SOME RESULTS ON THE INTEGRAL TRANSFORMS AND APPLICATIONS TO DIFFERENTIAL EQUATIONS

2010 ◽  
Author(s):  
Hassan Eltayeb ◽  
Adem Kiliçman
Keyword(s):  
Differential Equations ◽  
Integral Transforms

scholarly journals Some Schemata for Applications of the Integral Transforms of Mathematical Physics

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 254 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Integral Transform ◽  
First Integral ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Basic Properties ◽  
Survey Article ◽  
Laplace Integral ◽  
Generating Operators

In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-type operational calculi that are also discussed in the article. The general schemata for applications of the integral transforms of mathematical physics are illustrated on an example of the Laplace integral transform. Finally, the Mellin integral transform and its basic properties and applications are briefly discussed.

scholarly journals Some results of Watson, Plancherel type integral transforms related to the Hartley, Fourier convolutions and applications

2021 ◽  
Author(s):  
Tuan Trinh
Keyword(s):  
Differential Equations ◽  
Type Theorem ◽  
Sufficient Conditions ◽  
Integral Transforms ◽  
System Of Differential Equations ◽  
Symmetric Form ◽  
Original Function ◽  
Type Integral ◽  
Necessary And Sufficient ◽  
Sequence Of Functions

In this work, we study the Watson-type integral transforms for the convolutions related to the Hartley and Fourier transformations. We establish necessary and sufficient conditions for these operators to be unitary in the L 2 (R) space and get their inverse represented in the conjugate symmetric form. Furthermore, we also formulated the Plancherel-type theorem for the aforementioned operators and prove a sequence of functions that converge to the original function in the defined L 2 (R) norm. Next, we study the boundedness of the operators (T k ). Besides, showing the obtained results, we demonstrate how to use it to solve the class of integro-differential equations of Barbashin type, the differential equations, and the system of differential equations. And there are numerical examples given to illustrate these.

scholarly journals Aboodh and Mahgoub Transform in Boundary Value Problems of System of Ordinary Differential Equations

2021 ◽  
pp. 67-75
Author(s):  
Dr. D. P. Patil
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Value ◽  
Integral Transforms

Integral transforms plays important role in solving differential equations and integral equations. In this paper we discuss application of Aboodh transform and Mahgoub transform in solving boundary value problem of system of ordinary differential equations and result shows that Aboodh transform and Mahgoub transform are closely connected.

scholarly journals Integral transforms connected with differential systems with a singularity.

2019 ◽  
Vol 50 (3) ◽  
pp. 253-268 ◽  
Author(s):  
Mikhail Ignatiev
Keyword(s):  
Differential Equations ◽  
Integral Transforms ◽  
System Of Differential Equations ◽  
Differential Systems

We consider some integral transforms with the kernels expressed in terms of solutions of the system of differential equations \( y'=(x^{-1}A+B)y, \) where \(A\) and \(B\) are constant \(n\times n\), \(n>2\) , matrices. We study analytical and asymptotical properties of such transforms. We also study the transforms as operators acting in some functional spaces.

scholarly journals Jafari Transformation for Solving a System of Ordinary Differential Equations with Medical Application

2021 ◽  
Vol 5 (3) ◽  
pp. 130
Author(s):  
Ahmed Ibrahim El-Mesady ◽  
Yaser Salah Hamed ◽  
Abdullah M. Alsharif
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Transform ◽  
Inverse Operator ◽  
Integral Transforms ◽  
Delta Function ◽  
Medical Application ◽  
Heaviside Function ◽  
Algebraic Equations ◽  
General Integral

Integral transformations are essential for solving complex problems in business, engineering, natural sciences, computers, optical science, and modern mathematics. In this paper, we apply a general integral transform, called the Jafari transform, for solving a system of ordinary differential equations. After applying the Jafari transform, ordinary differential equations are converted to a simple system of algebraic equations that can be solved easily. Then, by using the inverse operator of the Jafari transform, we can solve the main system of ordinary differential equations. Jafari transform belongs to the class of Laplace transform and is considered a generalization to integral transforms such as Laplace, Elzaki, Sumudu, G\_transforms, Aboodh, Pourreza, etc. Jafari transform does not need a large computational work as the previous integral transforms. For the Jafari transform, we have studied some valuable properties and theories that have not been studied before. Such as the linearity property, scaling property, first and second shift properties, the transformation of periodic functions, Heaviside function, and the transformation of Dirac’s delta function, and so on. There is a mathematical model that describes the cell population dynamics in the colonic crypt and colorectal cancer. We have applied the Jafari transform for solving this model.

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