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Papers by Ashraf Fathallah

Research paper thumbnail of White-Noise Functional Solutions for Wick-Type Stochastic Time-Fractional Benjamin-Bona-Mahony Equation

International journal of differential equations and applications, Oct 3, 2014

A multiple families of White-noise functional solutions for Wick-type stochastic time-fractional ... more A multiple families of White-noise functional solutions for Wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equations modified with variable coefficients are proposed. The suggested technique is based on employing white noise analysis, Hermite and inverse Hermite transforms, improved homogeneous balance method and a modified tanh-coth method; to explore some exact solutions for time-fractional BBM equation with variable coefficients in random processes. Rigorous computations with implemented formal examples are explicitly introduced. \pagebreak

Research paper thumbnail of Enlarged integral inequalities through recent fractional generalized operators

Journal of Inequalities and Applications

This paper is devoted to proving some new fractional inequalities via recent generalized fraction... more This paper is devoted to proving some new fractional inequalities via recent generalized fractional operators. These inequalities are in the Hermite–Hadamard and Minkowski settings. Many previously documented inequalities may clearly be deduced as specific examples from our findings. Moreover, we give some comparative remarks to show the advantage and novelty of the obtained results.

Research paper thumbnail of Further Integral Inequalities through Some Generalized Fractional Integral Operators

Fractal and Fractional, 2021

In this article, we utilize recent generalized fractional operators to establish some fractional ... more In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the proofs of recently affiliated formulas by revealing the weak points and introducing more rigorous proofs amending and expanding the results.

Research paper thumbnail of P a White-Noise Functional Solutions for Wick-Type Stochastic Time-Fractional Benjamin-Bona-Mahony Equation

A multiple families of White-noise functional solutions for Wick-type stochastic time-fractional ... more A multiple families of White-noise functional solutions for Wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equations modified with variable coefficients are proposed. The suggested technique is based on employing white noise analysis, Hermite and inverse Hermite transforms, improved homogeneous balance method and a modified tanh-coth method; to explore some exact solutions for time-fractional BBM equation with variable coefficients in random processes. Rigorous computations with implemented formal examples are explicitly introduced.

Research paper thumbnail of Exact Solutions for KDV-Burger Equations with an Application of White-Noise Analysis

In this paper we will give exact solutions of the variable coefficient KdV-Burger equations ut +�... more In this paper we will give exact solutions of the variable coefficient KdV-Burger equations ut +�(t)uux +�(t)uxx +(t)uxxx = 0, where�(t), �(t) and(t) are bounded measurable or integrable functions on R+. Moreover, using the Hermite transform and the homogeneous balance principle, the white noise functional solutions for the Wick-type stochastic KdV-Burger equations are explicitly obtained.

Research paper thumbnail of Reproducing Kernel Function for Sturm-Liouville Problem with Variable Coefficients

Research paper thumbnail of State-dependent delay differential equations in population dynamics: Modeling and analysis

Topics in Functional Differential and Difference Equations, 2001

ABSTRACT The authors wrote a very readable and timely short survey on state-dependent delay diffe... more ABSTRACT The authors wrote a very readable and timely short survey on state-dependent delay differential population models. As suggested by the authors, most time delays in nature are not constant. The most likely scenarios are that they dependent on the history of the environment which includes the history of food supply and population density. It thus can be argued that population growth models must model food supply explicitly and allow maturation time delay to vary. This, however, can be a challenge. To have plausible and yet tractable models, the authors introduce a nice modification of the model due to Aiello, Freedman and Wu and find more plausible and new dynamics. Specifically, if the maturation time delay depends on the total population at birth, the model can exhibit oscillatory dynamics.

Research paper thumbnail of On a Model Representing Stage-Structured Population Growth with State-Dependent Time Delay

Journal of Biological Systems, 1997

A new model for the growth of a single-species with a two-stage structure is developed employing ... more A new model for the growth of a single-species with a two-stage structure is developed employing a state-dependent age of maturity. The paper discusses some remarks in a previous model where the time to maturity is state dependent. Then, we present a new model which is mainly based on making the maturation period of juveniles depend on the total population size not at the present time t but at an earlier time stage, namely, the time when they were born. The paper considers both biological and mathematical aspects. Emphasis is given to the basic theory of the solutions, such as local and global properties of existence, uniqueness, positivity and boundedness.

Research paper thumbnail of White Noise Functional Solutions for the Wick-Type Stochastic Fractional KDV-Burgers-Kuramoto Equations with Time-Fractional Derivatives

International Journal of Apllied Mathematics, 2013

The aim of this paper is to give some new approximations for the exact solutions of the Wick-type... more The aim of this paper is to give some new approximations for the exact solutions of the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. The homotopy analysis method (HAM) is employed to obtain approximate analytical solutions for the exact solutions of fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. Moreover, by using white noise functional analysis, Hermite transform and inverse Hermite transform we will obtained new exact solutions of the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. Finally, by the help of the mapping relation constructed between the general formal solutions of

Research paper thumbnail of White Noise Functional Solutions for the Wick-Type Stochastic Fractional KDV-Burgers-Kuramoto Equations with Time-Fractional Derivatives

International Journal of Apllied Mathematics, 2013

The aim of this paper is to give some new approximations for the exact solutions of the Wick-type... more The aim of this paper is to give some new approximations for the exact solutions of the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. The homotopy analysis method (HAM) is employed to obtain approximate analytical solutions for the exact solutions of fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. Moreover, by using white noise functional analysis, Hermite transform and inverse Hermite transform we will obtained new exact solutions of the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. Finally, by the help of the mapping relation constructed between the general formal solutions of

Research paper thumbnail of Generalized Bessel beams in modified axially symmetric graded index structures

Applied Optics, 2011

We explore a general type of stable Bessel beams in graded index media. The proposed axially symm... more We explore a general type of stable Bessel beams in graded index media. The proposed axially symmetric medium is characterized by an "α" index profile. Explicit solutions for the radial envelope of the field EðrÞ are derived in terms of generalized Bessel functions. Emphasis is given on illustrating how far the conditions of the proposed modified structure permit only a Bessel function of the first kind to be uniquely retained in the solution. This paper considers both the optical and mathematical aspects. Some numerical examples corroborating our theoretical results are included, showing the stability, propagation, and diffraction of such Bessel beams.

Research paper thumbnail of White-Noise Functional Solutions for Wick-Type Stochastic Time-Fractional Benjamin-Bona-Mahony Equation

International journal of differential equations and applications, Oct 3, 2014

A multiple families of White-noise functional solutions for Wick-type stochastic time-fractional ... more A multiple families of White-noise functional solutions for Wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equations modified with variable coefficients are proposed. The suggested technique is based on employing white noise analysis, Hermite and inverse Hermite transforms, improved homogeneous balance method and a modified tanh-coth method; to explore some exact solutions for time-fractional BBM equation with variable coefficients in random processes. Rigorous computations with implemented formal examples are explicitly introduced. \pagebreak

Research paper thumbnail of Enlarged integral inequalities through recent fractional generalized operators

Journal of Inequalities and Applications

This paper is devoted to proving some new fractional inequalities via recent generalized fraction... more This paper is devoted to proving some new fractional inequalities via recent generalized fractional operators. These inequalities are in the Hermite–Hadamard and Minkowski settings. Many previously documented inequalities may clearly be deduced as specific examples from our findings. Moreover, we give some comparative remarks to show the advantage and novelty of the obtained results.

Research paper thumbnail of Further Integral Inequalities through Some Generalized Fractional Integral Operators

Fractal and Fractional, 2021

In this article, we utilize recent generalized fractional operators to establish some fractional ... more In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the proofs of recently affiliated formulas by revealing the weak points and introducing more rigorous proofs amending and expanding the results.

Research paper thumbnail of P a White-Noise Functional Solutions for Wick-Type Stochastic Time-Fractional Benjamin-Bona-Mahony Equation

A multiple families of White-noise functional solutions for Wick-type stochastic time-fractional ... more A multiple families of White-noise functional solutions for Wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equations modified with variable coefficients are proposed. The suggested technique is based on employing white noise analysis, Hermite and inverse Hermite transforms, improved homogeneous balance method and a modified tanh-coth method; to explore some exact solutions for time-fractional BBM equation with variable coefficients in random processes. Rigorous computations with implemented formal examples are explicitly introduced.

Research paper thumbnail of Exact Solutions for KDV-Burger Equations with an Application of White-Noise Analysis

In this paper we will give exact solutions of the variable coefficient KdV-Burger equations ut +�... more In this paper we will give exact solutions of the variable coefficient KdV-Burger equations ut +�(t)uux +�(t)uxx +(t)uxxx = 0, where�(t), �(t) and(t) are bounded measurable or integrable functions on R+. Moreover, using the Hermite transform and the homogeneous balance principle, the white noise functional solutions for the Wick-type stochastic KdV-Burger equations are explicitly obtained.

Research paper thumbnail of Reproducing Kernel Function for Sturm-Liouville Problem with Variable Coefficients

Research paper thumbnail of State-dependent delay differential equations in population dynamics: Modeling and analysis

Topics in Functional Differential and Difference Equations, 2001

ABSTRACT The authors wrote a very readable and timely short survey on state-dependent delay diffe... more ABSTRACT The authors wrote a very readable and timely short survey on state-dependent delay differential population models. As suggested by the authors, most time delays in nature are not constant. The most likely scenarios are that they dependent on the history of the environment which includes the history of food supply and population density. It thus can be argued that population growth models must model food supply explicitly and allow maturation time delay to vary. This, however, can be a challenge. To have plausible and yet tractable models, the authors introduce a nice modification of the model due to Aiello, Freedman and Wu and find more plausible and new dynamics. Specifically, if the maturation time delay depends on the total population at birth, the model can exhibit oscillatory dynamics.

Research paper thumbnail of On a Model Representing Stage-Structured Population Growth with State-Dependent Time Delay

Journal of Biological Systems, 1997

A new model for the growth of a single-species with a two-stage structure is developed employing ... more A new model for the growth of a single-species with a two-stage structure is developed employing a state-dependent age of maturity. The paper discusses some remarks in a previous model where the time to maturity is state dependent. Then, we present a new model which is mainly based on making the maturation period of juveniles depend on the total population size not at the present time t but at an earlier time stage, namely, the time when they were born. The paper considers both biological and mathematical aspects. Emphasis is given to the basic theory of the solutions, such as local and global properties of existence, uniqueness, positivity and boundedness.

Research paper thumbnail of White Noise Functional Solutions for the Wick-Type Stochastic Fractional KDV-Burgers-Kuramoto Equations with Time-Fractional Derivatives

International Journal of Apllied Mathematics, 2013

The aim of this paper is to give some new approximations for the exact solutions of the Wick-type... more The aim of this paper is to give some new approximations for the exact solutions of the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. The homotopy analysis method (HAM) is employed to obtain approximate analytical solutions for the exact solutions of fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. Moreover, by using white noise functional analysis, Hermite transform and inverse Hermite transform we will obtained new exact solutions of the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. Finally, by the help of the mapping relation constructed between the general formal solutions of

Research paper thumbnail of White Noise Functional Solutions for the Wick-Type Stochastic Fractional KDV-Burgers-Kuramoto Equations with Time-Fractional Derivatives

International Journal of Apllied Mathematics, 2013

The aim of this paper is to give some new approximations for the exact solutions of the Wick-type... more The aim of this paper is to give some new approximations for the exact solutions of the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. The homotopy analysis method (HAM) is employed to obtain approximate analytical solutions for the exact solutions of fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. Moreover, by using white noise functional analysis, Hermite transform and inverse Hermite transform we will obtained new exact solutions of the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with time-fractional derivatives. Finally, by the help of the mapping relation constructed between the general formal solutions of

Research paper thumbnail of Generalized Bessel beams in modified axially symmetric graded index structures

Applied Optics, 2011

We explore a general type of stable Bessel beams in graded index media. The proposed axially symm... more We explore a general type of stable Bessel beams in graded index media. The proposed axially symmetric medium is characterized by an "α" index profile. Explicit solutions for the radial envelope of the field EðrÞ are derived in terms of generalized Bessel functions. Emphasis is given on illustrating how far the conditions of the proposed modified structure permit only a Bessel function of the first kind to be uniquely retained in the solution. This paper considers both the optical and mathematical aspects. Some numerical examples corroborating our theoretical results are included, showing the stability, propagation, and diffraction of such Bessel beams.