surface. We use the implicit difference scheme, the theta-method, to find the approximation solution of these equations in the long run. The proofs of stability of the difference scheme of each models are given. We compare the numerical results of these models for different values of the space and the time fractional orders and for different values of theta.

Research paper thumbnail of Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations

Applied Mathematics, 2013

Research paper thumbnail of Simulation of the Space–Time-Fractional Ultrasound Waves with Attenuation in Fractal Media

Springer Proceedings in Mathematics & Statistics, 2019

In this paper, we are interested in studying the propagation of the over diagnostic ultrasound wa... more In this paper, we are interested in studying the propagation of the over diagnostic ultrasound waves through complex biological vascular networks such as the tumor tissue. Evidence shows that the over diagnostic wave propagates through complex media with power law of non-integer order \(t^{-\nu }, \, 1< \nu <2\). Evidence shows also that the vascular morphology of the tumor is non-smooth and is a complex media that means it is a fractal media. The wave propagates through this fractal media which exhibits with extremely long jumps whose length is distributed according to the Levy long tail \(\sim |x|^{-1-\alpha }\), \(0<\alpha <2\). Therefore, the space–time-fractional forced wave equation with attenuation, or the so-called multi-term wave equation, mathematically models this medicine problem. This equation mathematically models many other physical, biological, chemical, and environmental problems. We get the approximate solution of this model to study the time evolution ...

Research paper thumbnail of On simulating the short and long memory of ergodic Markov and Non-Markov genetic diffusion processes on the long run

Chaos, Solitons & Fractals, 2021

Patient: A 66-year-old woman, who had a bilateral free-end edentulous mandible and no experience ... more Patient: A 66-year-old woman, who had a bilateral free-end edentulous mandible and no experience with dentures, was examined for the chief complaint of masticatory dysfunction on left side of dental arch. A unilateral distal extension removable partial denture (RPD) replacing lower-left molars was selected. Tomographic images were obtained using Fluorine-18 NaF positron emission computerized tomography (NaF-PET)/ computed tomography (CT) before the RPD use and at 1, 6, and 13 weeks after the RPD use to observe the metabolic changes in residual bone caused by the RPD use. PET standardized uptake values (SUVs) and CT values were calculated for lower-left edentulous site (test side) and lowerright edentulous site (control side). As a result, SUVs on the control side remained static after the RPD use, whereas those on the test side increased at 1 and 6 weeks after the RPD use and then decreased. However, CT images showed no obvious changes in the bone shape and structure beneath RPD, and CT values both on the control and test sides did not change either. Discussion: This report shows that NaF-PET could detect bone metabolic changes soon after the RPD use, which cannot be detected by clinical Xrays. The SUV changes may be a mechanobiological reaction to the pressure due to the RPD use, and wearing of the RPD may increase the bone turnover beneath denture. Conclusion: This report demonstrates that wearing of an RPD increases bone turnover beneath denture immediately after the RPD use without clinically detectable changes in bone structure or volume.

Research paper thumbnail of Time evolution of the approximate and stationary solutions of the Time-Fractional Forced-Damped-Wave equation

Tbilisi Mathematical Journal, 2017

In this paper, the simulation of the time-fractional-forced-damped-wave equation (the diffusion a... more In this paper, the simulation of the time-fractional-forced-damped-wave equation (the diffusion advection forced wave) is given for different parameters. The common finite difference rules beside the backward Grünwald-Letnikov scheme are used to find the approximation solution of this model. The paper discusses also the effects of the memory, the internal force (resistance) and the external force on the travelling wave. We follow the waves till they reach their stationary waves. The Von-Neumann stability condition is also considered and discussed. Besides the simulation of the time evolution of the approximation solution of the classical and time-fractional model, the stationary solutions are also simulated. All the numerical results are compared for different values of time.

Research paper thumbnail of Simulation of the approximate solutions of the time-fractional multi-term wave equations

Computers & Mathematics with Applications, 2017

In this paper, simulations of the approximation solutions of time-fractional wave, forced wave (s... more In this paper, simulations of the approximation solutions of time-fractional wave, forced wave (shear wave), and damped wave equations are given. The common finite difference rules besides the backward Grünwald-Letnikov scheme are used to find the approximation solution of these models. The paper discusses also the effects of the memory, the internal force (resistance) and the external force on the travelling wave. The Von-Neumann stability conditions are also considered and discussed for these models. Besides the simulations of the time evolutions of the approximation solutions, the stationary solutions are also simulated. The numerical results are obtained by the Mathematica software.

Research paper thumbnail of The need for the fractional operators

Journal of the Egyptian Mathematical Society, 2021

In this review paper, I focus on presenting the reasons of extending the partial differential equ... more In this review paper, I focus on presenting the reasons of extending the partial differential equations to space-time fractional differential equations. I believe that extending any partial differential equations or any system of equations to fractional systems without giving concrete reasons has no sense. The experiments agrees with the theoretical studies on extending the first order-time derivative to time-fractional derivative. The simulations of some processes also agrees with the theory of continuous time random walks for extending the second-order space fractional derivative to the Riesz–Feller fractional operators. For this aim, I give a condense review the theory of Brownian motion, Langevin equations, diffusion processes and the continuous time random walk. Some partial differential equations that are successfully extended to space-time-fractional differential equations are also presented.

Research paper thumbnail of Review of: "Time evolution and convergence of simple migration models

Research paper thumbnail of Modelling and Simulating of Classical and Non-Classical Diffusion Processes by Random Walks

In this thesis we discuss the equation of one-dimensional space-time fractional diffusion with dr... more In this thesis we discuss the equation of one-dimensional space-time fractional diffusion with drift. In Chapter 3, discrete approximations to time-fractional diffusion processes, (alpha = 2) with drift towards the origin are obtained as explicit and implicit difference schemes and as a random walk models. We have simulated these random walk models and given numerical results for the discrete approximations. Then we discuss the convergence of the discrete solutions to the stationary solutions of the model. Numerical solutions are displayed for central linear drift and for cubic central drift. Furthermore we discuss in detail the relations to the classical Ehrenfest model which is described carefully in Chapter 2. In Chapter 4, we give a survey of the theory of continuous time random walk. We show how the above space-time-time-fractional diffusion equation, with F(x)=0, can be obtained from the integral equation for a continuous time random walk or from that describing a cumulative r...

Research paper thumbnail of The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker–Planck Operator and Applications

Axioms, 2021

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Plan... more In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function G3(t) of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function 1F1, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The e...

Research paper thumbnail of A study on two-person zero-sum rough interval continuous differential games

OPSEARCH, 2019

In this paper, we concentrate on solving the zero-sum two-person continuous differential games us... more In this paper, we concentrate on solving the zero-sum two-person continuous differential games using rough programming approach. A new class defined as rough continuous differential games is resulted from the combination of rough programming and continuous differential games. An effective and simple technique is given for solving such problem. In addition, the trust measure and the expected value operator of rough interval are used to find the α-trust and expected equilibrium strategies for the rough zero-sum two-person continuous differential games. Moreover, sufficient and necessary conditions for an open loop saddle point solution of rough continuous differential games are also derived. Finally, a numerical example is given to confirm the theoretical results.