Description:[0001] Background description includes information that may be useful in understanding the present invention. It is not an admission that any of the information provided herein is prior art or relevant to the presently claimed invention, or that any publication specifically or implicitly referenced is prior art.
[0002] An equation is a mathematical statement with an 'equal to' symbol between two expressions that have equal values. For example, 3x + 5 = 15. There are different types of equations like linear, quadratic, cubic, etc. Let us learn more about equations in math in this article. Solving an equation containing variables consists of determining which values of the variables make the equality true.
[0003] A number of different types of differential equations analysis systems that are known in the prior art. For example, the following patents are provided for their supportive teachings and are all incorporated by reference.
[0004] Weak solutions of nonlinear boundary value problems of partial differential equations using variational method: This research delves into the examination of weak solutions for boundary value problems associated with nonlinear partial differential equations. Utilizing the variational method, we explore the conditions necessary and sufficient for the existence and uniqueness of these weak solutions. Furthermore, we provide practical demonstrations by solving specific examples of nonlinear problems involving partial differential equations. The study concludes with an analysis of the benefits and limitations inherent in employing the variational method for such investigations.
[0005] A differential equation is a mathematical equation that relates one or more unknown functions and their derivatives. The derivatives represent the rate of change of a function at a particular point. In applications, the functions generally represent physical quantities, and the differential equation defines a relationship between the two. Differential equations are used in various fields such as physics, engineering, and biology. The proposed invention focuses on analyzing the Weak Solutions through Variational Methods of Partial Differential Equations.
[0006] Above information is presented as background information only to assist with an understanding of the present disclosure. No determination has been made, no assertion is made, and as to whether any of the above might be applicable as prior art with regard to the present invention.
[0007] In the view of the foregoing disadvantages inherent in the known types of differential equations analysis systems now present in the prior art, the present invention provides an improved system. As such, the general purpose of the present invention, which will be described subsequently in greater detail, is to provide a new and improved system that focuses on understanding the functions of Nonlinear Boundary Value Problems of Differential Equations using deep learning algorithms that has all the advantages of the prior art and none of the disadvantages.
SUMMARY OF INVENTION
[0008] In the view of the foregoing disadvantages inherent in the known types of differential equations analysis systems now present in the prior art, the present invention provides an improved one. As such, the general purpose of the present invention, which will be described subsequently in greater detail, is to provide a new and improved system that focuses on understanding the functions of Nonlinear Boundary Value Problems of Differential Equations using deep learning algorithms which has all the advantages of the prior art and none of the disadvantages.
[0009] The Main objective of the proposed invention relates to the field of designing & implementing techniques for analyzing the parameters of Weak Solutions through Variational Methods of Partial Differential Equations. Nonlinear Boundary Value Problems of Partial Differential Equations is analyzed.
[0010] Yet another important aspect of the proposed invention relates to the field of designing & implementing techniques that will consider on understanding the functions of Nonlinear Boundary Value Problems of Differential Equations. Nonlinear Boundary Value Problems of Partial Differential Equations is analyzed by predictive unit. The results of prediction are displayed on the display unit.
[0011] In this respect, before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not limited in its application to the details of construction and to the arrangements of the components set forth in the following description or illustrated in the various ways. Also, it is to be understood that the phraseology and terminology employed herein are for the purpose of description and should not be regarded as limiting.
[0012] These together with other objects of the invention, along with the various features of novelty which characterize the invention, are pointed out with particularity in the disclosure. For a better understanding of the invention, its operating advantages and the specific objects attained by its uses, reference should be had to the accompanying drawings and descriptive matter in which there are illustrated preferred embodiments of the invention.
BRIEF DESCRIPTION OF DRAWINGS
[0013] The invention will be better understood and objects other than those set forth above will become apparent when consideration is given to the following detailed description thereof. Such description makes reference to the annexed drawings wherein:
Figure 1 illustrates the schematic view of exploring weak solutions through variational methods in nonlinear boundary value problems of partial differential equations, according to the embodiment herein.
DETAILED DESCRIPTION OF INVENTION
[0014] In the following detailed description, reference is made to the accompanying drawings which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention, and it is to be understood that the embodiments may be combined, or that other embodiments may be utilized and that structural and logical changes may be made without departing from the spirit and scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims and their equivalents.
[0015] While the present invention is described herein by way of example using several embodiments and illustrative drawings, those skilled in the art will recognize that the invention is neither intended to be limited to the embodiments of drawing or drawings described, nor intended to represent the scale of the various components. Further, some components that may form a part of the invention may not be illustrated in certain figures, for ease of illustration, and such omissions do not limit the embodiments outlined in any way. It should be understood that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the invention covers all modification/s, equivalents and alternatives falling within the spirit and scope of the present invention as defined by the appended claims. The headings are used for organizational purposes only and are not meant to limit the scope of the description or the claims. As used throughout this description, the word "may" be used in a permissive sense (i.e., meaning having the potential to), rather than the mandatory sense (i.e., meaning must). Further, the words "a" or "a" mean "at least one” and the word “plurality” means one or more, unless otherwise mentioned. Furthermore, the terminology and phraseology used herein is solely used for descriptive purposes and should not be construed as limiting in scope. Language such as "including," "comprising," "having," "containing," or "involving," and variations thereof, is intended to be broad and encompass the subject matter listed thereafter, equivalents, and any additional subject matter not recited, and is not intended to exclude any other additives, components, integers or steps. Likewise, the term "comprising" is considered synonymous with the terms "including" or "containing" for applicable legal purposes. Any discussion of documents, acts, materials, devices, articles and the like are included in the specification solely for the purpose of providing a context for the present invention.
[0016] In this disclosure, whenever an element or a group of elements is preceded with the transitional phrase "comprising", it is understood that we also contemplate the same element or group of elements with transitional phrases "consisting essentially of, "consisting", "selected from the group consisting of”, "including", or "is" preceding the recitation of the element or group of elements and vice versa.
[0017] A partial differential equation (PDE) is a mathematical equation that calculates a function between partial derivatives of a multivariable function. A partial differential equation (PDE) contains an unknown function of two or more variables and its partial derivatives. The order of a partial differential equation (PDE) is determined by the highest-order derivative in the equation.
[0018] Variational iteration method (VIM) is a technique used to solve linear and nonlinear partial differential equations. It reduces the size of calculations and helps with nonlinear terms. VIM is used by many researchers to study problems in physics, chemistry, biology, and engineering. Variational iteration method (VIM) is a novel method that solves nonlinear partial differential equations without small perturbations or linearization. The proposed invention focuses on implementing techniques for studying the functions of Nonlinear Boundary Value Problems of Differential Equations.
[0019] Reference will now be made in detail to the exemplary embodiment of the present disclosure. Before describing the detailed embodiments that are in accordance with the present disclosure, it should be observed that the embodiment resides primarily in combinations arrangement of the system according to an embodiment herein and as exemplified in FIG. 1
[0020] Figure 1 illustrates the schematic view of exploring weak solutions through variational methods in nonlinear boundary value problems of partial differential equations 100. The proposed invention 100 includes differential equations 101 which is analysing using the analysis unit 102. The analysis unit 102 verifies the weaker solutions 103. The deep learning unit 104 will run predictive algorithm 105 to predict the weaker solutions for the differential equations.
[0021] In the following description, for the purpose of explanation, numerous specific details are set forth in order to provide a thorough understanding of the arrangement of the system according to an embodiment herein. It will be apparent, however, to one skilled in the art that the present embodiment can be practiced without these specific details. In other instances, structures are shown in block diagram form only in order to avoid obscuring the present invention.
, Claims:1. Exploring Weak Solutions through Variational Methods in Nonlinear Boundary Value Problems of Partial Differential Equations, comprises of:
Analysis unit;
Predictive unit and
Deep learning unit.
2. Exploring Weak Solutions through Variational Methods in Nonlinear Boundary Value Problems of Partial Differential Equations, according to claim 1, includes an analysis unit, wherein the analysis unit will verify the weaker solutions.
3. Exploring Weak Solutions through Variational Methods in Nonlinear Boundary Value Problems of Partial Differential Equations, according to claim 1, includes a predictive unit, wherein the predictive unit will predict the weaker solutions for the differential equations.
4. Exploring Weak Solutions through Variational Methods in Nonlinear Boundary Value Problems of Partial Differential Equations, according to claim 1, includes a deep learning unit, wherein the deep learning unit will run the predictive unit.