Laplace Transform Example

Laplace Transform Example Abstract: This paper describes the Laplace transform used in solving the differential equation and the comparison with the other usual methods of solving the differential equation. The method of Laplace transform has the advantage of directly giving the solution of differential equation with given boundary values without the necessity of first finding the general solution and then evaluating from it the arbitrary constants. Moreover the ready formulas of the Laplace reduce the problem of solving differential equations to mere algebraic manipulation. Introduction: Differential equation is an equation which involves differential coefficients or differentials. It may be defined in a more refined way as an equation that defines a Relationship between a function and one or more derivatives of that function. Let y be some function of the independent variable t. Then following are some differential equations relating y to one or more of its derivatives. The equation states that the first derivative of the function y equals the product of and the function y itself. An additional, implicit statement in this differential equation is that the stated relationship holds only for all t for which both the function and its first derivative are defined. Some other differential equations: Differential equations arise from many problems in oscillations of mechanical and electrical systems, bending of beams conduction of heat, velocity of chemical reactions etc., and as such play a very important role in all modern scientific and engineering studies. There are many ways of solving the differential equation and the most effective way is to use the Laplace equation because it provides the easy path to solve the differential equation without involving any long process of finding out the complementary function and particular integral. Solution of differential equation: A solution of a differential equation is a relation between the variables which satisfy the given differential equation. A first order homogeneous differential equation involves only the first derivative of a function and the function itself, with constants only as multipliers. The equation is of the form and can be solved by the substitutio The solution which fits a specific physical situation is obtained by substituting the solution into the equation and evaluating the various constants by forcing the solution to fit the physical boundary conditions of the problem at hand. Substituting gives The general solution to a differential equation must satisfy both the homogeneous and non-homogeneous equations. It is the nature of the homogeneous solution that the equation gives a zero value. If you find a particular solution to the non-homogeneous equation, you can add the homogeneous solution to that solution and it will still be a solution since its net result will be to add zero. This does not mean that the homogeneous solution adds no meaning to the picture; the homogeneous part of the solution for a physical situation helps in the understanding of the physical system. A solution can be formed as the sum of the homogeneous and non-homogeneous solutions, and it will have a number of arbitrary (undetermined) constants. Such a solution is called the general solution to the differential equation. For application to a physical problem, the constants must be determined by forcing the solution to fit physical boundary conditions. Once a general solution is formed and then forced to fit the physical boundary conditions, one can be confident that it is the unique solution to the problem, as gauranteed by the uniqueness theorem. Uniqueness theorem: For the differential equations applicable to physical problems, it is often possible to start with a general form and force that form to fit the physical boundary conditions of the problem. This kind of approach is made possible by the fact that there is one and only one solution to the differential equation, i.e., the solution is unique. Stated in terms of a first order differential equation, if the problem meets the condition such that f(x,y) and the derivative of y is continuous in a given rectangle of (x,y) values, then there is one and only one solution to the equation which will meet the boundary conditions. Laplace in solving differential equation: The Laplace transform method of solving differential equations yields particular solutions without the necessity of first finding the general solution and then evaluating the arbitrary constants. This method is in general shorter than the above mentioned methods and is specially used for solving the linear differential equation with constant coefficients. Working procedure: Take the Laplace transform of both sides of the differential equation using the formulas of Laplace and the given initial conditions. Transpose the terms with minus sign to right. Divide by the coefficient of y, getting y as a known function of s. Resolve this function of s into partial fractions and take the inverse transform of both sides. This gives y as a function of t which is the desired solution satisfying the given conditions. Solving the algebraic equation in the mapped space Back transformation of the solution into the original space. Figure 1: Schema for solving differential equations using the Laplace transformation Some of the examples which demonstrate the use of the Laplace in solving the differential equation are as follows: Example no.1 Consider the differential equation with the initial conditions . Proceeding using the steps given above one has Step 1: Step 2: Step 3: The complex function must be decomposed into partial fractions in order to use the tables of correspondences. This gives By using the formulas of the inverse laplace transform we can convert these frequency domains back in the time domain and hence get the desired result as , Another example of the laplace involving trigonometric function is We want to solve with initial conditions f(0) = 0 and f †²(0)=0. We note that and we get So this is equivalent to We deduce So we apply the Laplace inverse transform and get Periodic functions: In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2Ï€. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. A function f is said to be periodic if for all values of x. The constant P is called the period, and is required to be nonzero. A function with period P will repeat on intervals of length P, and these intervals are sometimes also referred to as periods. For example, the sine function is periodic with period 2Ï€, since for all values of x. This function repeats on intervals of length 2Ï€ (see the graph to the right). Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.A function that is not periodic is called aperiodic. Laplace transform of periodic functions: If function f(t) is periodic with period p > 0, so that f(t + p) = f(t), and f1(t) is one period (i.e. one cycle) of the function, then the Laplace of this periodic function is given by The basic concept of the formula is the Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of one cycle of the function, divided by (1 − e-sp).Laplace transform of some of the common functions like the graph given below is given by Fig no3:continous graphical function From the graph, we see that the first period is given by: and that the period p = 2. Now So Hence, the Laplace transform of the periodic function, f(t) is given by: Other continuous wave forms and there Laplace transforms are This wave is an example of the full wave rectification which is obtained by the rectifier used in the electronic instruments. Here, and the period, p = Ï€. So the Laplace Transform of the periodic function is given by: Conclusion: The knowledge of Laplace transform has in recent years become an essential part of mathematical background required of engineers and scientists. This is because the transform method an easy and effective means for the solution of many problems arising in engineering. The method of laplace transformation is proving to be the most effective and easy way of solving differential equations and hence it is replacing other methods of solution of the differential equation. The most frequent function encompassed in electronics engineering is continuous function and most of the functions are in the time domain and we need to convert them in the frequency domain, this operation is performed excellently by the Laplace transform and hence its application is further enlarged using it in the solution of the continuous functions.

Unit case study

Technical Institute Skin cancer, otherwise known as skin melanoma, has become a serious epidemic in the United States over the last several years. Approximately one in 60 people will develop some form of skin cancer in their life time. With tanning beds and extended outdoor tanning becoming more common, as well as â€Å"being tan† become the new beauty tip. However, genetics still plays a very important role in skin cancer. It has been said that people with minimal melanin genes, the gene that determines hair and eye color, are more susceptible to skin cancer to do their light hair, eyes and fair kin.Most skin cancer cases are caused by ultraviolet rays, however, approximately five percent of cases are inherited. The risk of melanoma is doubled in the first degree relatives of those who have had melanoma. (Stamp-Posthumous J, Van Dunedin C, Schaffer E, Pink J, Bergman W, 2001 ;44:22-7). The most common type of skin cancer in the United States is Basal Cell (2014 Stanford medic ine) Pigments are what give your skin its color. Melanin, carotene and hemoglobin are three different types of pigment that affect your skin color. When you are in the sun the ultraviolet rays activate the melancholy's which produce melanin.The melanin Is what causes your skin to tan. This helps protect your skin from damage. Melanin absorbs ultraviolet rays and prevents the rays from damaging or affecting the DNA In epidermal cells. It has also been studied that due to the ozone layer depleting over the years, your own pigment can no longer handle the strength of the ultraviolet rays, also contributing to the increase in melanoma (Chickasaws T, Kodak K, Amounts J, Takeaway S). With all of the new Information and studies available to today there Is no reason to doubt the importance of wearing sunscreen whenever you are exposed to the sun for extended periods of time.

Difference Between Spirituality And Religion Essay

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Religion to many, is a symbol to who they are, what the stand for, and how they perceive their environment. †¢ What are some of the responses that people have or give for the belief in â€Å"Ultimate and Sacred Realities?† (pgs.7-13) In an ultimate reality the alone factor of your life is taken away. It is now about being part of the greater whole. They tell a story of universalism and becoming self- transcendence. †¢ What are the different names given to ultimate reality? (pg.8, para. 6) The different names for ultimate reality is immanent and transcendent. †¢Show MoreRelatedThank You For Your Letter Concerning The Content Of The1237 Words   |  5 Pagesprogram. Your worries about contemporary spirituality being taught in year 11 Study of Religion is understandable, but the focus on modern, contemporary spirituality expressed through video games is necessary and important for your child’s education. 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Journal Article Critique Essay - 1685 Words

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Age | 30 to 60 years | Nominal | | 2. BMI (Body mass index) | Equal or greater than 30kg/m2 | Interval | | 3. Stable body weight | lt; 3kg self-reported change during the previous 3 months | Interval | | 4. Well contr olled arterial hypertension | Systolic blood pressure less than 160 mmHg and/or Diastolic blood pressure less than 100mmHg | Interval | Research Concepts * The experimental group was composed of 13 men and 15 women, they were given one capsule of Green tea extract (GTE), that contained 379 mg of GTE (including 208 mg of EGCG). The control group was composed of 15 men and 13 women, they were given a placebo consisted of pure microcrystalline cellulose. Both groups were told to maintain an isocaloric diet and to continue normal eating habits and exercise pattern throughout the study. Data Collection * Among 183 registered patients with hypertension, researches picked 56 of them, equal number of males and females. 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