An Error Was Encountered

Error Number: 1054

Unknown column 'cache_on' in 'field list'

UPDATE `aff_pdf_cache` SET `cache` = 'a:10:{i:0;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"114587\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:4:\"loes\";s:9:\"author_id\";s:1:\"0\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:160:\"An Introduction to Partial Differential Equations with MATLAB (Chapman & Hall/Crc Applied Mathematics & Nonlinear Science) by Matthew P. Coleman\";s:11:\"description\";s:291:\"\n

  1. An Introduction to Partial Differential\nEquations with MATLAB (Chapman &\n Hall/Crc Applied Mathematics &\n Nonlinear Science) by Matthew P.\n Coleman\n\n\n\n\n Good And Clear\n\n\nAn Introduction to Partial Differential Equations with…\";s:5:\"thumb\";s:185:\"data/thumb/An-Introduction-to-Partial-Differential-Equations-with-MATLAB-Chapman-amp-HallCrc-Applied-Mathematics-amp-Nonlinear-Science-by-Matthew-P.-Coleman-Document-Transcript-6942.jpg\";s:6:\"thumb2\";s:186:\"data/thumb2/An-Introduction-to-Partial-Differential-Equations-with-MATLAB-Chapman-amp-HallCrc-Applied-Mathematics-amp-Nonlinear-Science-by-Matthew-P.-Coleman-Document-Transcript-6942.jpg\";s:9:\"permalink\";s:100:\"an-introduction-to-partial-differential-equations-with-matlab-chapman-amp-hall-crc-applied-mathemati\";s:5:\"pages\";s:1:\"2\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:1;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"115923\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:6:\"jayden\";s:9:\"author_id\";s:1:\"0\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:106:\"A Digital Signal Processing Primer: With Applications to Digital Audio and Computer Music by Ken Steiglitz\";s:11:\"description\";s:237:\"\n
    1. A Digital Signal Processing Primer:\n With Applications to Digital Audio\nand Computer Music by Ken Steiglitz\n\n\n\n\n Great Dsp Introduction, Focus On Computer Music\n\n\nCovers important topics such as phasors and…\";s:5:\"thumb\";s:146:\"data/thumb/A-Digital-Signal-Processing-Primer-With-Applications-to-Digital-Audio-and-Computer-Music-by-Ken-Steiglitz-Document-Transcript-10728.jpg\";s:6:\"thumb2\";s:147:\"data/thumb2/A-Digital-Signal-Processing-Primer-With-Applications-to-Digital-Audio-and-Computer-Music-by-Ken-Steiglitz-Document-Transcript-10728.jpg\";s:9:\"permalink\";s:100:\"a-digital-signal-processing-primer-with-applications-to-digital-audio-and-computer-music-by-ken-stei\";s:5:\"pages\";s:1:\"2\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:2;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"472873\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:3:\"ats\";s:9:\"author_id\";s:1:\"0\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:97:\"Stability behavior of second order neutral impulsive stochastic differential equations with delay\";s:11:\"description\";s:300:\"In this article, we study the existence and asymptotic stability in pth moment of mild solutions to second order neutral\nstochastic partial differential equations with delay. Our method of investigating the stability of solutions is based on fixed point\ntheorem and Lipchitz conditions being imposed.\";s:5:\"thumb\";s:115:\"images/t/4729/stability-behavior-of-second-order-neutral-impulsive-stochastic-differential-equations-with-delay.jpg\";s:6:\"thumb2\";s:116:\"images/t2/4729/stability-behavior-of-second-order-neutral-impulsive-stochastic-differential-equations-with-delay.jpg\";s:9:\"permalink\";s:97:\"stability-behavior-of-second-order-neutral-impulsive-stochastic-differential-equations-with-delay\";s:5:\"pages\";s:1:\"7\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:3;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"262547\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:12:\"vistateam123\";s:9:\"author_id\";s:6:\"100238\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:32:\"Solving Equations with Fractions\";s:11:\"description\";s:988:\"If we have any equation which has a fraction along with the variable and the constant value,\nthen we have a different method of Solving Equations with Fractions. To solve the equation\nwith the fraction, we will first make the denominators of all the terms same. For this we need\nto first find the lcm of all the denominators.\nIn case the term does not have any denominator, then we will write one as the denominator\nfor that term and thus find the LCM. Now once the LCM for all the terms is calculated, we will\nconvert the terms into their equivalent form, such that the denominator becomes equal to the\nLCM which we have already calculated.\nNow all the fractions are in form of like fractions and now the constant terms are taken on one\nside of the equation and variables are taken to another side of the equation.\nThus we separate the constant terms from the variables and we get the value of the variable.\nLet us understand it more clearly by the following example:\n2x + 1⁄2 = 4\n\";s:5:\"thumb\";s:50:\"images/t/2626/solving-equations-with-fractions.jpg\";s:6:\"thumb2\";s:51:\"images/t2/2626/solving-equations-with-fractions.jpg\";s:9:\"permalink\";s:32:\"solving-equations-with-fractions\";s:5:\"pages\";s:1:\"4\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:4;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"184769\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:10:\"nishagoyal\";s:9:\"author_id\";s:5:\"54357\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:47:\"Qualitative Solutions To Differential Equations\";s:11:\"description\";s:2029:\"n differential equation we study the equations which are changed when their\nparameters changed or it is the study of anything that changes. From the study\nof differential, we learn that the derivative of a function nothing but is the rate of\nchange of the function in calculus.\nSo that any quantity likes velocity, temperature or volume that varies can be\ndescribed by an equation involving its derivative. To study differential equations\nthere are three main methods.\nThe first method is analytic methods, in this method the solution of any\ndifferential equation is obtained by using a mathematical formula. The second\none is numerical techniques, it provide an approximate solution for differential\nequation by using a computer or programmable calculator.\nKnow More About Laplace Transform\nAnd the third one is qualitative technique; in this technique we determining the\ngeneral properties of solution without knowing the exact behavior. It is very\ndifficult to find the analytical solutions to differential equations, often the\nbehavior of the equations can be examined from qualitative types of behavior of\nquantity. Now we are going to the main topic that is Qualitative Solutions to\nDifferential Equations.\nWe investigate the behavior of the differential equation by the given expression:\ndM/dt = r M(K-M)/K, The above equation provides the basic information of a\npopulation where, K is the carrying capacity. Now, we rewrite the equation by\nintroducing the new variable as follows, a=M/K, We use the method of\nseparation of variables which helps to understand the solutions that is, dy/ dt =\nra(1-a).\nThis method is long and more complex, which involve integration by partial\nfractions and a lots of algebraic manipulation to reach the form of the solution\nand also difficult to understand the behavior of a (t) and also the whole\npopulation M (t).\nThere are many cases in which no need to go through the elaborate the\nprocess if we want the appreciation solution of Qualitative behavior except the\nexact result or value.\n\";s:5:\"thumb\";s:65:\"images/t/1848/qualitative-solutions-to-differential-equations.jpg\";s:6:\"thumb2\";s:66:\"images/t2/1848/qualitative-solutions-to-differential-equations.jpg\";s:9:\"permalink\";s:47:\"qualitative-solutions-to-differential-equations\";s:5:\"pages\";s:1:\"4\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:5;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"185451\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:10:\"nishagoyal\";s:9:\"author_id\";s:5:\"54357\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:49:\"Differential Equations and Mathematical Modelling\";s:11:\"description\";s:1659:\"Differential Equations and Mathematical Modelling\nA mathematical model for any system is the description of that system which\nuses the mathematical concepts related to the differential equation and some of\nthe languages.\nThis process of modeling is called as the mathematical modeling and when we\nuse differential equations and mathematical modelling together it is called as\nthe mathematical modeling of the differential equation.\nWe use the differential equations in the process of the modeling of the\nmathematics. To build the model we go through some of the process:\nStep 1: We clearly include the all assumptions for the model. These\nassumptions describe the relationship between the quantities used in modeling.\nKnow More About Exponential Growth and Decay\nStep 2: We describes all the parameters used in the model. Step 3: Now we\nuses all the assumptions to find the model which relates the parameters and\nvariables\nAntiderivatives\nAn antiderivative can be defined as an operation opposite to differentiation\noperation or we can say that to find antiderivative of a function we have to find\nthe integral of that function. Suppose we have a function.....\nIntegration using u-substitutions\nntegration by U substitution is a process used to simplify the integral\nexpressions. U substitution can only be applied to one type of integral, that is.-\n∫f(x)*d/dx(f(x)). We can apply this rule only .......\nExponential Growth and Decay\nExponential growth and decay are some of the real world problems and it\nincludes all those problems which increases or decreases exponential.\nGenerally the exponential growth of any mathematical function occ...Read More\n\";s:5:\"thumb\";s:67:\"images/t/1855/differential-equations-and-mathematical-modelling.jpg\";s:6:\"thumb2\";s:68:\"images/t2/1855/differential-equations-and-mathematical-modelling.jpg\";s:9:\"permalink\";s:49:\"differential-equations-and-mathematical-modelling\";s:5:\"pages\";s:1:\"3\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:6;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"268508\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:14:\"tutorvistateam\";s:9:\"author_id\";s:5:\"62572\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:37:\"How To Solve Equations With Fractions\";s:11:\"description\";s:999:\"TO SOLVE AN EQUATION WITH fractions, we transform it into an equation without fractions\n-- which we know how to solve. The technique is called clearing of fractions.\nDo you start to get nervous when you see fractions? Do you have to stop and review all the\nrules for adding, subtracting, multiplying and dividing fractions?\nIf so, you are just like almost every other math student out there! But... I am going to make\nyour life so much easier when it comes to solving equations with fractions!\nOur first step when solving these equations is to get rid of the fractions because they are not\neasy to work with!\nThat makes sense, right? We don\'t want to solve from here and end up having to subtract 9/4\nfrom 9.\nWe also don\'t want to multiply by the reciprocal yet, because we have so many terms, that\nagain it will create more fractions within the problem.\nSo... what do we do? We are going to get rid of just the denominator in the fraction, so we will\nbe left with the numerator, or just an integer!\";s:5:\"thumb\";s:55:\"images/t/2686/how-to-solve-equations-with-fractions.jpg\";s:6:\"thumb2\";s:56:\"images/t2/2686/how-to-solve-equations-with-fractions.jpg\";s:9:\"permalink\";s:37:\"how-to-solve-equations-with-fractions\";s:5:\"pages\";s:1:\"4\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:7;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"246776\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:19:\"tutorvistateam_team\";s:9:\"author_id\";s:5:\"78231\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:22:\"Differential Equations\";s:11:\"description\";s:1021:\"Today I am going to tell you about a very interesting field of mathematics: differential\nequations. A differential equation is an equation for the functions which are unknown and are\nconsisting of one or more variables.\nThese variables generally relates the values of these unknown functions itself and its\nderivatives. These derivatives can be of various orders depending on the dependent and\nindependent variables of the given equation.\nDifferential Equations are used in various fields like physics, economics etc. These equations\nare used in real world applications. One of the examples is the determination of the velocity of\na box which is falling through the air and we can only consider the resistance of air and the\ngravity.\nNow the acceleration the falling box towards the plane or ground is the acceleration due to the\ngravity minus the deceleration due to air resistance.\nHere the gravitation is assumed to be constant and the air resistance can be modeled as\nproportional to the velocity of the falling box.\n\";s:5:\"thumb\";s:42:\"images/t/2468/differential-equations-4.jpg\";s:6:\"thumb2\";s:43:\"images/t2/2468/differential-equations-4.jpg\";s:9:\"permalink\";s:24:\"differential-equations-4\";s:5:\"pages\";s:1:\"4\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:8;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"253386\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:19:\"tutorvistateam_team\";s:9:\"author_id\";s:5:\"78231\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:46:\"Solving Equations With Variables On Both Sides\";s:11:\"description\";s:778:\"Students in this session we will learn how to solve the Equations with the variables on both\nsides. This topic seems to be hard but if we know some basics it become quite easier.\nFor Solving Equations with the Variable on Each Side we should remember the following\nsteps\n1: first of all if there is any fraction in the equation just removes it by multiplying the whole\nequation by the denominator.\n2: if required use the distributive property\n3: keep all the variables on one side of the equation and all the constant on the other side of\nthe equation\nBy following the above steps we can easily Solve Equations with Variables on Both Sides.\nLet us try to solve some problems on the basis of the equations With the Variable on Each\nSide with the help of the steps written above.\n\";s:5:\"thumb\";s:64:\"images/t/2534/solving-equations-with-variables-on-both-sides.jpg\";s:6:\"thumb2\";s:65:\"images/t2/2534/solving-equations-with-variables-on-both-sides.jpg\";s:9:\"permalink\";s:46:\"solving-equations-with-variables-on-both-sides\";s:5:\"pages\";s:1:\"4\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}i:9;O:8:\"stdClass\":13:{s:2:\"id\";s:6:\"343710\";s:6:\"status\";s:8:\"verified\";s:11:\"author_name\";s:18:\"sougata chatterjee\";s:9:\"author_id\";s:1:\"0\";s:14:\"author_website\";s:0:\"\";s:5:\"title\";s:154:\"National Work Shop On Application of Fractional Calculus in Engineering Physical Law and Solving Extra Ordinary Differential Equations of Fractional Order\";s:11:\"description\";s:331:\"A modern approach to Solve Extra Ordinary Differential Equations\nSeries reaction of several internal-modes generated to external perturbation.\nNo Laplace Transformation.\nNo discretization required.\nNo perturbation required.\nGives the solution as analytical (approximate)\nClose to physical reasoning of principal of action reaction.\";s:5:\"thumb\";s:118:\"images/t/3438/national-work-shop-on-application-of-fractional-calculus-in-engineering-physical-law-and-solving-ext.jpg\";s:6:\"thumb2\";s:119:\"images/t2/3438/national-work-shop-on-application-of-fractional-calculus-in-engineering-physical-law-and-solving-ext.jpg\";s:9:\"permalink\";s:100:\"national-work-shop-on-application-of-fractional-calculus-in-engineering-physical-law-and-solving-ext\";s:5:\"pages\";s:2:\"54\";s:6:\"rating\";s:1:\"0\";s:5:\"voter\";s:1:\"0\";}}', `cache_on` = '2015-02-28 10:56:42' WHERE `aff_id` = '476345'